10049:
10041:
669:
2839:
8966:
2844:
which has reached a cycle of period 2. Within any subinterval of [0, 1), no matter how small, there are therefore an infinite number of points whose orbits are eventually periodic, and an infinite number of points whose orbits are never periodic. This sensitive dependence on initial conditions
1793:
iterations, our simulation has reached the fixed point zero, regardless of the true iterate values; thus we have suffered a complete loss of information. This illustrates sensitive dependence on initial conditionsâthe mapping from the truncated initial condition has deviated exponentially from the
4072:
Instead of looking at the orbits of individual points under the action of the map, it is equally worthwhile to explore how the map affects densities on the unit interval. That is, imagine sprinkling some dust on the unit interval; it is denser in some places than in others. What happens to this
2040:
8740:
20:
2400:
4641:
7633:
4802:
4059:
periodic! The aperiodic orbits correspond to the irrational numbers. This property also holds true in a more general setting. An open question is to what degree the behavior of the periodic orbits constrain the behavior of the system as a whole. Phenomena such as
569:
3502:
2730:
4021:
6197:
4339:
8795:
8538:
4938:
8258:
8378:
7882:
5847:
for the fair coin-toss. If there is just one bit specified in the string of arbitrary positions, the measure is 1/2. If there are two bits specified, the measure is 1/4, and so on. One can get fancier: given a real number
6667:
9511:
5772:
1849:
438:
8589:
2244:
5989:
1374:
5512:
3013:
2645:
points in the unit interval are), then the dynamics are non-periodicâthis follows directly from the definition of an irrational number as one with a non-repeating binary expansion. This is the chaotic case.
1159:
1024:
3146:
1794:
mapping from the true initial condition. And since our simulation has reached a fixed point, for almost all initial conditions it will not describe the dynamics in the qualitatively correct way as chaotic.
1703:
6955:
5010:
9329:
7957:
4539:
9164:
7497:
4030:
rational number can be expressed in this way: an initial "random" sequence, followed by a cycling repeat. That is, the periodic orbits of the map are in one-to-one correspondence with the rationals.
2624:
8121:
3760:
9243:
238:
7455:
4676:
3868:
7173:
6797:
1564:
7114:
3664:
3583:
3277:
646:
7242:
7076:
6990:
5628:
5562:
5346:
5309:
3192:
3063:
863:
776:
9592:
5162:
2565:
457:
2151:
166:
2236:
1463:
1827:
value 0.001101, we will be able to observe the truncated value 0.0011, which is more accurate than our predicted value 0.001. So we have received an information gain of one bit.
2834:{\displaystyle {\frac {11}{24}}\mapsto {\frac {11}{12}}\mapsto {\frac {5}{6}}\mapsto {\frac {2}{3}}\mapsto {\frac {1}{3}}\mapsto {\frac {2}{3}}\mapsto {\frac {1}{3}}\mapsto \cdots ,}
2486:
4120:
3305:
1751:
904:
713:
5572:. There is a potential clash of topologies; some care must be taken. However, as presented above, there is a homomorphism from the Cantor set into the reals; fortunately, it maps
1079:
582:
notation, the next iterate is obtained by shifting the binary point one bit to the right, and if the bit to the left of the new binary point is a "one", replacing it with a zero.
1425:
7991:
7315:
7041:
5841:
5394:
5235:
5193:
4836:
2903:
9357:
7724:
4155:
4412:
7346:
3879:
2090:
1619:
280:
9081:
9045:
7370:
6756:
6732:
6403:
4668:
1221:
6859:
6238:
8043:
6319:
7678:
7277:
5878:
4220:
9395:
5102:
4531:
6700:
6074:
4451:
7760:
9418:
9269:
9012:
352:
8581:
5843:
are a finite number of specific bit-values scattered in the infinite bit-string. These are the open sets of the topology. The canonical measure on this space is the
7485:
6477:
6066:
4485:
2634:
Because of the simple nature of the dynamics when the iterates are viewed in binary notation, it is easy to categorize the dynamics based on the initial condition:
316:
8290:
8069:
5036:
8443:
9534:
8989:
8787:
8767:
8561:
7010:
6817:
6032:
6012:
5795:
5122:
5056:
4370:
4175:
3780:
3297:
1265:
1241:
1182:
816:
796:
8961:{\displaystyle Z=\sum _{\{\sigma _{i}=\pm 1,\,i\in \mathbb {Z} _{N}\}}\prod _{i\in \mathbb {Z} _{N}}{\mathcal {R}}e^{-{\mathcal {R}}\sigma _{i}\sigma _{i+1}},}
6359:
6281:
4228:
590:
1797:
Equivalent to the concept of information loss is the concept of information gain. In practice some real-world process may generate a sequence of values (
8451:
4845:
8143:
8302:
7768:
6488:
9426:
2035:{\displaystyle x_{n+1}=f_{1}(x_{n})={\begin{cases}x_{n}&\mathrm {for} ~~x_{n}\leq 1/2\\1-x_{n}&\mathrm {for} ~~x_{n}\geq 1/2\end{cases}}}
865:
the space of all (semi-)infinite strings of binary bits. The word "infinite" is qualified with "semi-", as one can also define a different space
5652:
8735:{\displaystyle Z=\sum _{\{\sigma _{i}=\pm 1,\,i\in \mathbb {Z} _{2N}\}}\prod _{i\in \mathbb {Z} _{2N}}Ce^{-\beta g\sigma _{i}\sigma _{i+1}}.}
2395:{\displaystyle f_{1}^{n}=f_{1}\circ \cdots \circ f_{1}=S^{-1}\circ T\circ S\circ S^{-1}\circ \cdots \circ T\circ S=S^{-1}\circ T^{n}\circ S}
358:
10338:
10168:
5886:
3588:
Expressed as bit strings, the periodic orbits of the map can be seen to the rationals. That is, after an initial "chaotic" sequence of
1270:
5411:
2912:
1087:
923:
3077:
1634:
6867:
4943:
4636:{\displaystyle \rho (x)\mapsto {\frac {1}{2}}\rho \!\left({\frac {x}{2}}\right)+{\frac {1}{2}}\rho \!\left({\frac {x+1}{2}}\right)}
7628:{\displaystyle {\frac {1}{2}}B_{n}\!\left({\frac {y}{2}}\right)+{\frac {1}{2}}B_{n}\!\left({\frac {y+1}{2}}\right)=2^{-n}B_{n}(y)}
9274:
9665:
7893:
10048:
9808:
9089:
8403:
3585:
corresponds to the non-periodic part of the orbit, after which iteration settles down to all zeros (equivalently, all-ones).
2584:
9681:
A. RĂ©nyi, âRepresentations for real numbers and their ergodic propertiesâ, Acta Math Acad Sci
Hungary, 8, 1957, pp. 477â493.
8074:
4797:{\displaystyle \left(x)={\frac {1}{2}}\rho \!\left({\frac {x}{2}}\right)+{\frac {1}{2}}\rho \!\left({\frac {x+1}{2}}\right)}
3669:
1765:
One hallmark of chaotic dynamics is the loss of information as simulation occurs. If we start with information on the first
10001:
9172:
8383:
All of these different bases can be expressed as linear combinations of one-another. In this sense, they are equivalent.
172:
9948:
9613:
7393:
3788:
10281:
7122:
6761:
1478:
9711:
Dean J. Driebe, Fully
Chaotic Maps and Broken Time Symmetry, (1999) Kluwer Academic Publishers, Dordrecht Netherlands
7081:
3591:
3510:
3204:
596:
10516:
9771:
9716:
7201:
7046:
6960:
5164:
so the uniform density is invariant under the transformation. This is in fact the largest eigenvalue of the operator
10323:
10011:
5586:
5520:
5314:
5267:
3165:
3021:
821:
734:
564:{\displaystyle T(x)={\begin{cases}2x&0\leq x<{\frac {1}{2}}\\2x-1&{\frac {1}{2}}\leq x<1.\end{cases}}}
10198:
9542:
5127:
2502:
8398:(the blancmange curve). Perhaps not a surprise; the Cantor set has exactly the same set of symmetries (as do the
4345:
9748:
M. Bosschaert; C. Jepsen; F. Popov, âChaotic RG flow in tensor modelsâ, Physical Review D, 105, 2022, p. 065021.
2101:
1789:
bits of information remaining. Thus we lose information at the exponential rate of one bit per iteration. After
108:
5577:
3497:{\displaystyle b_{0},b_{1},b_{2},\dots ,b_{k-1},1,0,0,0,\dots =b_{0},b_{1},b_{2},\dots ,b_{k-1},0,1,1,1,\dots }
2159:
1430:
10016:
10006:
2420:
4079:
1715:
868:
679:
10363:
10271:
10136:
7187:
5196:
2571:
1836:
818:. These can be understood to be the flips of a coin, coming up heads or tails. Equivalently, one can write
1032:
2570:
There is also a semi-conjugacy between the dyadic transformation (here named angle doubling map) and the
1806:) over time, but we may only be able to observe these values in truncated form. Suppose for example that
1379:
9690:
A.O. Gelâfond, âA common property of number systemsâ, Izv Akad Nauk SSSR Ser Mat, 23, 1959, pp. 809â814.
7965:
7289:
7015:
5800:
5351:
5209:
5167:
4810:
4033:
This phenomenon is note-worthy, because something similar happens in many chaotic systems. For example,
4016:{\displaystyle \sum _{j=0}^{\infty }b_{k+j}2^{-j-1}=y\sum _{j=0}^{\infty }2^{-jm}={\frac {y}{1-2^{-m}}}}
2860:
10276:
9801:
9334:
7701:
4125:
657:
10343:
4375:
2697:
iterations the iterates reach the fixed point 0; if the initial condition is a rational number with a
10391:
7324:
2051:
1572:
449:
246:
9050:
9017:
7351:
6737:
6713:
6364:
4649:
1906:
1194:
481:
10095:
9699:
W. Parry, âOn the ÎČ -expansion of real numbersâ, Acta Math Acad Sci
Hungary, 11, 1960, pp. 401â416.
6828:
6205:
5253:. Working with any of these spaces is surprisingly difficult, although a spectrum can be obtained.
3201:
problem. The doubled representations hold in general: for any given finite-length initial sequence
10040:
8000:
6286:
6192:{\displaystyle (b_{0},b_{1},b_{2},\dots )\mapsto x=\sum _{n=0}^{\infty }{\frac {b_{n}}{2^{n+1}}}.}
10256:
10021:
9928:
7641:
7247:
5851:
4180:
2689:. Specifically, if the initial condition is a rational number with a finite binary expansion of
585:
The dyadic transformation provides an example of how a simple 1-dimensional map can give rise to
9996:
9362:
5072:
4490:
10296:
9908:
8134:
6675:
6241:
4417:
3782:. It is not hard to see that such repeating sequences correspond to rational numbers. Writing
2857:
The periodic and non-periodic orbits can be more easily understood not by working with the map
7729:
10615:
10446:
10353:
9978:
9913:
9888:
9794:
9400:
9251:
8994:
8746:
8583:
be the inverse temperature for the system, the partition function for this model is given by
8126:
7488:
2679:
324:
8566:
10456:
10208:
10108:
9953:
7463:
7381:
6702:
That is, it maps such that the measure on the unit interval is again the
Lebesgue measure.
6408:
6037:
5062:
4456:
4026:
Tacking on the initial non-repeating sequence, one clearly has a rational number. In fact,
2846:
289:
10286:
8266:
8125:
A complete basis can be given in other ways, as well; they may be written in terms of the
1760:
8:
10416:
10373:
10358:
10203:
10156:
10141:
10126:
10026:
9933:
9918:
9903:
8048:
7691:
6820:
5015:
99:
9958:
8425:
10594:
10461:
10291:
10178:
10173:
10065:
9943:
9845:
9616:, a random distribution on permutations given by applying the doubling map to a set of
9519:
8974:
8772:
8752:
8546:
8399:
6995:
6802:
6017:
5997:
5780:
5107:
5041:
4355:
4334:{\displaystyle \rho (x)\mapsto \sum _{y=T^{-1}(x)}{\frac {\rho (y)}{|T^{\prime }(y)|}}}
4160:
3765:
3282:
1250:
1226:
1167:
801:
781:
6324:
6246:
10466:
10431:
10421:
10318:
9938:
9860:
9767:
9712:
9662:
9603:
7373:
7195:
7179:
6479:
which still has the measure 1/2. That is, the embedding above preserves the measure.
5844:
5565:
5246:
5200:
2638:
728:
653:
649:
445:
95:
10481:
5061:
Viewed as a linear operator, the most obvious and pressing question is: what is its
10574:
10486:
10436:
10333:
10261:
10213:
10090:
10075:
10070:
9865:
9608:
8533:{\displaystyle H(\sigma )=g\sum _{i\in \mathbb {Z} _{2N}}\sigma _{i}\sigma _{i+1}.}
5635:
5631:
5569:
5405:
5401:
5242:
4933:{\displaystyle {\mathcal {L}}_{T}(f+g)={\mathcal {L}}_{T}(f)+{\mathcal {L}}_{T}(g)}
4061:
3151:
is basically a statement that the Cantor set can be mapped into the reals. It is a
2668:
716:
10506:
10401:
10328:
10161:
9973:
9963:
9669:
8395:
8253:{\displaystyle {\mbox{blanc}}_{w,k}(x)=\sum _{n=0}^{\infty }w^{n}s((2k+1)2^{n}x)}
8130:
4839:
3156:
2657:
1813:= 0.1001101, but we only observe the truncated value 0.1001. Our prediction for
1709:
38:
10496:
10441:
906:
consisting of all doubly-infinite (double-ended) strings; this will lead to the
10589:
10556:
10551:
10546:
10348:
10238:
10233:
10131:
10080:
9870:
9637:
8373:{\displaystyle {\mathcal {L}}_{T}{\mbox{blanc}}_{w,k}=w\;{\mbox{blanc}}_{w,k}.}
7877:{\displaystyle \psi _{z,k}(x)=\sum _{n=1}^{\infty }z^{n}\exp i\pi (2k+1)2^{n}x}
7318:
5238:
3159:
has not one, but two distinct representations in the Cantor set. For example,
1466:
907:
8769:
can also be equated with the partition function for a smaller system with but
6662:{\displaystyle (b_{0},b_{1},b_{2},\dots )\mapsto x=\sum _{n=0}^{\infty }\left}
1820:
is 0.001. If we wait until the real-world process has generated the true
648:. This map has been extensively studied by many authors. It was introduced by
10609:
10584:
10541:
10531:
10526:
10426:
10406:
10266:
10188:
10085:
9893:
8391:
8293:
7695:
7377:
5639:
5397:
4038:
1189:
579:
10536:
10501:
10411:
10368:
10223:
10218:
9817:
9506:{\displaystyle e^{-2\beta g}=i\tan {\big (}\pi (t-{\frac {1}{2}}){\big )},}
8407:
7687:
5643:
5515:
5262:
4048:
3069:
3016:
2575:
2410:
1244:
724:
586:
10183:
589:. This map readily generalizes to several others. An important one is the
10521:
10511:
10396:
10146:
9968:
9875:
9779:
The
Bernoulli Map, the Gauss-Kuzmin-Wirsing Operator and the Riemann Zeta
8419:
7191:
1185:
914:
5767:{\displaystyle (*,*,*,\dots ,*,b_{k},b_{k+1},*,\dots ,*,b_{m},*,\dots )}
2667:
contains a finite number of distinct values within [0, 1) and the
10579:
10476:
9923:
8386:
The fractal eigenfunctions show an explicit symmetry under the fractal
7683:
5250:
5066:
4349:
4052:
3152:
2642:
1761:
Rate of information loss and sensitive dependence on initial conditions
1754:
7384:! (This coincidence of naming was presumably not known to Bernoulli.)
5261:
A vast amount of simplification results if one instead works with the
433:{\displaystyle {\text{for all }}n\geq 0,\ x_{n+1}=(2x_{n}){\bmod {1}}}
10491:
10451:
10193:
9855:
9840:
2906:
2489:
9778:
1081:
is an infinite string of binary digits. Given such a string, write
668:
10228:
8387:
7491:. This follows because the Bernoulli polynomials obey the identity
5573:
4041:
4034:
3198:
1840:
444:
Equivalently, the dyadic transformation can also be defined as the
283:
5984:{\displaystyle \mu _{p}(*,\dots ,*,b_{k},*,\dots )=p^{n}(1-p)^{m}}
9898:
9850:
8749:
by integrating out every other spin. In so doing, one finds that
7994:
5797:
are arbitrary bit values (not necessarily all the same), and the
5630:, one must provide a topology for it; by convention, this is the
1369:{\displaystyle T(b_{0},b_{1},b_{2},\dots )=(b_{1},b_{2},\dots ),}
7372:
to the set of all polynomials. In this case, the operator has a
5507:{\displaystyle T(b_{0},b_{1},b_{2},\dots )=(b_{1},b_{2},\dots )}
5241:(on the unit interval) to work with. This might be the space of
3008:{\displaystyle T(b_{0},b_{1},b_{2},\dots )=(b_{1},b_{2},\dots )}
1154:{\displaystyle x=\sum _{n=0}^{\infty }{\frac {b_{n}}{2^{n+1}}}.}
1019:{\displaystyle T(b_{0},b_{1},b_{2},\dots )=(b_{1},b_{2},\dots )}
10471:
9786:
4414:. Also, there are obviously only two points in the preimage of
3141:{\displaystyle x=\sum _{n=0}^{\infty }{\frac {b_{n}}{2^{n+1}}}}
2678:
is eventually periodic, with period equal to the period of the
1698:{\displaystyle y=\sum _{n=0}^{\infty }{\frac {b_{n}}{3^{n+1}}}}
19:
6361:
and both of these intervals have a measure of 1/2. Similarly,
5237:
in greater detail, one must first limit oneself to a suitable
2713: > 1) that repeats itself infinitely, then after
9731:-adic one-dimensional maps and the Euler summation formula",
9572:
6950:{\displaystyle \left(f\circ T^{-1}\right)\!(x)=f(T^{-1}(x)).}
5377:
5005:{\displaystyle {\mathcal {L}}_{T}(af)=a{\mathcal {L}}_{T}(f)}
2886:
2601:
1405:
696:
629:
421:
8394:; this is developed in greater detail in the article on the
6710:
Denote the collection of all open sets on the Cantor set by
1712:, as conventionally defined. This is one reason why the set
9766:, (1999) Kluwer Academic Publishers, Dordrecht Netherlands
9653:, edited by A. V. Holden, Princeton University Press, 1986.
9516:
and the resulting renormalization group transformation for
9324:{\displaystyle \operatorname {Im} ={\frac {\pi }{2}}+\pi n}
2028:
557:
8445:
spins with periodic boundary conditions can be written as
7952:{\displaystyle {\mathcal {L}}_{T}\psi _{z,k}=z\psi _{z,k}}
5199:. The uniform density is, in fact, nothing other than the
9649:
Wolf, A. "Quantifying Chaos with
Lyapunov exponents," in
9159:{\displaystyle {\mathcal {R}}^{2}=4\cosh(2\beta g)C^{4},}
578:
arises because, if the value of an iterate is written in
8137:
function. The eigenfunctions are explicitly of the form
5634:. By adjoining set-complements, it can be extended to a
3666:, a periodic orbit settles down into a repeating string
1830:
9663:
Dynamical
Systems and Ergodic Theory â The Doubling Map
4047:
Keep in mind, however, that the rationals are a set of
2619:{\displaystyle \theta \mapsto 2\theta {\bmod {2}}\pi .}
778:
be the set of all semi-infinite strings of the letters
8349:
8322:
8149:
8116:{\displaystyle z={\frac {1}{2}},{\frac {1}{4}},\dots }
3755:{\displaystyle b_{k},b_{k+1},b_{k+2},\dots ,b_{k+m-1}}
652:
in 1957, and an invariant measure for it was given by
9545:
9522:
9429:
9403:
9365:
9337:
9277:
9254:
9238:{\displaystyle e^{-2{\mathcal {R}}}=\cosh(2\beta g).}
9175:
9092:
9053:
9020:
8997:
8977:
8798:
8775:
8755:
8592:
8569:
8549:
8454:
8428:
8305:
8269:
8146:
8077:
8051:
8003:
7968:
7896:
7771:
7732:
7704:
7644:
7500:
7466:
7396:
7354:
7327:
7292:
7250:
7204:
7125:
7084:
7049:
7018:
6998:
6963:
6870:
6831:
6805:
6764:
6740:
6716:
6678:
6491:
6411:
6367:
6327:
6289:
6249:
6208:
6077:
6040:
6020:
6000:
5889:
5854:
5803:
5783:
5655:
5589:
5523:
5414:
5354:
5317:
5270:
5212:
5170:
5130:
5110:
5075:
5044:
5018:
4946:
4848:
4813:
4679:
4652:
4542:
4493:
4459:
4420:
4378:
4358:
4231:
4183:
4163:
4128:
4082:
3882:
3791:
3768:
3672:
3594:
3513:
3308:
3285:
3207:
3197:
This is just the binary-string version of the famous
3168:
3080:
3024:
2915:
2863:
2733:
2717:
iterations the iterates reach a cycle of length
2587:
2505:
2423:
2247:
2162:
2104:
2054:
1852:
1718:
1637:
1575:
1481:
1433:
1382:
1273:
1253:
1229:
1197:
1170:
1090:
1035:
926:
871:
824:
804:
784:
737:
682:
599:
460:
361:
327:
292:
249:
175:
111:
4064:
suggest that the general answer is "not very much".
663:
233:{\displaystyle x\mapsto (x_{0},x_{1},x_{2},\ldots )}
8045:The Bernoulli polynomials are recovered by setting
7450:{\displaystyle {\mathcal {L}}_{T}B_{n}=2^{-n}B_{n}}
1843:. Recall that the unit-height tent map is given by
9586:
9528:
9505:
9412:
9389:
9351:
9323:
9263:
9237:
9158:
9075:
9039:
9006:
8983:
8960:
8781:
8761:
8734:
8575:
8555:
8532:
8437:
8418:The Hamiltonian of the zero-field one-dimensional
8372:
8284:
8252:
8115:
8063:
8037:
7985:
7951:
7876:
7754:
7718:
7672:
7627:
7479:
7449:
7364:
7340:
7309:
7271:
7236:
7194:is the invariant measure: in this case, it is the
7167:
7108:
7070:
7035:
7004:
6984:
6949:
6853:
6811:
6791:
6750:
6726:
6694:
6661:
6471:
6397:
6353:
6313:
6275:
6232:
6191:
6060:
6026:
6006:
5983:
5872:
5835:
5789:
5766:
5622:
5556:
5506:
5388:
5340:
5303:
5229:
5187:
5156:
5116:
5096:
5050:
5030:
5004:
4932:
4830:
4796:
4662:
4635:
4525:
4479:
4445:
4406:
4364:
4333:
4214:
4169:
4149:
4114:
4044:can have periodic orbits that behave in this way.
4015:
3863:{\displaystyle y=\sum _{j=0}^{m-1}b_{k+j}2^{-j-1}}
3862:
3774:
3754:
3658:
3577:
3496:
3291:
3271:
3186:
3140:
3057:
3007:
2897:
2833:
2629:
2618:
2559:
2480:
2394:
2230:
2145:
2084:
2034:
1745:
1697:
1613:
1558:
1457:
1419:
1368:
1259:
1235:
1215:
1176:
1153:
1073:
1018:
898:
857:
810:
790:
770:
707:
640:
563:
432:
346:
310:
274:
232:
160:
7563:
7521:
7168:{\displaystyle {\mathcal {L}}_{T}f=f\circ T^{-1}}
6900:
6792:{\displaystyle f:{\mathcal {B}}\to \mathbb {R} .}
4767:
4732:
4606:
4571:
1559:{\displaystyle (x,T(x),T^{2}(x),T^{3}(x),\dots )}
10607:
8563:be a suitably chosen normalization constant and
8402:.) This then leads elegantly into the theory of
7109:{\displaystyle f:{\mathcal {B}}\to \mathbb {R} }
5564:, which by convention is given a very different
3659:{\displaystyle b_{0},b_{1},b_{2},\dots ,b_{k-1}}
3578:{\displaystyle b_{0},b_{1},b_{2},\dots ,b_{k-1}}
3272:{\displaystyle b_{0},b_{1},b_{2},\dots ,b_{k-1}}
641:{\displaystyle T_{\beta }(x)=\beta x{\bmod {1}}}
7686:, and the functions spanning the space are the
7237:{\displaystyle {\mathcal {L}}_{T}(\rho )=\rho }
7071:{\displaystyle {\mathcal {B}}\to \mathbb {R} .}
6985:{\displaystyle {\mathcal {B}}\to \mathbb {R} .}
6068:is preferred, since it is preserved by the map
910:. The qualification "semi-" is dropped below.
8413:
5623:{\displaystyle \Omega =\{0,1\}^{\mathbb {N} }}
5576:into open sets, and thus preserves notions of
5557:{\displaystyle \Omega =\{0,1\}^{\mathbb {N} }}
5341:{\displaystyle \rho :\Omega \to \mathbb {R} .}
5304:{\displaystyle \Omega =\{0,1\}^{\mathbb {N} }}
4177:on this density, one needs to find all points
3187:{\displaystyle 0.1000000\dots =0.011111\dots }
3058:{\displaystyle \Omega =\{0,1\}^{\mathbb {N} }}
1472:The dyadic sequence is then just the sequence
858:{\displaystyle \Omega =\{0,1\}^{\mathbb {N} }}
771:{\displaystyle \Omega =\{H,T\}^{\mathbb {N} }}
9802:
9587:{\displaystyle {\mathcal {R}}=2t{\bmod {1}}.}
9495:
9463:
7190:, and in this case, it is 1. The associated
5157:{\displaystyle {\mathcal {L}}_{T}\rho =\rho }
2852:
9620:uniformly random points on the unit interval
9359:. In that case we can introduce a parameter
8854:
8810:
8651:
8604:
8129:. Another complete basis is provided by the
6705:
5609:
5596:
5543:
5530:
5290:
5277:
3044:
3031:
2724:For example, the forward orbit of 11/24 is:
2560:{\displaystyle z_{n}=\sin ^{2}(2\pi x_{n}).}
1732:
1719:
885:
872:
844:
831:
757:
744:
9764:Fully Chaotic Maps and Broken Time Symmetry
4348:of the transformation, here it is just the
2721:. Thus cycles of all lengths are possible.
2574:. Here, the map doubles angles measured in
2417: = 4 case of the logistic map is
9809:
9795:
8346:
2146:{\displaystyle f_{1}=S^{-1}\circ T\circ S}
161:{\displaystyle T:[0,1)\to [0,1)^{\infty }}
9672:, Corinna Ulcigrai, University of Bristol
9345:
8872:
8844:
8835:
8669:
8638:
8629:
8486:
7976:
7712:
7102:
7061:
6975:
6782:
5614:
5548:
5331:
5295:
4108:
3049:
2231:{\displaystyle f_{1}(x)=S^{-1}(T(S(x))).}
1737:
1458:{\displaystyle \Omega =2^{\mathbb {Z} },}
1446:
1427:For the doubly-infinite sequence of bits
890:
849:
762:
9707:
9705:
7993:This is a complete basis, in that every
6034:tails in the sequence. The measure with
5038:on the unit interval, and all constants
4646:By convention, such maps are denoted by
1769:bits of the initial iterate, then after
667:
18:
2481:{\displaystyle z_{n+1}=4z_{n}(1-z_{n})}
10608:
5646:. A cylinder set has the generic form
4115:{\displaystyle \rho :\to \mathbb {R} }
4067:
1746:{\displaystyle \{H,T\}^{\mathbb {N} }}
899:{\displaystyle \{0,1\}^{\mathbb {Z} }}
708:{\displaystyle x\mapsto 2x{\bmod {1}}}
9790:
9702:
7694:, consisting of the unit disk on the
7317:, one must provide a suitable set of
2045:The conjugacy is explicitly given by
1831:Relation to tent map and logistic map
676: : [0, 1) â [0, 1),
5408:on the reals. By contrast, the map
5348:Some caution is advised, as the map
4344:The denominator in the above is the
2405:It is also conjugate to the chaotic
1074:{\displaystyle (b_{0},b_{1},\dots )}
9949:Measure-preserving dynamical system
9831:
7387:Indeed, one can easily verify that
7178:This linear operator is called the
2238:This is stable under iteration, as
1420:{\displaystyle T(x)=2x{\bmod {1}}.}
656:in 1959 and again independently by
13:
9548:
9536:will be precisely the dyadic map:
9189:
9095:
9056:
9023:
8910:
8886:
8309:
8195:
7986:{\displaystyle k\in \mathbb {Z} .}
7900:
7816:
7400:
7357:
7330:
7310:{\displaystyle {\mathcal {L}}_{T}}
7296:
7208:
7129:
7093:
7052:
7036:{\displaystyle {\mathcal {L}}_{T}}
7022:
6966:
6773:
6743:
6719:
6565:
6151:
5836:{\displaystyle b_{k},b_{m},\dots }
5590:
5524:
5389:{\displaystyle T(x)=2x{\bmod {1}}}
5324:
5271:
5230:{\displaystyle {\mathcal {L}}_{T}}
5216:
5188:{\displaystyle {\mathcal {L}}_{T}}
5174:
5134:
4982:
4950:
4910:
4884:
4852:
4831:{\displaystyle {\mathcal {L}}_{T}}
4817:
4688:
4655:
4533:Putting it all together, one gets
4384:
4309:
3961:
3899:
3103:
3025:
2898:{\displaystyle T(x)=2x{\bmod {1}}}
1991:
1988:
1985:
1928:
1925:
1922:
1660:
1434:
1113:
825:
738:
267:
153:
14:
10627:
10517:Oleksandr Mykolayovych Sharkovsky
9352:{\displaystyle n\in \mathbb {Z} }
7719:{\displaystyle z\in \mathbb {C} }
7682:Another basis is provided by the
7186:. The largest eigenvalue is the
4150:{\displaystyle x\mapsto \rho (x)}
1623:
664:Relation to the Bernoulli process
16:Doubling map on the unit interval
10047:
10039:
9816:
7184:RuelleâFrobeniusâPerron operator
5249:functions, or perhaps even just
4407:{\displaystyle T^{\prime }(y)=2}
2578:. That is, the map is given by
1465:the induced homomorphism is the
7348:One such choice is to restrict
7341:{\displaystyle {\mathcal {F}}.}
2630:Periodicity and non-periodicity
2085:{\displaystyle S(x)=\sin \pi x}
1614:{\displaystyle x_{n}=T^{n}(x).}
275:{\displaystyle [0,1)^{\infty }}
10282:RabinovichâFabrikant equations
9742:
9721:
9693:
9684:
9675:
9656:
9643:
9631:
9559:
9553:
9490:
9471:
9384:
9372:
9296:
9284:
9229:
9217:
9203:
9194:
9140:
9128:
9107:
9100:
9076:{\displaystyle {\mathcal {R}}}
9070:
9061:
9040:{\displaystyle {\mathcal {R}}}
9034:
9028:
8924:
8915:
8897:
8891:
8464:
8458:
8279:
8273:
8247:
8231:
8216:
8213:
8173:
8167:
8019:
8004:
7858:
7843:
7794:
7788:
7742:
7734:
7661:
7655:
7622:
7616:
7365:{\displaystyle {\mathcal {F}}}
7260:
7254:
7225:
7219:
7098:
7057:
7043:on the space of all functions
6971:
6941:
6938:
6932:
6916:
6907:
6901:
6778:
6751:{\displaystyle {\mathcal {F}}}
6727:{\displaystyle {\mathcal {B}}}
6639:
6626:
6623:
6604:
6540:
6537:
6492:
6466:
6438:
6432:
6412:
6398:{\displaystyle (*,0,*,\dots )}
6392:
6368:
6348:
6328:
6308:
6290:
6270:
6250:
6227:
6209:
6126:
6123:
6078:
5972:
5959:
5943:
5900:
5761:
5656:
5501:
5469:
5463:
5418:
5364:
5358:
5327:
5256:
5203:of the dyadic transformation.
5085:
5079:
4999:
4993:
4970:
4961:
4927:
4921:
4901:
4895:
4875:
4863:
4713:
4707:
4663:{\displaystyle {\mathcal {L}}}
4555:
4552:
4546:
4512:
4500:
4440:
4434:
4395:
4389:
4324:
4320:
4314:
4300:
4294:
4288:
4277:
4271:
4244:
4241:
4235:
4209:
4203:
4144:
4138:
4132:
4104:
4101:
4089:
3002:
2970:
2964:
2919:
2905:directly, but rather with the
2873:
2867:
2822:
2809:
2796:
2783:
2770:
2757:
2744:
2705: â„ 0) followed by a
2591:
2551:
2532:
2475:
2456:
2222:
2219:
2216:
2210:
2204:
2198:
2179:
2173:
2064:
2058:
1895:
1882:
1605:
1599:
1553:
1544:
1538:
1522:
1516:
1500:
1494:
1482:
1392:
1386:
1360:
1328:
1322:
1277:
1267:, on the unit interval. Since
1216:{\displaystyle 0\leq x\leq 1.}
1068:
1036:
1013:
981:
975:
930:
686:
616:
610:
470:
464:
417:
401:
305:
293:
263:
250:
227:
182:
179:
149:
136:
133:
130:
118:
1:
9755:
6854:{\displaystyle f\circ T^{-1}}
6233:{\displaystyle (0,*,\cdots )}
5243:Lebesgue measurable functions
1835:The dyadic transformation is
723:The map can be obtained as a
8038:{\displaystyle (2k+1)2^{n}.}
7690:. In this case, one finds a
6957:This is again some function
6672:which preserves the measure
6314:{\displaystyle (1,*,\dots )}
5583:To work with the Cantor set
4670:so that in this case, write
2637:If the initial condition is
1837:topologically semi-conjugate
7:
10017:Poincaré recurrence theorem
9614:GilbertâShannonâReeds model
9597:
8414:Relation to the Ising model
7997:can be written in the form
7673:{\displaystyle B_{0}(x)=1.}
7281:
7272:{\displaystyle \rho (x)=1.}
7188:FrobeniusâPerron eigenvalue
6758:of all arbitrary functions
6482:An alternative is to write
5873:{\displaystyle 0<p<1}
5206:To explore the spectrum of
5197:FrobeniusâPerron eigenvalue
4215:{\displaystyle y=T^{-1}(x)}
2409: = 4 case of the
10:
10632:
10012:PoincarĂ©âBendixson theorem
9739:(letter) L483-L485 (1992).
9390:{\displaystyle t\in [0,1)}
9248:Suppose now that we allow
7726:in the unit disk, so that
7286:To obtain the spectrum of
5642:. The topology is that of
5245:, or perhaps the space of
5097:{\displaystyle \rho (x)=1}
4842:, as one easily sees that
4526:{\displaystyle y=(x+1)/2.}
4157:. To obtain the action of
2853:Periodicity via bit shifts
10565:
10382:
10364:Swinging Atwood's machine
10309:
10247:
10117:
10104:
10056:
10037:
10007:KrylovâBogolyubov theorem
9987:
9884:
9824:
9083:satisfying the equations
9014:with renormalized values
6706:FrobeniusâPerron operator
6695:{\displaystyle \mu _{p}.}
5880:one can define a measure
4446:{\displaystyle T^{-1}(x)}
4122:as this density, so that
4073:density as one iterates?
2488:; this is related to the
913:This space has a natural
450:piecewise linear function
10272:LotkaâVolterra equations
10096:Synchronization of chaos
9899:axiom A dynamical system
9624:
7755:{\displaystyle |z|<1}
1753:is sometimes called the
1376:one can easily see that
10257:Double scroll attractor
10022:Stable manifold theorem
9929:False nearest neighbors
9413:{\displaystyle \beta g}
9271:to be complex and that
9264:{\displaystyle \beta g}
9007:{\displaystyle \beta g}
5124:then one obviously has
2845:is a characteristic of
347:{\displaystyle x_{0}=x}
318:) produced by the rule
10297:Van der Pol oscillator
10277:MackeyâGlass equations
9909:Box-counting dimension
9588:
9530:
9507:
9414:
9391:
9353:
9325:
9265:
9239:
9160:
9077:
9041:
9008:
8985:
8962:
8783:
8763:
8736:
8577:
8576:{\displaystyle \beta }
8557:
8534:
8439:
8374:
8286:
8254:
8199:
8135:differentiable-nowhere
8117:
8065:
8039:
7987:
7953:
7878:
7820:
7756:
7720:
7674:
7629:
7481:
7451:
7366:
7342:
7311:
7273:
7238:
7169:
7110:
7072:
7037:
7006:
6986:
6951:
6855:
6813:
6793:
6752:
6728:
6696:
6663:
6569:
6473:
6399:
6355:
6315:
6277:
6234:
6193:
6155:
6062:
6028:
6008:
5985:
5874:
5837:
5791:
5768:
5624:
5558:
5508:
5390:
5342:
5305:
5231:
5189:
5158:
5118:
5098:
5052:
5032:
5006:
4934:
4832:
4798:
4664:
4637:
4527:
4481:
4447:
4408:
4366:
4335:
4216:
4171:
4151:
4116:
4017:
3965:
3903:
3864:
3824:
3776:
3756:
3660:
3579:
3498:
3293:
3273:
3188:
3142:
3107:
3059:
3009:
2899:
2835:
2620:
2561:
2482:
2396:
2232:
2147:
2086:
2036:
1773:simulated iterations (
1747:
1699:
1664:
1615:
1560:
1459:
1421:
1370:
1261:
1237:
1217:
1178:
1155:
1117:
1075:
1020:
900:
859:
812:
792:
772:
720:
709:
642:
565:
434:
348:
312:
276:
234:
162:
59:
10447:Svetlana Jitomirskaya
10354:Multiscroll attractor
10199:Interval exchange map
10152:Dyadic transformation
10137:Complex quadratic map
9979:Topological conjugacy
9914:Correlation dimension
9889:Anosov diffeomorphism
9589:
9531:
9508:
9415:
9392:
9354:
9326:
9266:
9240:
9161:
9078:
9042:
9009:
8986:
8963:
8784:
8764:
8747:renormalization group
8745:We can implement the
8737:
8578:
8558:
8535:
8440:
8375:
8287:
8255:
8179:
8133:. This is a fractal,
8127:Hurwitz zeta function
8118:
8066:
8040:
7988:
7954:
7879:
7800:
7757:
7721:
7675:
7630:
7489:Bernoulli polynomials
7482:
7480:{\displaystyle B_{n}}
7452:
7382:Bernoulli polynomials
7367:
7343:
7312:
7274:
7239:
7170:
7111:
7073:
7038:
7007:
6992:In this way, the map
6987:
6952:
6856:
6814:
6794:
6753:
6734:and consider the set
6729:
6697:
6664:
6549:
6474:
6472:{\displaystyle \cup }
6405:maps to the interval
6400:
6356:
6321:maps to the interval
6316:
6278:
6235:
6194:
6135:
6063:
6061:{\displaystyle p=1/2}
6029:
6009:
5986:
5875:
5838:
5792:
5769:
5625:
5559:
5509:
5391:
5343:
5306:
5232:
5190:
5159:
5119:
5099:
5053:
5033:
5007:
4935:
4833:
4799:
4665:
4638:
4528:
4482:
4480:{\displaystyle y=x/2}
4448:
4409:
4367:
4336:
4217:
4172:
4152:
4117:
4018:
3945:
3883:
3873:one then clearly has
3865:
3798:
3777:
3757:
3661:
3580:
3507:The initial sequence
3499:
3294:
3274:
3189:
3143:
3087:
3060:
3010:
2900:
2836:
2621:
2562:
2483:
2397:
2233:
2148:
2087:
2037:
1748:
1700:
1644:
1616:
1561:
1460:
1422:
1371:
1262:
1238:
1218:
1179:
1156:
1097:
1076:
1021:
901:
860:
813:
793:
773:
710:
671:
643:
566:
435:
349:
313:
311:{\displaystyle [0,1)}
277:
235:
163:
64:dyadic transformation
22:
10457:Edward Norton Lorenz
9733:Journal of Physics A
9543:
9520:
9427:
9401:
9363:
9335:
9275:
9252:
9173:
9090:
9051:
9018:
8995:
8975:
8971:provided we replace
8796:
8773:
8753:
8590:
8567:
8547:
8452:
8426:
8303:
8285:{\displaystyle s(x)}
8267:
8144:
8075:
8049:
8001:
7966:
7894:
7769:
7730:
7702:
7642:
7498:
7464:
7394:
7380:are (curiously) the
7352:
7325:
7290:
7248:
7202:
7123:
7082:
7078:That is, given some
7047:
7016:
7012:induces another map
6996:
6961:
6868:
6829:
6803:
6762:
6738:
6714:
6676:
6489:
6409:
6365:
6325:
6287:
6247:
6206:
6075:
6038:
6018:
5998:
5887:
5852:
5801:
5781:
5653:
5587:
5521:
5412:
5352:
5315:
5268:
5210:
5168:
5128:
5108:
5073:
5042:
5016:
4944:
4846:
4811:
4677:
4650:
4540:
4491:
4457:
4418:
4376:
4356:
4346:Jacobian determinant
4229:
4181:
4161:
4126:
4080:
3880:
3789:
3766:
3670:
3592:
3511:
3306:
3283:
3205:
3166:
3078:
3022:
2913:
2861:
2731:
2585:
2572:quadratic polynomial
2503:
2421:
2245:
2160:
2102:
2052:
1850:
1716:
1635:
1573:
1479:
1431:
1380:
1271:
1251:
1227:
1195:
1168:
1088:
1033:
924:
869:
822:
802:
782:
735:
680:
597:
458:
359:
325:
290:
247:
173:
109:
80: mod 1 map
10417:Mitchell Feigenbaum
10359:Population dynamics
10344:HĂ©nonâHeiles system
10204:Irrational rotation
10157:Dynamical billiards
10142:Coupled map lattice
10002:Liouville's theorem
9934:Hausdorff dimension
9919:Conservative system
9904:Bifurcation diagram
8400:continued fractions
8064:{\displaystyle k=0}
7692:continuous spectrum
5031:{\displaystyle f,g}
4068:Density formulation
2262:
1839:to the unit-height
591:beta transformation
100:recurrence relation
66:(also known as the
10595:Santa Fe Institute
10462:Aleksandr Lyapunov
10292:Three-body problem
10179:Gingerbreadman map
10066:Bifurcation theory
9944:Lyapunov stability
9668:2013-02-12 at the
9584:
9526:
9503:
9410:
9387:
9349:
9321:
9261:
9235:
9156:
9073:
9037:
9004:
8981:
8958:
8883:
8858:
8779:
8759:
8732:
8683:
8655:
8573:
8553:
8530:
8500:
8438:{\displaystyle 2N}
8435:
8404:elliptic equations
8370:
8353:
8326:
8296:. One has, again,
8282:
8250:
8153:
8113:
8061:
8035:
7983:
7949:
7874:
7752:
7716:
7670:
7625:
7477:
7447:
7362:
7338:
7307:
7269:
7234:
7165:
7106:
7068:
7033:
7002:
6982:
6947:
6851:
6809:
6789:
6748:
6724:
6692:
6659:
6469:
6395:
6351:
6311:
6273:
6230:
6189:
6058:
6024:
6004:
5981:
5870:
5833:
5787:
5764:
5620:
5554:
5514:is defined on the
5504:
5396:is defined on the
5386:
5338:
5301:
5239:space of functions
5227:
5185:
5154:
5114:
5094:
5048:
5028:
5012:for all functions
5002:
4930:
4828:
4794:
4660:
4633:
4523:
4477:
4443:
4404:
4362:
4331:
4281:
4212:
4167:
4147:
4112:
4013:
3860:
3772:
3752:
3656:
3575:
3494:
3289:
3269:
3184:
3138:
3055:
3005:
2895:
2831:
2616:
2557:
2478:
2392:
2248:
2228:
2143:
2082:
2032:
2027:
1743:
1695:
1628:Note that the sum
1611:
1556:
1455:
1417:
1366:
1257:
1233:
1213:
1174:
1151:
1071:
1016:
896:
855:
808:
788:
768:
721:
705:
638:
561:
556:
430:
344:
308:
272:
230:
158:
60:
10603:
10602:
10467:BenoĂźt Mandelbrot
10432:Martin Gutzwiller
10422:Peter Grassberger
10305:
10304:
10287:Rössler attractor
10035:
10034:
9939:Invariant measure
9861:Lyapunov exponent
9727:Pierre Gaspard, "
9604:Bernoulli process
9529:{\displaystyle t}
9488:
9420:via the equation
9310:
8984:{\displaystyle C}
8859:
8805:
8782:{\displaystyle N}
8762:{\displaystyle Z}
8656:
8599:
8556:{\displaystyle C}
8473:
8352:
8325:
8152:
8105:
8092:
7584:
7551:
7534:
7509:
7374:discrete spectrum
7196:Bernoulli measure
7180:transfer operator
7005:{\displaystyle T}
6812:{\displaystyle T}
6202:So, for example,
6184:
6027:{\displaystyle m}
6007:{\displaystyle n}
5845:Bernoulli measure
5790:{\displaystyle *}
5247:square integrable
5201:invariant measure
5117:{\displaystyle x}
5051:{\displaystyle a}
4788:
4762:
4745:
4727:
4627:
4601:
4584:
4566:
4365:{\displaystyle T}
4329:
4247:
4170:{\displaystyle T}
4011:
3775:{\displaystyle m}
3292:{\displaystyle k}
3136:
2820:
2807:
2794:
2781:
2768:
2755:
2742:
2693:bits, then after
2000:
1997:
1937:
1934:
1693:
1260:{\displaystyle T}
1236:{\displaystyle T}
1177:{\displaystyle x}
1146:
811:{\displaystyle T}
791:{\displaystyle H}
729:Bernoulli process
654:Alexander Gelfond
540:
512:
446:iterated function
381:
365:
37: â is
10623:
10575:Butterfly effect
10487:Itamar Procaccia
10437:Brosl Hasslacher
10334:Elastic pendulum
10262:Duffing equation
10209:KaplanâYorke map
10127:Arnold's cat map
10115:
10114:
10091:Stability theory
10076:Dynamical system
10071:Control of chaos
10051:
10043:
10027:Takens's theorem
9959:Poincaré section
9829:
9828:
9811:
9804:
9797:
9788:
9787:
9762:Dean J. Driebe,
9749:
9746:
9740:
9725:
9719:
9709:
9700:
9697:
9691:
9688:
9682:
9679:
9673:
9660:
9654:
9647:
9641:
9640:, Evgeny Demidov
9635:
9609:Bernoulli scheme
9593:
9591:
9590:
9585:
9580:
9579:
9552:
9551:
9535:
9533:
9532:
9527:
9512:
9510:
9509:
9504:
9499:
9498:
9489:
9481:
9467:
9466:
9448:
9447:
9419:
9417:
9416:
9411:
9396:
9394:
9393:
9388:
9358:
9356:
9355:
9350:
9348:
9330:
9328:
9327:
9322:
9311:
9303:
9270:
9268:
9267:
9262:
9244:
9242:
9241:
9236:
9207:
9206:
9193:
9192:
9165:
9163:
9162:
9157:
9152:
9151:
9115:
9114:
9099:
9098:
9082:
9080:
9079:
9074:
9060:
9059:
9046:
9044:
9043:
9038:
9027:
9026:
9013:
9011:
9010:
9005:
8990:
8988:
8987:
8982:
8967:
8965:
8964:
8959:
8954:
8953:
8952:
8951:
8936:
8935:
8914:
8913:
8890:
8889:
8882:
8881:
8880:
8875:
8857:
8853:
8852:
8847:
8822:
8821:
8788:
8786:
8785:
8780:
8768:
8766:
8765:
8760:
8741:
8739:
8738:
8733:
8728:
8727:
8726:
8725:
8710:
8709:
8682:
8681:
8680:
8672:
8654:
8650:
8649:
8641:
8616:
8615:
8582:
8580:
8579:
8574:
8562:
8560:
8559:
8554:
8539:
8537:
8536:
8531:
8526:
8525:
8510:
8509:
8499:
8498:
8497:
8489:
8444:
8442:
8441:
8436:
8379:
8377:
8376:
8371:
8366:
8365:
8354:
8350:
8339:
8338:
8327:
8323:
8319:
8318:
8313:
8312:
8291:
8289:
8288:
8283:
8259:
8257:
8256:
8251:
8243:
8242:
8209:
8208:
8198:
8193:
8166:
8165:
8154:
8150:
8122:
8120:
8119:
8114:
8106:
8098:
8093:
8085:
8070:
8068:
8067:
8062:
8044:
8042:
8041:
8036:
8031:
8030:
7992:
7990:
7989:
7984:
7979:
7958:
7956:
7955:
7950:
7948:
7947:
7926:
7925:
7910:
7909:
7904:
7903:
7883:
7881:
7880:
7875:
7870:
7869:
7830:
7829:
7819:
7814:
7787:
7786:
7762:, the functions
7761:
7759:
7758:
7753:
7745:
7737:
7725:
7723:
7722:
7717:
7715:
7679:
7677:
7676:
7671:
7654:
7653:
7634:
7632:
7631:
7626:
7615:
7614:
7605:
7604:
7589:
7585:
7580:
7569:
7562:
7561:
7552:
7544:
7539:
7535:
7527:
7520:
7519:
7510:
7502:
7486:
7484:
7483:
7478:
7476:
7475:
7456:
7454:
7453:
7448:
7446:
7445:
7436:
7435:
7420:
7419:
7410:
7409:
7404:
7403:
7371:
7369:
7368:
7363:
7361:
7360:
7347:
7345:
7344:
7339:
7334:
7333:
7316:
7314:
7313:
7308:
7306:
7305:
7300:
7299:
7278:
7276:
7275:
7270:
7243:
7241:
7240:
7235:
7218:
7217:
7212:
7211:
7174:
7172:
7171:
7166:
7164:
7163:
7139:
7138:
7133:
7132:
7115:
7113:
7112:
7107:
7105:
7097:
7096:
7077:
7075:
7074:
7069:
7064:
7056:
7055:
7042:
7040:
7039:
7034:
7032:
7031:
7026:
7025:
7011:
7009:
7008:
7003:
6991:
6989:
6988:
6983:
6978:
6970:
6969:
6956:
6954:
6953:
6948:
6931:
6930:
6899:
6895:
6894:
6893:
6860:
6858:
6857:
6852:
6850:
6849:
6818:
6816:
6815:
6810:
6798:
6796:
6795:
6790:
6785:
6777:
6776:
6757:
6755:
6754:
6749:
6747:
6746:
6733:
6731:
6730:
6725:
6723:
6722:
6701:
6699:
6698:
6693:
6688:
6687:
6668:
6666:
6665:
6660:
6658:
6654:
6653:
6652:
6622:
6621:
6600:
6599:
6584:
6583:
6568:
6563:
6530:
6529:
6517:
6516:
6504:
6503:
6478:
6476:
6475:
6470:
6462:
6448:
6428:
6404:
6402:
6401:
6396:
6360:
6358:
6357:
6354:{\displaystyle }
6352:
6338:
6320:
6318:
6317:
6312:
6282:
6280:
6279:
6276:{\displaystyle }
6274:
6266:
6239:
6237:
6236:
6231:
6198:
6196:
6195:
6190:
6185:
6183:
6182:
6167:
6166:
6157:
6154:
6149:
6116:
6115:
6103:
6102:
6090:
6089:
6067:
6065:
6064:
6059:
6054:
6033:
6031:
6030:
6025:
6013:
6011:
6010:
6005:
5990:
5988:
5987:
5982:
5980:
5979:
5958:
5957:
5930:
5929:
5899:
5898:
5879:
5877:
5876:
5871:
5842:
5840:
5839:
5834:
5826:
5825:
5813:
5812:
5796:
5794:
5793:
5788:
5773:
5771:
5770:
5765:
5748:
5747:
5717:
5716:
5698:
5697:
5632:product topology
5629:
5627:
5626:
5621:
5619:
5618:
5617:
5570:product topology
5563:
5561:
5560:
5555:
5553:
5552:
5551:
5513:
5511:
5510:
5505:
5494:
5493:
5481:
5480:
5456:
5455:
5443:
5442:
5430:
5429:
5406:natural topology
5402:real number line
5395:
5393:
5392:
5387:
5385:
5384:
5347:
5345:
5344:
5339:
5334:
5311:, and functions
5310:
5308:
5307:
5302:
5300:
5299:
5298:
5236:
5234:
5233:
5228:
5226:
5225:
5220:
5219:
5194:
5192:
5191:
5186:
5184:
5183:
5178:
5177:
5163:
5161:
5160:
5155:
5144:
5143:
5138:
5137:
5123:
5121:
5120:
5115:
5103:
5101:
5100:
5095:
5057:
5055:
5054:
5049:
5037:
5035:
5034:
5029:
5011:
5009:
5008:
5003:
4992:
4991:
4986:
4985:
4960:
4959:
4954:
4953:
4939:
4937:
4936:
4931:
4920:
4919:
4914:
4913:
4894:
4893:
4888:
4887:
4862:
4861:
4856:
4855:
4837:
4835:
4834:
4829:
4827:
4826:
4821:
4820:
4803:
4801:
4800:
4795:
4793:
4789:
4784:
4773:
4763:
4755:
4750:
4746:
4738:
4728:
4720:
4706:
4702:
4698:
4697:
4692:
4691:
4669:
4667:
4666:
4661:
4659:
4658:
4642:
4640:
4639:
4634:
4632:
4628:
4623:
4612:
4602:
4594:
4589:
4585:
4577:
4567:
4559:
4532:
4530:
4529:
4524:
4519:
4486:
4484:
4483:
4478:
4473:
4452:
4450:
4449:
4444:
4433:
4432:
4413:
4411:
4410:
4405:
4388:
4387:
4371:
4369:
4368:
4363:
4340:
4338:
4337:
4332:
4330:
4328:
4327:
4313:
4312:
4303:
4297:
4283:
4280:
4270:
4269:
4221:
4219:
4218:
4213:
4202:
4201:
4176:
4174:
4173:
4168:
4156:
4154:
4153:
4148:
4121:
4119:
4118:
4113:
4111:
4062:Arnold diffusion
4022:
4020:
4019:
4014:
4012:
4010:
4009:
4008:
3986:
3981:
3980:
3964:
3959:
3938:
3937:
3919:
3918:
3902:
3897:
3869:
3867:
3866:
3861:
3859:
3858:
3840:
3839:
3823:
3812:
3781:
3779:
3778:
3773:
3761:
3759:
3758:
3753:
3751:
3750:
3720:
3719:
3701:
3700:
3682:
3681:
3665:
3663:
3662:
3657:
3655:
3654:
3630:
3629:
3617:
3616:
3604:
3603:
3584:
3582:
3581:
3576:
3574:
3573:
3549:
3548:
3536:
3535:
3523:
3522:
3503:
3501:
3500:
3495:
3463:
3462:
3438:
3437:
3425:
3424:
3412:
3411:
3369:
3368:
3344:
3343:
3331:
3330:
3318:
3317:
3298:
3296:
3295:
3290:
3278:
3276:
3275:
3270:
3268:
3267:
3243:
3242:
3230:
3229:
3217:
3216:
3193:
3191:
3190:
3185:
3147:
3145:
3144:
3139:
3137:
3135:
3134:
3119:
3118:
3109:
3106:
3101:
3064:
3062:
3061:
3056:
3054:
3053:
3052:
3014:
3012:
3011:
3006:
2995:
2994:
2982:
2981:
2957:
2956:
2944:
2943:
2931:
2930:
2904:
2902:
2901:
2896:
2894:
2893:
2840:
2838:
2837:
2832:
2821:
2813:
2808:
2800:
2795:
2787:
2782:
2774:
2769:
2761:
2756:
2748:
2743:
2735:
2701:-bit transient (
2625:
2623:
2622:
2617:
2609:
2608:
2566:
2564:
2563:
2558:
2550:
2549:
2528:
2527:
2515:
2514:
2492:map in variable
2487:
2485:
2484:
2479:
2474:
2473:
2455:
2454:
2439:
2438:
2401:
2399:
2398:
2393:
2385:
2384:
2372:
2371:
2338:
2337:
2310:
2309:
2294:
2293:
2275:
2274:
2261:
2256:
2237:
2235:
2234:
2229:
2197:
2196:
2172:
2171:
2152:
2150:
2149:
2144:
2130:
2129:
2114:
2113:
2091:
2089:
2088:
2083:
2041:
2039:
2038:
2033:
2031:
2030:
2021:
2010:
2009:
1998:
1995:
1994:
1981:
1980:
1958:
1947:
1946:
1935:
1932:
1931:
1918:
1917:
1894:
1893:
1881:
1880:
1868:
1867:
1777: <
1752:
1750:
1749:
1744:
1742:
1741:
1740:
1704:
1702:
1701:
1696:
1694:
1692:
1691:
1676:
1675:
1666:
1663:
1658:
1620:
1618:
1617:
1612:
1598:
1597:
1585:
1584:
1565:
1563:
1562:
1557:
1537:
1536:
1515:
1514:
1464:
1462:
1461:
1456:
1451:
1450:
1449:
1426:
1424:
1423:
1418:
1413:
1412:
1375:
1373:
1372:
1367:
1353:
1352:
1340:
1339:
1315:
1314:
1302:
1301:
1289:
1288:
1266:
1264:
1263:
1258:
1242:
1240:
1239:
1234:
1222:
1220:
1219:
1214:
1183:
1181:
1180:
1175:
1160:
1158:
1157:
1152:
1147:
1145:
1144:
1129:
1128:
1119:
1116:
1111:
1080:
1078:
1077:
1072:
1061:
1060:
1048:
1047:
1025:
1023:
1022:
1017:
1006:
1005:
993:
992:
968:
967:
955:
954:
942:
941:
905:
903:
902:
897:
895:
894:
893:
864:
862:
861:
856:
854:
853:
852:
817:
815:
814:
809:
797:
795:
794:
789:
777:
775:
774:
769:
767:
766:
765:
717:Lebesgue measure
714:
712:
711:
706:
704:
703:
647:
645:
644:
639:
637:
636:
609:
608:
570:
568:
567:
562:
560:
559:
541:
533:
513:
505:
439:
437:
436:
431:
429:
428:
416:
415:
397:
396:
379:
366:
363:
353:
351:
350:
345:
337:
336:
317:
315:
314:
309:
281:
279:
278:
273:
271:
270:
239:
237:
236:
231:
220:
219:
207:
206:
194:
193:
167:
165:
164:
159:
157:
156:
10631:
10630:
10626:
10625:
10624:
10622:
10621:
10620:
10606:
10605:
10604:
10599:
10567:
10561:
10507:Caroline Series
10402:Mary Cartwright
10384:
10378:
10329:Double pendulum
10311:
10301:
10250:
10243:
10169:Exponential map
10120:
10106:
10100:
10058:
10052:
10045:
10031:
9997:Ergodic theorem
9990:
9983:
9974:Stable manifold
9964:Recurrence plot
9880:
9834:
9820:
9815:
9785:
9776:Linas Vepstas,
9758:
9753:
9752:
9747:
9743:
9726:
9722:
9710:
9703:
9698:
9694:
9689:
9685:
9680:
9676:
9670:Wayback Machine
9661:
9657:
9648:
9644:
9638:Chaotic 1D maps
9636:
9632:
9627:
9600:
9575:
9571:
9547:
9546:
9544:
9541:
9540:
9521:
9518:
9517:
9494:
9493:
9480:
9462:
9461:
9434:
9430:
9428:
9425:
9424:
9402:
9399:
9398:
9364:
9361:
9360:
9344:
9336:
9333:
9332:
9302:
9276:
9273:
9272:
9253:
9250:
9249:
9188:
9187:
9180:
9176:
9174:
9171:
9170:
9147:
9143:
9110:
9106:
9094:
9093:
9091:
9088:
9087:
9055:
9054:
9052:
9049:
9048:
9022:
9021:
9019:
9016:
9015:
8996:
8993:
8992:
8976:
8973:
8972:
8941:
8937:
8931:
8927:
8909:
8908:
8904:
8900:
8885:
8884:
8876:
8871:
8870:
8863:
8848:
8843:
8842:
8817:
8813:
8809:
8797:
8794:
8793:
8774:
8771:
8770:
8754:
8751:
8750:
8715:
8711:
8705:
8701:
8691:
8687:
8673:
8668:
8667:
8660:
8642:
8637:
8636:
8611:
8607:
8603:
8591:
8588:
8587:
8568:
8565:
8564:
8548:
8545:
8544:
8515:
8511:
8505:
8501:
8490:
8485:
8484:
8477:
8453:
8450:
8449:
8427:
8424:
8423:
8416:
8396:Takagi function
8355:
8348:
8347:
8328:
8321:
8320:
8314:
8308:
8307:
8306:
8304:
8301:
8300:
8268:
8265:
8264:
8238:
8234:
8204:
8200:
8194:
8183:
8155:
8148:
8147:
8145:
8142:
8141:
8131:Takagi function
8097:
8084:
8076:
8073:
8072:
8050:
8047:
8046:
8026:
8022:
8002:
7999:
7998:
7975:
7967:
7964:
7963:
7937:
7933:
7915:
7911:
7905:
7899:
7898:
7897:
7895:
7892:
7891:
7865:
7861:
7825:
7821:
7815:
7804:
7776:
7772:
7770:
7767:
7766:
7741:
7733:
7731:
7728:
7727:
7711:
7703:
7700:
7699:
7649:
7645:
7643:
7640:
7639:
7610:
7606:
7597:
7593:
7570:
7568:
7564:
7557:
7553:
7543:
7526:
7522:
7515:
7511:
7501:
7499:
7496:
7495:
7471:
7467:
7465:
7462:
7461:
7441:
7437:
7428:
7424:
7415:
7411:
7405:
7399:
7398:
7397:
7395:
7392:
7391:
7356:
7355:
7353:
7350:
7349:
7329:
7328:
7326:
7323:
7322:
7319:basis functions
7301:
7295:
7294:
7293:
7291:
7288:
7287:
7284:
7249:
7246:
7245:
7213:
7207:
7206:
7205:
7203:
7200:
7199:
7156:
7152:
7134:
7128:
7127:
7126:
7124:
7121:
7120:
7101:
7092:
7091:
7083:
7080:
7079:
7060:
7051:
7050:
7048:
7045:
7044:
7027:
7021:
7020:
7019:
7017:
7014:
7013:
6997:
6994:
6993:
6974:
6965:
6964:
6962:
6959:
6958:
6923:
6919:
6886:
6882:
6875:
6871:
6869:
6866:
6865:
6842:
6838:
6830:
6827:
6826:
6804:
6801:
6800:
6781:
6772:
6771:
6763:
6760:
6759:
6742:
6741:
6739:
6736:
6735:
6718:
6717:
6715:
6712:
6711:
6708:
6683:
6679:
6677:
6674:
6673:
6642:
6638:
6617:
6613:
6589:
6585:
6579:
6575:
6574:
6570:
6564:
6553:
6525:
6521:
6512:
6508:
6499:
6495:
6490:
6487:
6486:
6458:
6444:
6424:
6410:
6407:
6406:
6366:
6363:
6362:
6334:
6326:
6323:
6322:
6288:
6285:
6284:
6262:
6248:
6245:
6244:
6207:
6204:
6203:
6172:
6168:
6162:
6158:
6156:
6150:
6139:
6111:
6107:
6098:
6094:
6085:
6081:
6076:
6073:
6072:
6050:
6039:
6036:
6035:
6019:
6016:
6015:
5999:
5996:
5995:
5975:
5971:
5953:
5949:
5925:
5921:
5894:
5890:
5888:
5885:
5884:
5853:
5850:
5849:
5821:
5817:
5808:
5804:
5802:
5799:
5798:
5782:
5779:
5778:
5743:
5739:
5706:
5702:
5693:
5689:
5654:
5651:
5650:
5613:
5612:
5608:
5588:
5585:
5584:
5547:
5546:
5542:
5522:
5519:
5518:
5489:
5485:
5476:
5472:
5451:
5447:
5438:
5434:
5425:
5421:
5413:
5410:
5409:
5404:, assuming the
5380:
5376:
5353:
5350:
5349:
5330:
5316:
5313:
5312:
5294:
5293:
5289:
5269:
5266:
5265:
5259:
5221:
5215:
5214:
5213:
5211:
5208:
5207:
5179:
5173:
5172:
5171:
5169:
5166:
5165:
5139:
5133:
5132:
5131:
5129:
5126:
5125:
5109:
5106:
5105:
5074:
5071:
5070:
5069:is obvious: if
5043:
5040:
5039:
5017:
5014:
5013:
4987:
4981:
4980:
4979:
4955:
4949:
4948:
4947:
4945:
4942:
4941:
4915:
4909:
4908:
4907:
4889:
4883:
4882:
4881:
4857:
4851:
4850:
4849:
4847:
4844:
4843:
4840:linear operator
4822:
4816:
4815:
4814:
4812:
4809:
4808:
4774:
4772:
4768:
4754:
4737:
4733:
4719:
4693:
4687:
4686:
4685:
4684:
4680:
4678:
4675:
4674:
4654:
4653:
4651:
4648:
4647:
4613:
4611:
4607:
4593:
4576:
4572:
4558:
4541:
4538:
4537:
4515:
4492:
4489:
4488:
4469:
4458:
4455:
4454:
4425:
4421:
4419:
4416:
4415:
4383:
4379:
4377:
4374:
4373:
4357:
4354:
4353:
4323:
4308:
4304:
4299:
4298:
4284:
4282:
4262:
4258:
4251:
4230:
4227:
4226:
4194:
4190:
4182:
4179:
4178:
4162:
4159:
4158:
4127:
4124:
4123:
4107:
4081:
4078:
4077:
4070:
4001:
3997:
3990:
3985:
3970:
3966:
3960:
3949:
3924:
3920:
3908:
3904:
3898:
3887:
3881:
3878:
3877:
3845:
3841:
3829:
3825:
3813:
3802:
3790:
3787:
3786:
3767:
3764:
3763:
3734:
3730:
3709:
3705:
3690:
3686:
3677:
3673:
3671:
3668:
3667:
3644:
3640:
3625:
3621:
3612:
3608:
3599:
3595:
3593:
3590:
3589:
3563:
3559:
3544:
3540:
3531:
3527:
3518:
3514:
3512:
3509:
3508:
3452:
3448:
3433:
3429:
3420:
3416:
3407:
3403:
3358:
3354:
3339:
3335:
3326:
3322:
3313:
3309:
3307:
3304:
3303:
3284:
3281:
3280:
3257:
3253:
3238:
3234:
3225:
3221:
3212:
3208:
3206:
3203:
3202:
3167:
3164:
3163:
3157:dyadic rational
3124:
3120:
3114:
3110:
3108:
3102:
3091:
3079:
3076:
3075:
3048:
3047:
3043:
3023:
3020:
3019:
3015:defined on the
2990:
2986:
2977:
2973:
2952:
2948:
2939:
2935:
2926:
2922:
2914:
2911:
2910:
2889:
2885:
2862:
2859:
2858:
2855:
2812:
2799:
2786:
2773:
2760:
2747:
2734:
2732:
2729:
2728:
2709:-bit sequence (
2688:
2677:
2666:
2655:
2632:
2604:
2600:
2586:
2583:
2582:
2545:
2541:
2523:
2519:
2510:
2506:
2504:
2501:
2500:
2469:
2465:
2450:
2446:
2428:
2424:
2422:
2419:
2418:
2380:
2376:
2364:
2360:
2330:
2326:
2302:
2298:
2289:
2285:
2270:
2266:
2257:
2252:
2246:
2243:
2242:
2189:
2185:
2167:
2163:
2161:
2158:
2157:
2122:
2118:
2109:
2105:
2103:
2100:
2099:
2053:
2050:
2049:
2026:
2025:
2017:
2005:
2001:
1984:
1982:
1976:
1972:
1963:
1962:
1954:
1942:
1938:
1921:
1919:
1913:
1909:
1902:
1901:
1889:
1885:
1876:
1872:
1857:
1853:
1851:
1848:
1847:
1833:
1826:
1819:
1812:
1805:
1781:) we only have
1763:
1736:
1735:
1731:
1717:
1714:
1713:
1710:Cantor function
1681:
1677:
1671:
1667:
1665:
1659:
1648:
1636:
1633:
1632:
1626:
1593:
1589:
1580:
1576:
1574:
1571:
1570:
1532:
1528:
1510:
1506:
1480:
1477:
1476:
1445:
1444:
1440:
1432:
1429:
1428:
1408:
1404:
1381:
1378:
1377:
1348:
1344:
1335:
1331:
1310:
1306:
1297:
1293:
1284:
1280:
1272:
1269:
1268:
1252:
1249:
1248:
1228:
1225:
1224:
1196:
1193:
1192:
1169:
1166:
1165:
1134:
1130:
1124:
1120:
1118:
1112:
1101:
1089:
1086:
1085:
1056:
1052:
1043:
1039:
1034:
1031:
1030:
1001:
997:
988:
984:
963:
959:
950:
946:
937:
933:
925:
922:
921:
915:shift operation
889:
888:
884:
870:
867:
866:
848:
847:
843:
823:
820:
819:
803:
800:
799:
783:
780:
779:
761:
760:
756:
736:
733:
732:
699:
695:
681:
678:
677:
666:
632:
628:
604:
600:
598:
595:
594:
555:
554:
532:
530:
515:
514:
504:
490:
477:
476:
459:
456:
455:
424:
420:
411:
407:
386:
382:
362:
360:
357:
356:
332:
328:
326:
323:
322:
291:
288:
287:
266:
262:
248:
245:
244:
215:
211:
202:
198:
189:
185:
174:
171:
170:
152:
148:
110:
107:
106:
53:
36:
17:
12:
11:
5:
10629:
10619:
10618:
10601:
10600:
10598:
10597:
10592:
10590:Predictability
10587:
10582:
10577:
10571:
10569:
10563:
10562:
10560:
10559:
10557:Lai-Sang Young
10554:
10552:James A. Yorke
10549:
10547:Amie Wilkinson
10544:
10539:
10534:
10529:
10524:
10519:
10514:
10509:
10504:
10499:
10494:
10489:
10484:
10482:Henri Poincaré
10479:
10474:
10469:
10464:
10459:
10454:
10449:
10444:
10439:
10434:
10429:
10424:
10419:
10414:
10409:
10404:
10399:
10394:
10388:
10386:
10380:
10379:
10377:
10376:
10371:
10366:
10361:
10356:
10351:
10349:Kicked rotator
10346:
10341:
10336:
10331:
10326:
10321:
10319:Chua's circuit
10315:
10313:
10307:
10306:
10303:
10302:
10300:
10299:
10294:
10289:
10284:
10279:
10274:
10269:
10264:
10259:
10253:
10251:
10248:
10245:
10244:
10242:
10241:
10239:Zaslavskii map
10236:
10234:Tinkerbell map
10231:
10226:
10221:
10216:
10211:
10206:
10201:
10196:
10191:
10186:
10181:
10176:
10171:
10166:
10165:
10164:
10154:
10149:
10144:
10139:
10134:
10129:
10123:
10121:
10118:
10112:
10102:
10101:
10099:
10098:
10093:
10088:
10083:
10081:Ergodic theory
10078:
10073:
10068:
10062:
10060:
10054:
10053:
10038:
10036:
10033:
10032:
10030:
10029:
10024:
10019:
10014:
10009:
10004:
9999:
9993:
9991:
9988:
9985:
9984:
9982:
9981:
9976:
9971:
9966:
9961:
9956:
9951:
9946:
9941:
9936:
9931:
9926:
9921:
9916:
9911:
9906:
9901:
9896:
9891:
9885:
9882:
9881:
9879:
9878:
9873:
9871:Periodic point
9868:
9863:
9858:
9853:
9848:
9843:
9837:
9835:
9832:
9826:
9822:
9821:
9814:
9813:
9806:
9799:
9791:
9784:
9783:
9774:
9759:
9757:
9754:
9751:
9750:
9741:
9720:
9701:
9692:
9683:
9674:
9655:
9642:
9629:
9628:
9626:
9623:
9622:
9621:
9611:
9606:
9599:
9596:
9595:
9594:
9583:
9578:
9574:
9570:
9567:
9564:
9561:
9558:
9555:
9550:
9525:
9514:
9513:
9502:
9497:
9492:
9487:
9484:
9479:
9476:
9473:
9470:
9465:
9460:
9457:
9454:
9451:
9446:
9443:
9440:
9437:
9433:
9409:
9406:
9386:
9383:
9380:
9377:
9374:
9371:
9368:
9347:
9343:
9340:
9320:
9317:
9314:
9309:
9306:
9301:
9298:
9295:
9292:
9289:
9286:
9283:
9280:
9260:
9257:
9246:
9245:
9234:
9231:
9228:
9225:
9222:
9219:
9216:
9213:
9210:
9205:
9202:
9199:
9196:
9191:
9186:
9183:
9179:
9167:
9166:
9155:
9150:
9146:
9142:
9139:
9136:
9133:
9130:
9127:
9124:
9121:
9118:
9113:
9109:
9105:
9102:
9097:
9072:
9069:
9066:
9063:
9058:
9036:
9033:
9030:
9025:
9003:
9000:
8980:
8969:
8968:
8957:
8950:
8947:
8944:
8940:
8934:
8930:
8926:
8923:
8920:
8917:
8912:
8907:
8903:
8899:
8896:
8893:
8888:
8879:
8874:
8869:
8866:
8862:
8856:
8851:
8846:
8841:
8838:
8834:
8831:
8828:
8825:
8820:
8816:
8812:
8808:
8804:
8801:
8778:
8758:
8743:
8742:
8731:
8724:
8721:
8718:
8714:
8708:
8704:
8700:
8697:
8694:
8690:
8686:
8679:
8676:
8671:
8666:
8663:
8659:
8653:
8648:
8645:
8640:
8635:
8632:
8628:
8625:
8622:
8619:
8614:
8610:
8606:
8602:
8598:
8595:
8572:
8552:
8541:
8540:
8529:
8524:
8521:
8518:
8514:
8508:
8504:
8496:
8493:
8488:
8483:
8480:
8476:
8472:
8469:
8466:
8463:
8460:
8457:
8434:
8431:
8415:
8412:
8381:
8380:
8369:
8364:
8361:
8358:
8345:
8342:
8337:
8334:
8331:
8317:
8311:
8281:
8278:
8275:
8272:
8261:
8260:
8249:
8246:
8241:
8237:
8233:
8230:
8227:
8224:
8221:
8218:
8215:
8212:
8207:
8203:
8197:
8192:
8189:
8186:
8182:
8178:
8175:
8172:
8169:
8164:
8161:
8158:
8112:
8109:
8104:
8101:
8096:
8091:
8088:
8083:
8080:
8060:
8057:
8054:
8034:
8029:
8025:
8021:
8018:
8015:
8012:
8009:
8006:
7982:
7978:
7974:
7971:
7960:
7959:
7946:
7943:
7940:
7936:
7932:
7929:
7924:
7921:
7918:
7914:
7908:
7902:
7885:
7884:
7873:
7868:
7864:
7860:
7857:
7854:
7851:
7848:
7845:
7842:
7839:
7836:
7833:
7828:
7824:
7818:
7813:
7810:
7807:
7803:
7799:
7796:
7793:
7790:
7785:
7782:
7779:
7775:
7751:
7748:
7744:
7740:
7736:
7714:
7710:
7707:
7669:
7666:
7663:
7660:
7657:
7652:
7648:
7636:
7635:
7624:
7621:
7618:
7613:
7609:
7603:
7600:
7596:
7592:
7588:
7583:
7579:
7576:
7573:
7567:
7560:
7556:
7550:
7547:
7542:
7538:
7533:
7530:
7525:
7518:
7514:
7508:
7505:
7474:
7470:
7458:
7457:
7444:
7440:
7434:
7431:
7427:
7423:
7418:
7414:
7408:
7402:
7378:eigenfunctions
7359:
7337:
7332:
7321:for the space
7304:
7298:
7283:
7280:
7268:
7265:
7262:
7259:
7256:
7253:
7233:
7230:
7227:
7224:
7221:
7216:
7210:
7176:
7175:
7162:
7159:
7155:
7151:
7148:
7145:
7142:
7137:
7131:
7116:, one defines
7104:
7100:
7095:
7090:
7087:
7067:
7063:
7059:
7054:
7030:
7024:
7001:
6981:
6977:
6973:
6968:
6946:
6943:
6940:
6937:
6934:
6929:
6926:
6922:
6918:
6915:
6912:
6909:
6906:
6903:
6898:
6892:
6889:
6885:
6881:
6878:
6874:
6862:
6861:
6848:
6845:
6841:
6837:
6834:
6808:
6788:
6784:
6780:
6775:
6770:
6767:
6745:
6721:
6707:
6704:
6691:
6686:
6682:
6670:
6669:
6657:
6651:
6648:
6645:
6641:
6637:
6634:
6631:
6628:
6625:
6620:
6616:
6612:
6609:
6606:
6603:
6598:
6595:
6592:
6588:
6582:
6578:
6573:
6567:
6562:
6559:
6556:
6552:
6548:
6545:
6542:
6539:
6536:
6533:
6528:
6524:
6520:
6515:
6511:
6507:
6502:
6498:
6494:
6468:
6465:
6461:
6457:
6454:
6451:
6447:
6443:
6440:
6437:
6434:
6431:
6427:
6423:
6420:
6417:
6414:
6394:
6391:
6388:
6385:
6382:
6379:
6376:
6373:
6370:
6350:
6347:
6344:
6341:
6337:
6333:
6330:
6310:
6307:
6304:
6301:
6298:
6295:
6292:
6272:
6269:
6265:
6261:
6258:
6255:
6252:
6229:
6226:
6223:
6220:
6217:
6214:
6211:
6200:
6199:
6188:
6181:
6178:
6175:
6171:
6165:
6161:
6153:
6148:
6145:
6142:
6138:
6134:
6131:
6128:
6125:
6122:
6119:
6114:
6110:
6106:
6101:
6097:
6093:
6088:
6084:
6080:
6057:
6053:
6049:
6046:
6043:
6023:
6003:
5992:
5991:
5978:
5974:
5970:
5967:
5964:
5961:
5956:
5952:
5948:
5945:
5942:
5939:
5936:
5933:
5928:
5924:
5920:
5917:
5914:
5911:
5908:
5905:
5902:
5897:
5893:
5869:
5866:
5863:
5860:
5857:
5832:
5829:
5824:
5820:
5816:
5811:
5807:
5786:
5775:
5774:
5763:
5760:
5757:
5754:
5751:
5746:
5742:
5738:
5735:
5732:
5729:
5726:
5723:
5720:
5715:
5712:
5709:
5705:
5701:
5696:
5692:
5688:
5685:
5682:
5679:
5676:
5673:
5670:
5667:
5664:
5661:
5658:
5616:
5611:
5607:
5604:
5601:
5598:
5595:
5592:
5550:
5545:
5541:
5538:
5535:
5532:
5529:
5526:
5503:
5500:
5497:
5492:
5488:
5484:
5479:
5475:
5471:
5468:
5465:
5462:
5459:
5454:
5450:
5446:
5441:
5437:
5433:
5428:
5424:
5420:
5417:
5383:
5379:
5375:
5372:
5369:
5366:
5363:
5360:
5357:
5337:
5333:
5329:
5326:
5323:
5320:
5297:
5292:
5288:
5285:
5282:
5279:
5276:
5273:
5258:
5255:
5224:
5218:
5182:
5176:
5153:
5150:
5147:
5142:
5136:
5113:
5093:
5090:
5087:
5084:
5081:
5078:
5047:
5027:
5024:
5021:
5001:
4998:
4995:
4990:
4984:
4978:
4975:
4972:
4969:
4966:
4963:
4958:
4952:
4929:
4926:
4923:
4918:
4912:
4906:
4903:
4900:
4897:
4892:
4886:
4880:
4877:
4874:
4871:
4868:
4865:
4860:
4854:
4825:
4819:
4805:
4804:
4792:
4787:
4783:
4780:
4777:
4771:
4766:
4761:
4758:
4753:
4749:
4744:
4741:
4736:
4731:
4726:
4723:
4718:
4715:
4712:
4709:
4705:
4701:
4696:
4690:
4683:
4657:
4644:
4643:
4631:
4626:
4622:
4619:
4616:
4610:
4605:
4600:
4597:
4592:
4588:
4583:
4580:
4575:
4570:
4565:
4562:
4557:
4554:
4551:
4548:
4545:
4522:
4518:
4514:
4511:
4508:
4505:
4502:
4499:
4496:
4476:
4472:
4468:
4465:
4462:
4442:
4439:
4436:
4431:
4428:
4424:
4403:
4400:
4397:
4394:
4391:
4386:
4382:
4361:
4342:
4341:
4326:
4322:
4319:
4316:
4311:
4307:
4302:
4296:
4293:
4290:
4287:
4279:
4276:
4273:
4268:
4265:
4261:
4257:
4254:
4250:
4246:
4243:
4240:
4237:
4234:
4211:
4208:
4205:
4200:
4197:
4193:
4189:
4186:
4166:
4146:
4143:
4140:
4137:
4134:
4131:
4110:
4106:
4103:
4100:
4097:
4094:
4091:
4088:
4085:
4069:
4066:
4051:in the reals.
4024:
4023:
4007:
4004:
4000:
3996:
3993:
3989:
3984:
3979:
3976:
3973:
3969:
3963:
3958:
3955:
3952:
3948:
3944:
3941:
3936:
3933:
3930:
3927:
3923:
3917:
3914:
3911:
3907:
3901:
3896:
3893:
3890:
3886:
3871:
3870:
3857:
3854:
3851:
3848:
3844:
3838:
3835:
3832:
3828:
3822:
3819:
3816:
3811:
3808:
3805:
3801:
3797:
3794:
3771:
3749:
3746:
3743:
3740:
3737:
3733:
3729:
3726:
3723:
3718:
3715:
3712:
3708:
3704:
3699:
3696:
3693:
3689:
3685:
3680:
3676:
3653:
3650:
3647:
3643:
3639:
3636:
3633:
3628:
3624:
3620:
3615:
3611:
3607:
3602:
3598:
3572:
3569:
3566:
3562:
3558:
3555:
3552:
3547:
3543:
3539:
3534:
3530:
3526:
3521:
3517:
3505:
3504:
3493:
3490:
3487:
3484:
3481:
3478:
3475:
3472:
3469:
3466:
3461:
3458:
3455:
3451:
3447:
3444:
3441:
3436:
3432:
3428:
3423:
3419:
3415:
3410:
3406:
3402:
3399:
3396:
3393:
3390:
3387:
3384:
3381:
3378:
3375:
3372:
3367:
3364:
3361:
3357:
3353:
3350:
3347:
3342:
3338:
3334:
3329:
3325:
3321:
3316:
3312:
3288:
3266:
3263:
3260:
3256:
3252:
3249:
3246:
3241:
3237:
3233:
3228:
3224:
3220:
3215:
3211:
3195:
3194:
3183:
3180:
3177:
3174:
3171:
3149:
3148:
3133:
3130:
3127:
3123:
3117:
3113:
3105:
3100:
3097:
3094:
3090:
3086:
3083:
3051:
3046:
3042:
3039:
3036:
3033:
3030:
3027:
3004:
3001:
2998:
2993:
2989:
2985:
2980:
2976:
2972:
2969:
2966:
2963:
2960:
2955:
2951:
2947:
2942:
2938:
2934:
2929:
2925:
2921:
2918:
2892:
2888:
2884:
2881:
2878:
2875:
2872:
2869:
2866:
2854:
2851:
2842:
2841:
2830:
2827:
2824:
2819:
2816:
2811:
2806:
2803:
2798:
2793:
2790:
2785:
2780:
2777:
2772:
2767:
2764:
2759:
2754:
2751:
2746:
2741:
2738:
2686:
2675:
2664:
2653:
2631:
2628:
2627:
2626:
2615:
2612:
2607:
2603:
2599:
2596:
2593:
2590:
2568:
2567:
2556:
2553:
2548:
2544:
2540:
2537:
2534:
2531:
2526:
2522:
2518:
2513:
2509:
2477:
2472:
2468:
2464:
2461:
2458:
2453:
2449:
2445:
2442:
2437:
2434:
2431:
2427:
2403:
2402:
2391:
2388:
2383:
2379:
2375:
2370:
2367:
2363:
2359:
2356:
2353:
2350:
2347:
2344:
2341:
2336:
2333:
2329:
2325:
2322:
2319:
2316:
2313:
2308:
2305:
2301:
2297:
2292:
2288:
2284:
2281:
2278:
2273:
2269:
2265:
2260:
2255:
2251:
2227:
2224:
2221:
2218:
2215:
2212:
2209:
2206:
2203:
2200:
2195:
2192:
2188:
2184:
2181:
2178:
2175:
2170:
2166:
2154:
2153:
2142:
2139:
2136:
2133:
2128:
2125:
2121:
2117:
2112:
2108:
2093:
2092:
2081:
2078:
2075:
2072:
2069:
2066:
2063:
2060:
2057:
2043:
2042:
2029:
2024:
2020:
2016:
2013:
2008:
2004:
1993:
1990:
1987:
1983:
1979:
1975:
1971:
1968:
1965:
1964:
1961:
1957:
1953:
1950:
1945:
1941:
1930:
1927:
1924:
1920:
1916:
1912:
1908:
1907:
1905:
1900:
1897:
1892:
1888:
1884:
1879:
1875:
1871:
1866:
1863:
1860:
1856:
1832:
1829:
1824:
1817:
1810:
1801:
1762:
1759:
1739:
1734:
1730:
1727:
1724:
1721:
1706:
1705:
1690:
1687:
1684:
1680:
1674:
1670:
1662:
1657:
1654:
1651:
1647:
1643:
1640:
1625:
1624:The Cantor set
1622:
1610:
1607:
1604:
1601:
1596:
1592:
1588:
1583:
1579:
1567:
1566:
1555:
1552:
1549:
1546:
1543:
1540:
1535:
1531:
1527:
1524:
1521:
1518:
1513:
1509:
1505:
1502:
1499:
1496:
1493:
1490:
1487:
1484:
1454:
1448:
1443:
1439:
1436:
1416:
1411:
1407:
1403:
1400:
1397:
1394:
1391:
1388:
1385:
1365:
1362:
1359:
1356:
1351:
1347:
1343:
1338:
1334:
1330:
1327:
1324:
1321:
1318:
1313:
1309:
1305:
1300:
1296:
1292:
1287:
1283:
1279:
1276:
1256:
1247:, also called
1232:
1212:
1209:
1206:
1203:
1200:
1173:
1164:The resulting
1162:
1161:
1150:
1143:
1140:
1137:
1133:
1127:
1123:
1115:
1110:
1107:
1104:
1100:
1096:
1093:
1070:
1067:
1064:
1059:
1055:
1051:
1046:
1042:
1038:
1027:
1026:
1015:
1012:
1009:
1004:
1000:
996:
991:
987:
983:
980:
977:
974:
971:
966:
962:
958:
953:
949:
945:
940:
936:
932:
929:
892:
887:
883:
880:
877:
874:
851:
846:
842:
839:
836:
833:
830:
827:
807:
787:
764:
759:
755:
752:
749:
746:
743:
740:
715:preserves the
702:
698:
694:
691:
688:
685:
665:
662:
635:
631:
627:
624:
621:
618:
615:
612:
607:
603:
572:
571:
558:
553:
550:
547:
544:
539:
536:
531:
529:
526:
523:
520:
517:
516:
511:
508:
503:
500:
497:
494:
491:
489:
486:
483:
482:
480:
475:
472:
469:
466:
463:
442:
441:
427:
423:
419:
414:
410:
406:
403:
400:
395:
392:
389:
385:
378:
375:
372:
369:
354:
343:
340:
335:
331:
307:
304:
301:
298:
295:
282:is the set of
269:
265:
261:
258:
255:
252:
241:
240:
229:
226:
223:
218:
214:
210:
205:
201:
197:
192:
188:
184:
181:
178:
168:
155:
151:
147:
144:
141:
138:
135:
132:
129:
126:
123:
120:
117:
114:
49:
34:
15:
9:
6:
4:
3:
2:
10628:
10617:
10614:
10613:
10611:
10596:
10593:
10591:
10588:
10586:
10585:Edge of chaos
10583:
10581:
10578:
10576:
10573:
10572:
10570:
10564:
10558:
10555:
10553:
10550:
10548:
10545:
10543:
10542:Marcelo Viana
10540:
10538:
10535:
10533:
10532:Audrey Terras
10530:
10528:
10527:Floris Takens
10525:
10523:
10520:
10518:
10515:
10513:
10510:
10508:
10505:
10503:
10500:
10498:
10495:
10493:
10490:
10488:
10485:
10483:
10480:
10478:
10475:
10473:
10470:
10468:
10465:
10463:
10460:
10458:
10455:
10453:
10450:
10448:
10445:
10443:
10440:
10438:
10435:
10433:
10430:
10428:
10427:Celso Grebogi
10425:
10423:
10420:
10418:
10415:
10413:
10410:
10408:
10407:Chen Guanrong
10405:
10403:
10400:
10398:
10395:
10393:
10392:Michael Berry
10390:
10389:
10387:
10381:
10375:
10372:
10370:
10367:
10365:
10362:
10360:
10357:
10355:
10352:
10350:
10347:
10345:
10342:
10340:
10337:
10335:
10332:
10330:
10327:
10325:
10322:
10320:
10317:
10316:
10314:
10308:
10298:
10295:
10293:
10290:
10288:
10285:
10283:
10280:
10278:
10275:
10273:
10270:
10268:
10267:Lorenz system
10265:
10263:
10260:
10258:
10255:
10254:
10252:
10246:
10240:
10237:
10235:
10232:
10230:
10227:
10225:
10222:
10220:
10217:
10215:
10214:Langton's ant
10212:
10210:
10207:
10205:
10202:
10200:
10197:
10195:
10192:
10190:
10189:Horseshoe map
10187:
10185:
10182:
10180:
10177:
10175:
10172:
10170:
10167:
10163:
10160:
10159:
10158:
10155:
10153:
10150:
10148:
10145:
10143:
10140:
10138:
10135:
10133:
10130:
10128:
10125:
10124:
10122:
10116:
10113:
10110:
10103:
10097:
10094:
10092:
10089:
10087:
10086:Quantum chaos
10084:
10082:
10079:
10077:
10074:
10072:
10069:
10067:
10064:
10063:
10061:
10055:
10050:
10046:
10042:
10028:
10025:
10023:
10020:
10018:
10015:
10013:
10010:
10008:
10005:
10003:
10000:
9998:
9995:
9994:
9992:
9986:
9980:
9977:
9975:
9972:
9970:
9967:
9965:
9962:
9960:
9957:
9955:
9952:
9950:
9947:
9945:
9942:
9940:
9937:
9935:
9932:
9930:
9927:
9925:
9922:
9920:
9917:
9915:
9912:
9910:
9907:
9905:
9902:
9900:
9897:
9895:
9894:Arnold tongue
9892:
9890:
9887:
9886:
9883:
9877:
9874:
9872:
9869:
9867:
9864:
9862:
9859:
9857:
9854:
9852:
9849:
9847:
9844:
9842:
9839:
9838:
9836:
9830:
9827:
9823:
9819:
9812:
9807:
9805:
9800:
9798:
9793:
9792:
9789:
9781:
9780:
9775:
9773:
9772:0-7923-5564-4
9769:
9765:
9761:
9760:
9745:
9738:
9734:
9730:
9724:
9718:
9717:0-7923-5564-4
9714:
9708:
9706:
9696:
9687:
9678:
9671:
9667:
9664:
9659:
9652:
9646:
9639:
9634:
9630:
9619:
9615:
9612:
9610:
9607:
9605:
9602:
9601:
9581:
9576:
9568:
9565:
9562:
9556:
9539:
9538:
9537:
9523:
9500:
9485:
9482:
9477:
9474:
9468:
9458:
9455:
9452:
9449:
9444:
9441:
9438:
9435:
9431:
9423:
9422:
9421:
9407:
9404:
9381:
9378:
9375:
9369:
9366:
9341:
9338:
9318:
9315:
9312:
9307:
9304:
9299:
9293:
9290:
9287:
9281:
9278:
9258:
9255:
9232:
9226:
9223:
9220:
9214:
9211:
9208:
9200:
9197:
9184:
9181:
9177:
9169:
9168:
9153:
9148:
9144:
9137:
9134:
9131:
9125:
9122:
9119:
9116:
9111:
9103:
9086:
9085:
9084:
9067:
9064:
9031:
9001:
8998:
8978:
8955:
8948:
8945:
8942:
8938:
8932:
8928:
8921:
8918:
8905:
8901:
8894:
8877:
8867:
8864:
8860:
8849:
8839:
8836:
8832:
8829:
8826:
8823:
8818:
8814:
8806:
8802:
8799:
8792:
8791:
8790:
8776:
8756:
8748:
8729:
8722:
8719:
8716:
8712:
8706:
8702:
8698:
8695:
8692:
8688:
8684:
8677:
8674:
8664:
8661:
8657:
8646:
8643:
8633:
8630:
8626:
8623:
8620:
8617:
8612:
8608:
8600:
8596:
8593:
8586:
8585:
8584:
8570:
8550:
8527:
8522:
8519:
8516:
8512:
8506:
8502:
8494:
8491:
8481:
8478:
8474:
8470:
8467:
8461:
8455:
8448:
8447:
8446:
8432:
8429:
8421:
8411:
8409:
8408:modular forms
8405:
8401:
8397:
8393:
8392:modular group
8389:
8384:
8367:
8362:
8359:
8356:
8343:
8340:
8335:
8332:
8329:
8315:
8299:
8298:
8297:
8295:
8294:triangle wave
8276:
8270:
8244:
8239:
8235:
8228:
8225:
8222:
8219:
8210:
8205:
8201:
8190:
8187:
8184:
8180:
8176:
8170:
8162:
8159:
8156:
8140:
8139:
8138:
8136:
8132:
8128:
8123:
8110:
8107:
8102:
8099:
8094:
8089:
8086:
8081:
8078:
8058:
8055:
8052:
8032:
8027:
8023:
8016:
8013:
8010:
8007:
7996:
7980:
7972:
7969:
7944:
7941:
7938:
7934:
7930:
7927:
7922:
7919:
7916:
7912:
7906:
7890:
7889:
7888:
7871:
7866:
7862:
7855:
7852:
7849:
7846:
7840:
7837:
7834:
7831:
7826:
7822:
7811:
7808:
7805:
7801:
7797:
7791:
7783:
7780:
7777:
7773:
7765:
7764:
7763:
7749:
7746:
7738:
7708:
7705:
7697:
7696:complex plane
7693:
7689:
7688:Haar wavelets
7685:
7680:
7667:
7664:
7658:
7650:
7646:
7619:
7611:
7607:
7601:
7598:
7594:
7590:
7586:
7581:
7577:
7574:
7571:
7565:
7558:
7554:
7548:
7545:
7540:
7536:
7531:
7528:
7523:
7516:
7512:
7506:
7503:
7494:
7493:
7492:
7490:
7472:
7468:
7442:
7438:
7432:
7429:
7425:
7421:
7416:
7412:
7406:
7390:
7389:
7388:
7385:
7383:
7379:
7375:
7335:
7320:
7302:
7279:
7266:
7263:
7257:
7251:
7231:
7228:
7222:
7214:
7197:
7193:
7189:
7185:
7181:
7160:
7157:
7153:
7149:
7146:
7143:
7140:
7135:
7119:
7118:
7117:
7088:
7085:
7065:
7028:
6999:
6979:
6944:
6935:
6927:
6924:
6920:
6913:
6910:
6904:
6896:
6890:
6887:
6883:
6879:
6876:
6872:
6846:
6843:
6839:
6835:
6832:
6825:
6824:
6823:
6822:
6806:
6786:
6768:
6765:
6703:
6689:
6684:
6680:
6655:
6649:
6646:
6643:
6635:
6632:
6629:
6618:
6614:
6610:
6607:
6601:
6596:
6593:
6590:
6586:
6580:
6576:
6571:
6560:
6557:
6554:
6550:
6546:
6543:
6534:
6531:
6526:
6522:
6518:
6513:
6509:
6505:
6500:
6496:
6485:
6484:
6483:
6480:
6463:
6459:
6455:
6452:
6449:
6445:
6441:
6435:
6429:
6425:
6421:
6418:
6415:
6389:
6386:
6383:
6380:
6377:
6374:
6371:
6345:
6342:
6339:
6335:
6331:
6305:
6302:
6299:
6296:
6293:
6267:
6263:
6259:
6256:
6253:
6243:
6224:
6221:
6218:
6215:
6212:
6186:
6179:
6176:
6173:
6169:
6163:
6159:
6146:
6143:
6140:
6136:
6132:
6129:
6120:
6117:
6112:
6108:
6104:
6099:
6095:
6091:
6086:
6082:
6071:
6070:
6069:
6055:
6051:
6047:
6044:
6041:
6021:
6001:
5994:if there are
5976:
5968:
5965:
5962:
5954:
5950:
5946:
5940:
5937:
5934:
5931:
5926:
5922:
5918:
5915:
5912:
5909:
5906:
5903:
5895:
5891:
5883:
5882:
5881:
5867:
5864:
5861:
5858:
5855:
5846:
5830:
5827:
5822:
5818:
5814:
5809:
5805:
5784:
5758:
5755:
5752:
5749:
5744:
5740:
5736:
5733:
5730:
5727:
5724:
5721:
5718:
5713:
5710:
5707:
5703:
5699:
5694:
5690:
5686:
5683:
5680:
5677:
5674:
5671:
5668:
5665:
5662:
5659:
5649:
5648:
5647:
5645:
5644:cylinder sets
5641:
5640:sigma algebra
5638:, that is, a
5637:
5633:
5605:
5602:
5599:
5593:
5581:
5579:
5575:
5571:
5567:
5539:
5536:
5533:
5527:
5517:
5498:
5495:
5490:
5486:
5482:
5477:
5473:
5466:
5460:
5457:
5452:
5448:
5444:
5439:
5435:
5431:
5426:
5422:
5415:
5407:
5403:
5399:
5398:unit interval
5381:
5373:
5370:
5367:
5361:
5355:
5335:
5321:
5318:
5286:
5283:
5280:
5274:
5264:
5254:
5252:
5248:
5244:
5240:
5222:
5204:
5202:
5198:
5180:
5151:
5148:
5145:
5140:
5111:
5091:
5088:
5082:
5076:
5068:
5064:
5059:
5045:
5025:
5022:
5019:
4996:
4988:
4976:
4973:
4967:
4964:
4956:
4924:
4916:
4904:
4898:
4890:
4878:
4872:
4869:
4866:
4858:
4841:
4823:
4790:
4785:
4781:
4778:
4775:
4769:
4764:
4759:
4756:
4751:
4747:
4742:
4739:
4734:
4729:
4724:
4721:
4716:
4710:
4703:
4699:
4694:
4681:
4673:
4672:
4671:
4629:
4624:
4620:
4617:
4614:
4608:
4603:
4598:
4595:
4590:
4586:
4581:
4578:
4573:
4568:
4563:
4560:
4549:
4543:
4536:
4535:
4534:
4520:
4516:
4509:
4506:
4503:
4497:
4494:
4474:
4470:
4466:
4463:
4460:
4437:
4429:
4426:
4422:
4401:
4398:
4392:
4380:
4359:
4351:
4347:
4317:
4305:
4291:
4285:
4274:
4266:
4263:
4259:
4255:
4252:
4248:
4238:
4232:
4225:
4224:
4223:
4206:
4198:
4195:
4191:
4187:
4184:
4164:
4141:
4135:
4129:
4098:
4095:
4092:
4086:
4083:
4074:
4065:
4063:
4058:
4054:
4050:
4045:
4043:
4040:
4036:
4031:
4029:
4005:
4002:
3998:
3994:
3991:
3987:
3982:
3977:
3974:
3971:
3967:
3956:
3953:
3950:
3946:
3942:
3939:
3934:
3931:
3928:
3925:
3921:
3915:
3912:
3909:
3905:
3894:
3891:
3888:
3884:
3876:
3875:
3874:
3855:
3852:
3849:
3846:
3842:
3836:
3833:
3830:
3826:
3820:
3817:
3814:
3809:
3806:
3803:
3799:
3795:
3792:
3785:
3784:
3783:
3769:
3747:
3744:
3741:
3738:
3735:
3731:
3727:
3724:
3721:
3716:
3713:
3710:
3706:
3702:
3697:
3694:
3691:
3687:
3683:
3678:
3674:
3651:
3648:
3645:
3641:
3637:
3634:
3631:
3626:
3622:
3618:
3613:
3609:
3605:
3600:
3596:
3586:
3570:
3567:
3564:
3560:
3556:
3553:
3550:
3545:
3541:
3537:
3532:
3528:
3524:
3519:
3515:
3491:
3488:
3485:
3482:
3479:
3476:
3473:
3470:
3467:
3464:
3459:
3456:
3453:
3449:
3445:
3442:
3439:
3434:
3430:
3426:
3421:
3417:
3413:
3408:
3404:
3400:
3397:
3394:
3391:
3388:
3385:
3382:
3379:
3376:
3373:
3370:
3365:
3362:
3359:
3355:
3351:
3348:
3345:
3340:
3336:
3332:
3327:
3323:
3319:
3314:
3310:
3302:
3301:
3300:
3286:
3264:
3261:
3258:
3254:
3250:
3247:
3244:
3239:
3235:
3231:
3226:
3222:
3218:
3213:
3209:
3200:
3181:
3178:
3175:
3172:
3169:
3162:
3161:
3160:
3158:
3154:
3131:
3128:
3125:
3121:
3115:
3111:
3098:
3095:
3092:
3088:
3084:
3081:
3074:
3073:
3072:
3071:
3068:That is, the
3066:
3040:
3037:
3034:
3028:
3018:
2999:
2996:
2991:
2987:
2983:
2978:
2974:
2967:
2961:
2958:
2953:
2949:
2945:
2940:
2936:
2932:
2927:
2923:
2916:
2908:
2890:
2882:
2879:
2876:
2870:
2864:
2850:
2848:
2828:
2825:
2817:
2814:
2804:
2801:
2791:
2788:
2778:
2775:
2765:
2762:
2752:
2749:
2739:
2736:
2727:
2726:
2725:
2722:
2720:
2716:
2712:
2708:
2704:
2700:
2696:
2692:
2685:
2682:expansion of
2681:
2674:
2670:
2669:forward orbit
2663:
2660:the image of
2659:
2652:
2647:
2644:
2640:
2635:
2613:
2610:
2605:
2597:
2594:
2588:
2581:
2580:
2579:
2577:
2573:
2554:
2546:
2542:
2538:
2535:
2529:
2524:
2520:
2516:
2511:
2507:
2499:
2498:
2497:
2495:
2491:
2470:
2466:
2462:
2459:
2451:
2447:
2443:
2440:
2435:
2432:
2429:
2425:
2416:
2412:
2408:
2389:
2386:
2381:
2377:
2373:
2368:
2365:
2361:
2357:
2354:
2351:
2348:
2345:
2342:
2339:
2334:
2331:
2327:
2323:
2320:
2317:
2314:
2311:
2306:
2303:
2299:
2295:
2290:
2286:
2282:
2279:
2276:
2271:
2267:
2263:
2258:
2253:
2249:
2241:
2240:
2239:
2225:
2213:
2207:
2201:
2193:
2190:
2186:
2182:
2176:
2168:
2164:
2140:
2137:
2134:
2131:
2126:
2123:
2119:
2115:
2110:
2106:
2098:
2097:
2096:
2079:
2076:
2073:
2070:
2067:
2061:
2055:
2048:
2047:
2046:
2022:
2018:
2014:
2011:
2006:
2002:
1977:
1973:
1969:
1966:
1959:
1955:
1951:
1948:
1943:
1939:
1914:
1910:
1903:
1898:
1890:
1886:
1877:
1873:
1869:
1864:
1861:
1858:
1854:
1846:
1845:
1844:
1842:
1838:
1828:
1823:
1816:
1809:
1804:
1800:
1795:
1792:
1788:
1785: â
1784:
1780:
1776:
1772:
1768:
1758:
1756:
1728:
1725:
1722:
1711:
1688:
1685:
1682:
1678:
1672:
1668:
1655:
1652:
1649:
1645:
1641:
1638:
1631:
1630:
1629:
1621:
1608:
1602:
1594:
1590:
1586:
1581:
1577:
1550:
1547:
1541:
1533:
1529:
1525:
1519:
1511:
1507:
1503:
1497:
1491:
1488:
1485:
1475:
1474:
1473:
1470:
1468:
1452:
1441:
1437:
1414:
1409:
1401:
1398:
1395:
1389:
1383:
1363:
1357:
1354:
1349:
1345:
1341:
1336:
1332:
1325:
1319:
1316:
1311:
1307:
1303:
1298:
1294:
1290:
1285:
1281:
1274:
1254:
1246:
1230:
1210:
1207:
1204:
1201:
1198:
1191:
1190:unit interval
1187:
1171:
1148:
1141:
1138:
1135:
1131:
1125:
1121:
1108:
1105:
1102:
1098:
1094:
1091:
1084:
1083:
1082:
1065:
1062:
1057:
1053:
1049:
1044:
1040:
1010:
1007:
1002:
998:
994:
989:
985:
978:
972:
969:
964:
960:
956:
951:
947:
943:
938:
934:
927:
920:
919:
918:
916:
911:
909:
881:
878:
875:
840:
837:
834:
828:
805:
785:
753:
750:
747:
741:
730:
726:
718:
700:
692:
689:
683:
675:
670:
661:
659:
655:
651:
633:
625:
622:
619:
613:
605:
601:
593:, defined as
592:
588:
583:
581:
577:
576:bit shift map
551:
548:
545:
542:
537:
534:
527:
524:
521:
518:
509:
506:
501:
498:
495:
492:
487:
484:
478:
473:
467:
461:
454:
453:
452:
451:
447:
425:
412:
408:
404:
398:
393:
390:
387:
383:
376:
373:
370:
367:
364:for all
355:
341:
338:
333:
329:
321:
320:
319:
302:
299:
296:
285:
259:
256:
253:
224:
221:
216:
212:
208:
203:
199:
195:
190:
186:
176:
169:
145:
142:
139:
127:
124:
121:
115:
112:
105:
104:
103:
101:
97:
93:
89:
85:
84:Bernoulli map
81:
79:
73:
72:bit shift map
69:
65:
57:
54:for all
52:
48:
45: =
44:
40:
33:
30: =
29:
25:
21:
10616:Chaotic maps
10537:Mary Tsingou
10502:David Ruelle
10497:Otto Rössler
10442:Michel HĂ©non
10412:Leon O. Chua
10369:Tilt-A-Whirl
10339:FPUT problem
10224:Standard map
10219:Logistic map
10151:
10044:
9818:Chaos theory
9777:
9763:
9744:
9736:
9732:
9728:
9723:
9695:
9686:
9677:
9658:
9650:
9645:
9633:
9617:
9515:
9247:
8970:
8744:
8542:
8417:
8385:
8382:
8262:
8124:
7961:
7886:
7681:
7637:
7459:
7386:
7285:
7183:
7177:
6863:
6709:
6671:
6481:
6240:maps to the
6201:
5993:
5776:
5582:
5516:Cantor space
5263:Cantor space
5260:
5205:
5195:, it is the
5060:
4806:
4645:
4453:, these are
4343:
4222:and write
4075:
4071:
4056:
4049:measure zero
4046:
4032:
4027:
4025:
3872:
3587:
3506:
3199:0.999... = 1
3196:
3150:
3070:homomorphism
3067:
3017:Cantor space
2856:
2847:chaotic maps
2843:
2723:
2718:
2714:
2710:
2706:
2702:
2698:
2694:
2690:
2683:
2672:
2661:
2650:
2648:
2636:
2633:
2569:
2493:
2414:
2411:logistic map
2406:
2404:
2155:
2094:
2044:
1834:
1821:
1814:
1807:
1802:
1798:
1796:
1790:
1786:
1782:
1778:
1774:
1770:
1766:
1764:
1707:
1627:
1568:
1471:
1245:homomorphism
1163:
1028:
912:
725:homomorphism
722:
673:
650:Alfréd Rényi
584:
575:
573:
443:
242:
92:sawtooth map
91:
88:doubling map
87:
83:
77:
75:
71:
67:
63:
61:
55:
50:
46:
42:
31:
27:
23:
10522:Nina Snaith
10512:Yakov Sinai
10397:Rufus Bowen
10147:Duffing map
10132:Baker's map
10057:Theoretical
9969:SRB measure
9876:Phase space
9846:Bifurcation
9397:related to
8420:Ising model
7192:eigenvector
6864:defined by
6821:pushforward
5636:Borel space
5257:Borel space
5251:polynomials
4055:orbits are
1467:Baker's map
1186:real number
917:, given by
908:Baker's map
448:map of the
26:plot where
10580:Complexity
10477:Edward Ott
10324:Convection
10249:Continuous
9924:Ergodicity
9756:References
7684:Haar basis
7638:Note that
7460:where the
7376:, and the
6819:induces a
6799:The shift
6014:heads and
5777:where the
5578:continuity
5067:eigenvalue
4350:derivative
4053:Almost all
3762:of length
3299:, one has
3279:of length
3153:surjection
2643:almost all
2639:irrational
1755:Cantor set
1708:gives the
1243:induces a
1223:The shift
658:Bill Parry
68:dyadic map
10492:Mary Rees
10452:Bryna Kra
10385:theorists
10194:Ikeda map
10184:HĂ©non map
10174:Gauss map
9856:Limit set
9841:Attractor
9478:−
9469:π
9459:
9442:β
9436:−
9405:β
9370:∈
9342:∈
9331:for some
9316:π
9305:π
9291:β
9282:
9256:β
9224:β
9215:
9198:β
9182:−
9135:β
9126:
9065:β
8999:β
8939:σ
8929:σ
8919:β
8906:−
8868:∈
8861:∏
8840:∈
8827:±
8815:σ
8807:∑
8713:σ
8703:σ
8696:β
8693:−
8665:∈
8658:∏
8634:∈
8621:±
8609:σ
8601:∑
8571:β
8513:σ
8503:σ
8482:∈
8475:∑
8462:σ
8196:∞
8181:∑
8111:…
7973:∈
7935:ψ
7913:ψ
7841:π
7835:
7817:∞
7802:∑
7774:ψ
7709:∈
7599:−
7430:−
7252:ρ
7232:ρ
7223:ρ
7198:. Again,
7158:−
7150:∘
7099:→
7058:→
6972:→
6925:−
6888:−
6880:∘
6844:−
6836:∘
6779:→
6681:μ
6633:−
6611:−
6566:∞
6551:∑
6541:↦
6535:…
6436:∪
6390:…
6384:∗
6372:∗
6306:…
6300:∗
6225:⋯
6219:∗
6152:∞
6137:∑
6127:↦
6121:…
5966:−
5941:…
5935:∗
5916:∗
5910:…
5904:∗
5892:μ
5831:…
5785:∗
5759:…
5753:∗
5734:∗
5728:…
5722:∗
5684:∗
5678:…
5672:∗
5666:∗
5660:∗
5591:Ω
5574:open sets
5525:Ω
5499:…
5461:…
5328:→
5325:Ω
5319:ρ
5272:Ω
5152:ρ
5146:ρ
5077:ρ
4765:ρ
4730:ρ
4700:ρ
4604:ρ
4569:ρ
4556:↦
4544:ρ
4427:−
4385:′
4310:′
4286:ρ
4264:−
4249:∑
4245:↦
4233:ρ
4196:−
4136:ρ
4133:↦
4105:→
4084:ρ
4042:manifolds
4035:geodesics
4003:−
3995:−
3972:−
3962:∞
3947:∑
3932:−
3926:−
3900:∞
3885:∑
3853:−
3847:−
3818:−
3800:∑
3745:−
3725:…
3649:−
3635:…
3568:−
3554:…
3492:…
3457:−
3443:…
3398:⋯
3363:−
3349:…
3262:−
3248:…
3182:…
3173:⋯
3170:0.1000000
3104:∞
3089:∑
3026:Ω
3000:…
2962:…
2907:bit shift
2826:⋯
2823:↦
2810:↦
2797:↦
2784:↦
2771:↦
2758:↦
2745:↦
2611:π
2598:θ
2592:↦
2589:θ
2539:π
2530:
2490:bit shift
2463:−
2387:∘
2374:∘
2366:−
2352:∘
2346:∘
2343:⋯
2340:∘
2332:−
2324:∘
2318:∘
2312:∘
2304:−
2283:∘
2280:⋯
2277:∘
2191:−
2156:That is,
2138:∘
2132:∘
2124:−
2077:π
2074:
2012:≥
1970:−
1949:≤
1661:∞
1646:∑
1569:That is,
1551:…
1435:Ω
1358:…
1320:…
1208:≤
1202:≤
1114:∞
1099:∑
1066:…
1011:…
973:…
826:Ω
739:Ω
687:↦
660:in 1960.
623:β
606:β
574:The name
543:≤
525:−
496:≤
371:≥
284:sequences
268:∞
225:…
180:↦
154:∞
134:→
94:) is the
10610:Category
10568:articles
10310:Physical
10229:Tent map
10119:Discrete
10059:branches
9989:Theorems
9825:Concepts
9782:, (2004)
9666:Archived
9598:See also
8543:Letting
8388:groupoid
7698:. Given
7487:are the
7282:Spectrum
6242:interval
5566:topology
5104:for all
5063:spectrum
4807:The map
3179:0.011111
3155:: every
2658:rational
2413:. The
2095:so that
1841:tent map
672:The map
39:rational
10566:Related
10374:Weather
10312:systems
10105:Chaotic
9851:Fractal
8789:spins,
8390:of the
8292:is the
7995:integer
7182:or the
5400:of the
4372:and so
4039:compact
1188:in the
727:on the
243:(where
98:(i.e.,
96:mapping
10472:Hee Oh
10107:maps (
9954:Mixing
9770:
9715:
8263:where
5568:, the
5065:? One
4076:Write
2680:binary
1999:
1996:
1936:
1933:
1029:where
731:. Let
580:binary
380:
10383:Chaos
10162:outer
9866:Orbit
9651:Chaos
9625:Notes
8351:blanc
8324:blanc
8151:blanc
7887:obey
7244:when
4838:is a
4028:every
2576:turns
1184:is a
587:chaos
286:from
10109:list
9833:Core
9768:ISBN
9713:ISBN
9212:cosh
9123:cosh
9047:and
8991:and
8406:and
8071:and
7962:for
7747:<
6283:and
5865:<
5859:<
4940:and
4487:and
2909:map
2641:(as
798:and
549:<
502:<
62:The
41:and
9573:mod
9456:tan
8422:of
7832:exp
5378:mod
4352:of
4057:not
4037:on
2887:mod
2671:of
2656:is
2649:If
2602:mod
2521:sin
2496:by
2071:sin
1406:mod
697:mod
630:mod
422:mod
90:or
10612::
9737:25
9735:,
9704:^
9279:Im
8410:.
7668:1.
7267:1.
5580:.
5058:.
4521:2.
3065:.
2849:.
2753:12
2750:11
2740:24
2737:11
1757:.
1469:.
1211:1.
552:1.
102:)
86:,
82:,
74:,
70:,
24:xy
10111:)
9810:e
9803:t
9796:v
9729:r
9618:n
9582:.
9577:1
9569:t
9566:2
9563:=
9560:]
9557:t
9554:[
9549:R
9524:t
9501:,
9496:)
9491:)
9486:2
9483:1
9475:t
9472:(
9464:(
9453:i
9450:=
9445:g
9439:2
9432:e
9408:g
9385:)
9382:1
9379:,
9376:0
9373:[
9367:t
9346:Z
9339:n
9319:n
9313:+
9308:2
9300:=
9297:]
9294:g
9288:2
9285:[
9259:g
9233:.
9230:)
9227:g
9221:2
9218:(
9209:=
9204:]
9201:g
9195:[
9190:R
9185:2
9178:e
9154:,
9149:4
9145:C
9141:)
9138:g
9132:2
9129:(
9120:4
9117:=
9112:2
9108:]
9104:C
9101:[
9096:R
9071:]
9068:g
9062:[
9057:R
9035:]
9032:C
9029:[
9024:R
9002:g
8979:C
8956:,
8949:1
8946:+
8943:i
8933:i
8925:]
8922:g
8916:[
8911:R
8902:e
8898:]
8895:C
8892:[
8887:R
8878:N
8873:Z
8865:i
8855:}
8850:N
8845:Z
8837:i
8833:,
8830:1
8824:=
8819:i
8811:{
8803:=
8800:Z
8777:N
8757:Z
8730:.
8723:1
8720:+
8717:i
8707:i
8699:g
8689:e
8685:C
8678:N
8675:2
8670:Z
8662:i
8652:}
8647:N
8644:2
8639:Z
8631:i
8627:,
8624:1
8618:=
8613:i
8605:{
8597:=
8594:Z
8551:C
8528:.
8523:1
8520:+
8517:i
8507:i
8495:N
8492:2
8487:Z
8479:i
8471:g
8468:=
8465:)
8459:(
8456:H
8433:N
8430:2
8368:.
8363:k
8360:,
8357:w
8344:w
8341:=
8336:k
8333:,
8330:w
8316:T
8310:L
8280:)
8277:x
8274:(
8271:s
8248:)
8245:x
8240:n
8236:2
8232:)
8229:1
8226:+
8223:k
8220:2
8217:(
8214:(
8211:s
8206:n
8202:w
8191:0
8188:=
8185:n
8177:=
8174:)
8171:x
8168:(
8163:k
8160:,
8157:w
8108:,
8103:4
8100:1
8095:,
8090:2
8087:1
8082:=
8079:z
8059:0
8056:=
8053:k
8033:.
8028:n
8024:2
8020:)
8017:1
8014:+
8011:k
8008:2
8005:(
7981:.
7977:Z
7970:k
7945:k
7942:,
7939:z
7931:z
7928:=
7923:k
7920:,
7917:z
7907:T
7901:L
7872:x
7867:n
7863:2
7859:)
7856:1
7853:+
7850:k
7847:2
7844:(
7838:i
7827:n
7823:z
7812:1
7809:=
7806:n
7798:=
7795:)
7792:x
7789:(
7784:k
7781:,
7778:z
7750:1
7743:|
7739:z
7735:|
7713:C
7706:z
7665:=
7662:)
7659:x
7656:(
7651:0
7647:B
7623:)
7620:y
7617:(
7612:n
7608:B
7602:n
7595:2
7591:=
7587:)
7582:2
7578:1
7575:+
7572:y
7566:(
7559:n
7555:B
7549:2
7546:1
7541:+
7537:)
7532:2
7529:y
7524:(
7517:n
7513:B
7507:2
7504:1
7473:n
7469:B
7443:n
7439:B
7433:n
7426:2
7422:=
7417:n
7413:B
7407:T
7401:L
7358:F
7336:.
7331:F
7303:T
7297:L
7264:=
7261:)
7258:x
7255:(
7229:=
7226:)
7220:(
7215:T
7209:L
7161:1
7154:T
7147:f
7144:=
7141:f
7136:T
7130:L
7103:R
7094:B
7089::
7086:f
7066:.
7062:R
7053:B
7029:T
7023:L
7000:T
6980:.
6976:R
6967:B
6945:.
6942:)
6939:)
6936:x
6933:(
6928:1
6921:T
6917:(
6914:f
6911:=
6908:)
6905:x
6902:(
6897:)
6891:1
6884:T
6877:f
6873:(
6847:1
6840:T
6833:f
6807:T
6787:.
6783:R
6774:B
6769::
6766:f
6744:F
6720:B
6690:.
6685:p
6656:]
6650:1
6647:+
6644:n
6640:)
6636:p
6630:1
6627:(
6624:)
6619:n
6615:b
6608:1
6605:(
6602:+
6597:1
6594:+
6591:n
6587:p
6581:n
6577:b
6572:[
6561:0
6558:=
6555:n
6547:=
6544:x
6538:)
6532:,
6527:2
6523:b
6519:,
6514:1
6510:b
6506:,
6501:0
6497:b
6493:(
6467:]
6464:4
6460:/
6456:3
6453:,
6450:2
6446:/
6442:1
6439:[
6433:]
6430:4
6426:/
6422:1
6419:,
6416:0
6413:[
6393:)
6387:,
6381:,
6378:0
6375:,
6369:(
6349:]
6346:1
6343:,
6340:2
6336:/
6332:1
6329:[
6309:)
6303:,
6297:,
6294:1
6291:(
6271:]
6268:2
6264:/
6260:1
6257:,
6254:0
6251:[
6228:)
6222:,
6216:,
6213:0
6210:(
6187:.
6180:1
6177:+
6174:n
6170:2
6164:n
6160:b
6147:0
6144:=
6141:n
6133:=
6130:x
6124:)
6118:,
6113:2
6109:b
6105:,
6100:1
6096:b
6092:,
6087:0
6083:b
6079:(
6056:2
6052:/
6048:1
6045:=
6042:p
6022:m
6002:n
5977:m
5973:)
5969:p
5963:1
5960:(
5955:n
5951:p
5947:=
5944:)
5938:,
5932:,
5927:k
5923:b
5919:,
5913:,
5907:,
5901:(
5896:p
5868:1
5862:p
5856:0
5828:,
5823:m
5819:b
5815:,
5810:k
5806:b
5762:)
5756:,
5750:,
5745:m
5741:b
5737:,
5731:,
5725:,
5719:,
5714:1
5711:+
5708:k
5704:b
5700:,
5695:k
5691:b
5687:,
5681:,
5675:,
5669:,
5663:,
5657:(
5615:N
5610:}
5606:1
5603:,
5600:0
5597:{
5594:=
5549:N
5544:}
5540:1
5537:,
5534:0
5531:{
5528:=
5502:)
5496:,
5491:2
5487:b
5483:,
5478:1
5474:b
5470:(
5467:=
5464:)
5458:,
5453:2
5449:b
5445:,
5440:1
5436:b
5432:,
5427:0
5423:b
5419:(
5416:T
5382:1
5374:x
5371:2
5368:=
5365:)
5362:x
5359:(
5356:T
5336:.
5332:R
5322::
5296:N
5291:}
5287:1
5284:,
5281:0
5278:{
5275:=
5223:T
5217:L
5181:T
5175:L
5149:=
5141:T
5135:L
5112:x
5092:1
5089:=
5086:)
5083:x
5080:(
5046:a
5026:g
5023:,
5020:f
5000:)
4997:f
4994:(
4989:T
4983:L
4977:a
4974:=
4971:)
4968:f
4965:a
4962:(
4957:T
4951:L
4928:)
4925:g
4922:(
4917:T
4911:L
4905:+
4902:)
4899:f
4896:(
4891:T
4885:L
4879:=
4876:)
4873:g
4870:+
4867:f
4864:(
4859:T
4853:L
4824:T
4818:L
4791:)
4786:2
4782:1
4779:+
4776:x
4770:(
4760:2
4757:1
4752:+
4748:)
4743:2
4740:x
4735:(
4725:2
4722:1
4717:=
4714:)
4711:x
4708:(
4704:]
4695:T
4689:L
4682:[
4656:L
4630:)
4625:2
4621:1
4618:+
4615:x
4609:(
4599:2
4596:1
4591:+
4587:)
4582:2
4579:x
4574:(
4564:2
4561:1
4553:)
4550:x
4547:(
4517:/
4513:)
4510:1
4507:+
4504:x
4501:(
4498:=
4495:y
4475:2
4471:/
4467:x
4464:=
4461:y
4441:)
4438:x
4435:(
4430:1
4423:T
4402:2
4399:=
4396:)
4393:y
4390:(
4381:T
4360:T
4325:|
4321:)
4318:y
4315:(
4306:T
4301:|
4295:)
4292:y
4289:(
4278:)
4275:x
4272:(
4267:1
4260:T
4256:=
4253:y
4242:)
4239:x
4236:(
4210:)
4207:x
4204:(
4199:1
4192:T
4188:=
4185:y
4165:T
4145:)
4142:x
4139:(
4130:x
4109:R
4102:]
4099:1
4096:,
4093:0
4090:[
4087::
4006:m
3999:2
3992:1
3988:y
3983:=
3978:m
3975:j
3968:2
3957:0
3954:=
3951:j
3943:y
3940:=
3935:1
3929:j
3922:2
3916:j
3913:+
3910:k
3906:b
3895:0
3892:=
3889:j
3856:1
3850:j
3843:2
3837:j
3834:+
3831:k
3827:b
3821:1
3815:m
3810:0
3807:=
3804:j
3796:=
3793:y
3770:m
3748:1
3742:m
3739:+
3736:k
3732:b
3728:,
3722:,
3717:2
3714:+
3711:k
3707:b
3703:,
3698:1
3695:+
3692:k
3688:b
3684:,
3679:k
3675:b
3652:1
3646:k
3642:b
3638:,
3632:,
3627:2
3623:b
3619:,
3614:1
3610:b
3606:,
3601:0
3597:b
3571:1
3565:k
3561:b
3557:,
3551:,
3546:2
3542:b
3538:,
3533:1
3529:b
3525:,
3520:0
3516:b
3489:,
3486:1
3483:,
3480:1
3477:,
3474:1
3471:,
3468:0
3465:,
3460:1
3454:k
3450:b
3446:,
3440:,
3435:2
3431:b
3427:,
3422:1
3418:b
3414:,
3409:0
3405:b
3401:=
3395:,
3392:0
3389:,
3386:0
3383:,
3380:0
3377:,
3374:1
3371:,
3366:1
3360:k
3356:b
3352:,
3346:,
3341:2
3337:b
3333:,
3328:1
3324:b
3320:,
3315:0
3311:b
3287:k
3265:1
3259:k
3255:b
3251:,
3245:,
3240:2
3236:b
3232:,
3227:1
3223:b
3219:,
3214:0
3210:b
3176:=
3132:1
3129:+
3126:n
3122:2
3116:n
3112:b
3099:0
3096:=
3093:n
3085:=
3082:x
3050:N
3045:}
3041:1
3038:,
3035:0
3032:{
3029:=
3003:)
2997:,
2992:2
2988:b
2984:,
2979:1
2975:b
2971:(
2968:=
2965:)
2959:,
2954:2
2950:b
2946:,
2941:1
2937:b
2933:,
2928:0
2924:b
2920:(
2917:T
2891:1
2883:x
2880:2
2877:=
2874:)
2871:x
2868:(
2865:T
2829:,
2818:3
2815:1
2805:3
2802:2
2792:3
2789:1
2779:3
2776:2
2766:6
2763:5
2719:q
2715:k
2711:q
2707:q
2703:k
2699:k
2695:k
2691:k
2687:0
2684:x
2676:0
2673:x
2665:0
2662:x
2654:0
2651:x
2614:.
2606:2
2595:2
2555:.
2552:)
2547:n
2543:x
2536:2
2533:(
2525:2
2517:=
2512:n
2508:z
2494:x
2476:)
2471:n
2467:z
2460:1
2457:(
2452:n
2448:z
2444:4
2441:=
2436:1
2433:+
2430:n
2426:z
2415:r
2407:r
2390:S
2382:n
2378:T
2369:1
2362:S
2358:=
2355:S
2349:T
2335:1
2328:S
2321:S
2315:T
2307:1
2300:S
2296:=
2291:1
2287:f
2272:1
2268:f
2264:=
2259:n
2254:1
2250:f
2226:.
2223:)
2220:)
2217:)
2214:x
2211:(
2208:S
2205:(
2202:T
2199:(
2194:1
2187:S
2183:=
2180:)
2177:x
2174:(
2169:1
2165:f
2141:S
2135:T
2127:1
2120:S
2116:=
2111:1
2107:f
2080:x
2068:=
2065:)
2062:x
2059:(
2056:S
2023:2
2019:/
2015:1
2007:n
2003:x
1992:r
1989:o
1986:f
1978:n
1974:x
1967:1
1960:2
1956:/
1952:1
1944:n
1940:x
1929:r
1926:o
1923:f
1915:n
1911:x
1904:{
1899:=
1896:)
1891:n
1887:x
1883:(
1878:1
1874:f
1870:=
1865:1
1862:+
1859:n
1855:x
1825:1
1822:x
1818:1
1815:x
1811:0
1808:x
1803:n
1799:x
1791:s
1787:m
1783:s
1779:s
1775:m
1771:m
1767:s
1738:N
1733:}
1729:T
1726:,
1723:H
1720:{
1689:1
1686:+
1683:n
1679:3
1673:n
1669:b
1656:0
1653:=
1650:n
1642:=
1639:y
1609:.
1606:)
1603:x
1600:(
1595:n
1591:T
1587:=
1582:n
1578:x
1554:)
1548:,
1545:)
1542:x
1539:(
1534:3
1530:T
1526:,
1523:)
1520:x
1517:(
1512:2
1508:T
1504:,
1501:)
1498:x
1495:(
1492:T
1489:,
1486:x
1483:(
1453:,
1447:Z
1442:2
1438:=
1415:.
1410:1
1402:x
1399:2
1396:=
1393:)
1390:x
1387:(
1384:T
1364:,
1361:)
1355:,
1350:2
1346:b
1342:,
1337:1
1333:b
1329:(
1326:=
1323:)
1317:,
1312:2
1308:b
1304:,
1299:1
1295:b
1291:,
1286:0
1282:b
1278:(
1275:T
1255:T
1231:T
1205:x
1199:0
1172:x
1149:.
1142:1
1139:+
1136:n
1132:2
1126:n
1122:b
1109:0
1106:=
1103:n
1095:=
1092:x
1069:)
1063:,
1058:1
1054:b
1050:,
1045:0
1041:b
1037:(
1014:)
1008:,
1003:2
999:b
995:,
990:1
986:b
982:(
979:=
976:)
970:,
965:2
961:b
957:,
952:1
948:b
944:,
939:0
935:b
931:(
928:T
891:Z
886:}
882:1
879:,
876:0
873:{
850:N
845:}
841:1
838:,
835:0
832:{
829:=
806:T
786:H
763:N
758:}
754:T
751:,
748:H
745:{
742:=
719:.
701:1
693:x
690:2
684:x
674:T
634:1
626:x
620:=
617:)
614:x
611:(
602:T
546:x
538:2
535:1
528:1
522:x
519:2
510:2
507:1
499:x
493:0
488:x
485:2
479:{
474:=
471:)
468:x
465:(
462:T
440:.
426:1
418:)
413:n
409:x
405:2
402:(
399:=
394:1
391:+
388:n
384:x
377:,
374:0
368:n
342:x
339:=
334:0
330:x
306:)
303:1
300:,
297:0
294:[
264:)
260:1
257:,
254:0
251:[
228:)
222:,
217:2
213:x
209:,
204:1
200:x
196:,
191:0
187:x
183:(
177:x
150:)
146:1
143:,
140:0
137:[
131:)
128:1
125:,
122:0
119:[
116::
113:T
78:x
76:2
58:.
56:n
51:n
47:x
43:y
35:0
32:x
28:x
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
