33:
8561:, has, however, within these few months come to his knowledge, in which the same is explained at a considerably earlier date. He, however, does not seem to have noticed the convenience of applying this idea to the inverse functions tan, etc., nor does he appear at all aware of the inverse calculus of functions to which it gives rise." Herschel adds, "The symmetry of this notation and above all the new and most extensive views it opens of the nature of analytical operations seem to authorize its universal adoption." §535.
1181:
4106:
2899:
1950:
2421:
2665:
3435:
4129:
envelope triangle represents the limiting null iterate, the sawtooth function serving as the starting point leading to the sine function. The dashed line is the negative first iterate, i.e. the inverse of sine (arcsin). (From the general pedagogy web-site. For the notation, see
7151:
3764:
6938:
6762:
2152:
3112:
1713:
7896:
3231:
8332:(1952) . "§472. The power of a logarithm / §473. Iterated logarithms / §533. John Herschel's notation for inverse functions / §535. Persistence of rival notations for inverse functions / §537. Powers of trigonometric functions".
2677:
7282:
321:
3526:
2177:
6228:
5191:
403:
6122:
5804:
5098:
3173:
5562:
2432:
5645:
5723:
6352:
716:
6958:
5475:
2952:
7996:
5012:
4117:), i.e., the sine's functional square root; the functional square root of that, the quarter-iterate (black) above it; and further fractional iterates up to the 1/64th. The functions below the (
6773:
6594:
4736:
7641:
4874:
4814:
169:
5940:
5869:
4612:
6492:
4926:
7729:
6418:
7395:
2904:
This can be carried on indefinitely, although inefficiently, as the latter terms become increasingly complicated. A more systematic procedure is outlined in the following section on
7569:
5245:
1957:
1149:. It can be visualized as the behavior of a point-cloud or dust-cloud under repeated iteration. The invariant measure is an eigenstate of the Ruelle-Frobenius-Perron operator or
6271:
6021:
5980:
4663:
4542:
74:. In this process, starting from some initial object, the result of applying a given function is fed again into the function as input, and this process is repeated.
4433:
8213:
450:
7744:
4492:
4396:
259:
4462:
1945:{\displaystyle f^{n}(x)=f^{n}(a)+(x-a)\left.{\frac {d}{dx}}f^{n}(x)\right|_{x=a}+{\frac {(x-a)^{2}}{2}}\left.{\frac {d^{2}}{dx^{2}}}f^{n}(x)\right|_{x=a}+\cdots }
1086:
Upon iteration, one may find that there are sets that shrink and converge towards a single point. In such a case, the point that is converged to is known as an
1532:
If a function is bijective (and so possesses an inverse function), then negative iterates correspond to function inverses and their compositions. For example,
326:
7199:
1145:
If one considers the evolution of a density distribution, rather than that of individual point dynamics, then the limiting behavior is given by the
8782:
8617:). As functions of the last type do not ordinarily present themselves, the danger of misinterpretation is very much less than in case of log
3430:{\displaystyle {\sqrt {2}}^{{\sqrt {2}}^{{\sqrt {2}}^{\cdots }}}=f^{n}(1)=2-(\ln 2)^{n}+{\frac {(\ln 2)^{n+1}((\ln 2)^{n}-1)}{4(\ln 2-1)}}-\cdots }
8699:
8274:
6128:
3844:. Thus, if one can solve for one iterated function system, one also has solutions for all topologically conjugate systems. For example, the
2894:{\displaystyle f^{n}(x)=x+{\frac {(x-a)^{2}}{2}}(nf''(a))+{\frac {(x-a)^{3}}{6}}\left({\frac {3}{2}}n(n-1)f''(a)^{2}+nf'''(a)\right)+\cdots }
5108:
6029:
8068:
5729:
1127:
5020:
3125:
5485:
5568:
3759:{\displaystyle f^{n}(x)=1+b^{n}(x-1)+{\frac {1}{2}}b^{n}(b^{n}-1)(x-1)^{2}+{\frac {1}{3!}}b^{n}(b^{n}-1)(b^{n}-2)(x-1)^{3}+\cdots ~,}
7905:
suffices to determine the entire flow, given this exponential realization which automatically provides the general solution to the
7146:{\displaystyle {\frac {\delta f^{N}(x)}{\delta f(y)}}=f'(f^{N-1}(x)){\frac {\delta f^{N-1}(x)}{\delta f(y)}}+\delta (f^{N-1}(x)-y)}
5655:
2416:{\displaystyle f^{n}(x)=a+(x-a)f'(a)^{n}+{\frac {(x-a)^{2}}{2}}(f''(a)f'(a)^{n-1})\left(1+f'(a)+\cdots +f'(a)^{n-1}\right)+\cdots }
6277:
8644:(xviii+367+1 pages including 1 addenda page) (NB. ISBN and link for reprint of 2nd edition by Cosimo, Inc., New York, USA, 2013.)
630:
1368:
s domain can be extended sufficiently, cf. picture. The roots chosen are normally the ones belonging to the orbit under study.
8347:
4356:-th iterate. The table below lists some that do. Note that all these expressions are valid even for non-integer and negative
5256:
1090:. Conversely, iteration may give the appearance of points diverging away from a single point; this would be the case for an
70:
another function with itself two or several times. The process of repeatedly applying the same function is called
8445:
7915:
6933:{\displaystyle \left\{b+1,\prod _{i=a}^{b}g(i)\right\}\equiv \left(\{i,x\}\rightarrow \{i+1,xg(i)\}\right)^{b-a+1}\{a,1\}}
6757:{\displaystyle \left\{b+1,\sum _{i=a}^{b}g(i)\right\}\equiv \left(\{i,x\}\rightarrow \{i+1,x+g(i)\}\right)^{b-a+1}\{a,0\}}
9335:
2660:{\displaystyle f^{n}(x)=a+(x-a)f'(a)^{n}+{\frac {(x-a)^{2}}{2}}(f''(a)f'(a)^{n-1}){\frac {f'(a)^{n}-1}{f'(a)-1}}+\cdots }
1649:
One of several methods of finding a series formula for fractional iteration, making use of a fixed point, is as follows.
4936:
9282:
4669:
17:
7580:
6421:
4822:
4744:
1688:
belonging to the reals. This, in some ways, is the most natural extra condition to place upon the fractional iterates.
84:
8833:
8809:
8766:
8665:
7186:
5880:
9137:
7738:, then, now written as a subscript, this amounts to Lie's celebrated exponential realization of a continuous group,
5812:
4548:
4184:
8979:
5248:
9340:
8430:
8262:
8204:
6433:
479:
6529:, respectively, further reduces them to the nonchaotic and chaotic logistic cases discussed prior to the table.
4884:
7652:
6360:
4145:) is equivalent to the algorithm of the preceding section, albeit, in practice, more powerful and systematic.
3107:{\displaystyle f^{n}(x)={\frac {D}{1-C}}+\left(x-{\frac {D}{1-C}}\right)C^{n}=C^{n}x+{\frac {1-C^{n}}{1-C}}D~,}
9345:
9330:
9176:
8339:
7301:
6541:
7162:
6561:
1156:
In general, because repeated iteration corresponds to a shift, the transfer operator, and its adjoint, the
7502:
5201:
1123:
1056:
6236:
5986:
4090:
no longer needs be integer or positive, and is a continuous "time" of evolution for the full orbit: the
9320:
8225:, printed by W. Bulmer and Co., Cleveland-Row, St. James's, sold by G. and W. Nicol, Pall-Mall: 8–26 .
5948:
4620:
4338:
1052:
4500:
4178:
1032:
9078:
8063:
8058:
8043:
3994:
1357:
does not denote a unique function, just as numbers have multiple algebraic roots. A trivial root of
1153:, corresponding to an eigenvalue of 1. Smaller eigenvalues correspond to unstable, decaying states.
754:
486:
for it, but without giving a specific reference to the work of Bürmann, which remains undiscovered.
32:
9221:
Curtright, T. L.; Zachos, C. K. (2010). "Chaotic maps, Hamiltonian flows and holographic methods".
8170:
8149:
8013:
6581:
4095:
1169:
1063:
8558:
8293:
502:
483:
8222:
8028:
7190:
1581:. Fractional negative iterates are defined analogously to fractional positive ones; for example,
511:
891:
8162:
8110:
8048:
6569:
4349:
1372:
1292:
1087:
1075:
230:
4402:
7891:{\displaystyle e^{t~{\frac {\partial ~~}{\partial h(x)}}}g(x)=g(h^{-1}(h(x)+t))=g(f_{t}(x)).}
6949:
4157:, that is, a matrix whose rows or columns sum to one, then the iterated system is known as a
3820:
2426:
2147:{\displaystyle f^{n}(x)=f^{n}(a)+(x-a)f'(a)f'(f(a))f'(f^{2}(a))\cdots f'(f^{n-1}(a))+\cdots }
1114:
The ideas of attraction and repulsion generalize similarly; one may categorize iterates into
1067:
766:
188:
8655:
8266:
4131:
435:
9240:
9025:
8883:
8153:
8145:
4170:
1091:
475:
67:
50:
39:
4468:
4372:
1172:
provides general insight into many iterated functions, especially those leading to chaos.
8:
9299:
9252:
9122:
9037:
8083:
8038:
8033:
8008:
4441:
1048:
42:
4, because that is the smallest positive exponent that produces the identity. Below is a
9244:
9029:
8887:
8581:—Three principal notations have been used to denote, say, the square of sin
8286:
9256:
9230:
9157:
9041:
9015:
8960:
8909:
8873:
8776:
8755:
8693:
8244:
8236:
8073:
3985:
Even in the absence of a strict homeomorphism, near a fixed point, here taken to be at
1522:
1098:
8798:
8273:. Cambridge, UK: Printed by J. Smith, sold by J. Deighton & sons. pp. 1–13 .
988:
problem of finding the first periodic point in an orbit, and the period of the orbit.
9325:
9303:
9278:
9260:
9161:
9045:
8913:
8829:
8805:
8762:
8713:
8661:
8365:
8343:
8248:
8126:
6545:
4154:
3437:
which, taking just the first three terms, is correct to the first decimal place when
1150:
1146:
590:
417:
200:
9248:
9209:
9149:
9033:
8999:
8940:
8899:
8891:
8226:
8018:
7167:
Iterated functions crop up in the series expansion of combined functions, such as
6557:
4099:
1515:
1157:
922:
184:
176:
8601:, though the first is least likely to be misinterpreted. In the case of sin
8338:. Vol. 2 (3rd corrected printing of 1929 issue, 2nd ed.). Chicago, USA:
8271:
A Collection of
Examples of the Applications of the Calculus of Finite Differences
1051:
that guarantee the existence of fixed points in various situations, including the
9304:"A Primer on the Elementary Theory of Infinite Compositions of Complex Functions"
9184:
8983:
8864:
Aldrovandi, R.; Freitas, L. P. (1998). "Continuous
Iteration of Dynamical Maps".
8681:
8566:
8333:
8023:
6565:
4142:
1137:
are points that move away, and never come back even close to where they started.
1134:
1115:
981:
586:
8557:, etc., "as he then supposed for the first time. The work of a German Analyst,
8329:
8306:
7464:
7277:{\displaystyle v(x)=\left.{\frac {\partial f^{n}(x)}{\partial n}}\right|_{n=0}}
4174:
1161:
977:
722:
4121:) sine are six integral iterates below it, starting with the second iterate (
9314:
9214:
9197:
9003:
8053:
7493:
5872:
4138:
3978:
3787:
1707:
1071:
758:
316:{\displaystyle f^{0}~{\stackrel {\mathrm {def} }{=}}~\operatorname {id} _{X}}
43:
8945:
8928:
8735:
8231:
8208:
6532:
Some of these examples are related among themselves by simple conjugacies.
4188:
4158:
3849:
1119:
8569:'s books, to remove the chief objection to them; Peirce wrote: "cos
6223:{\displaystyle g^{-1}{\Bigl (}a^{n}g(x)+{\frac {a^{n}-1}{a-1}}b{\Bigr )}}
4341:
of a group element on a set, then the iterated function corresponds to a
1165:
59:
8974:
8878:
8742:. Monografie Matematyczne. Warszawa: PWN – Polish Scientific Publishers.
1097:
When the points of the orbit converge to one or more limits, the set of
741:
In general, for arbitrary general (negative, non-integer, etc.) indices
9153:
8904:
8717:
8369:
8130:
4342:
3228:
expanded around the fixed point value of 2 is then an infinite series,
1175:
1036:
of the iterated sequence. The set of fixed points is often denoted as
616:
In general, the following identity holds for all non-negative integers
594:
8240:
5186:{\displaystyle \alpha ={\frac {2ax+b\pm {\sqrt {(2ax+b)^{2}-16}}}{4}}}
8973:
Kimura, Tosihusa (1971). "On the
Iteration of Analytic Functions",
8895:
8565:— The use of Herschel's notation underwent a slight change in
8166:
8078:
6585:
4052:
Thus, its iteration orbit, or flow, under suitable provisions (e.g.,
3442:
1180:
1103:
985:
398:{\displaystyle f^{n+1}~{\stackrel {\mathrm {def} }{=}}~f\circ f^{n},}
71:
6117:{\displaystyle g^{-1}{\Bigl (}a\ g(x)+b{\Bigr )}\ (a\neq 1\vee b=0)}
1492:, and so forth, all based on the principle, mentioned earlier, that
9077:, not necessarily integer, where Ψ is the solution of the relevant
8846:
8267:"Part III. Section I. Examples of the Direct Method of Differences"
8180:
5799:{\displaystyle g^{-1}{\Bigl (}h^{n}{\bigl (}g(x){\bigr )}{\Bigr )}}
3845:
1371:
Fractional iteration of a function can be defined: for instance, a
906:
180:
9235:
9020:
8633:) are of frequent occurrence in analysis. The notation sin
5093:{\displaystyle {\frac {2\alpha ^{2^{n}}+2\alpha ^{-2^{n}}-b}{2a}}}
4105:
3168:{\displaystyle {\sqrt {2}}^{{\sqrt {2}}^{{\sqrt {2}}^{\cdots }}}}
77:
For example, on the image on the right:
8473:), but he justifies his own notation by pointing out that since
5557:{\displaystyle \alpha ={\frac {a+d+{\sqrt {(a-d)^{2}+4bc}}}{2}}}
3826:
Clearly, topological conjugacy is preserved under iteration, as
5640:{\displaystyle \beta ={\frac {a+d-{\sqrt {(a-d)^{2}+4bc}}}{2}}}
4091:
8929:"Analysis of Carleman Representation of Analytical Recursions"
8537:.=c. Some years later Herschel explained that in 1813 he used
6513:
are the only cases that have a closed-form solution. Choosing
1295:
of the functional roots of the identity map. For example, for
862:
also holds, analogous to the property of exponentiation that
9174:
Brand, Louis, "A sequence defined by a difference equation,"
6575:
5718:{\displaystyle g^{-1}{\Big (}h{\bigl (}g(x){\bigr )}{\Big )}}
4250:
A nonchaotic case Schröder also illustrated with his method,
9198:"Semigroups of analytic functions and composition operators"
6347:{\displaystyle {\sqrt {a^{n}x^{2}+{\frac {a^{n}-1}{a-1}}b}}}
8722:
Encyclopédie des sciences mathématiques pures et appliquées
7220:
6952:
of an iterated function is given by the recursive formula:
1868:
1778:
1506:. This idea can be generalized so that the iteration count
711:{\displaystyle f^{m}\circ f^{n}=f^{n}\circ f^{m}=f^{m+n}~.}
611:
8926:
4059:), amounts to the conjugate of the orbit of the monomial,
1291:
has multiple solutions, which is normally the case, as in
495:
may refer to both iteration (composition) of the function
8605:
two interpretations suggest themselves; first, sin
8214:
6584:
can be defined in terms of iterated functions. These are
6564:, which in turn anchor the study of such broad topics as
4137:
This method (perturbative determination of the principal
1653:
First determine a fixed point for the function such that
1644:
8804:. Universitext: Tracts in Mathematics. Springer-Verlag.
4084:
functional iteration has been reduced to multiplication!
9275:
Lectures on
Functional Equations and Their Applications
9058:
For explicit instance, example 2 above amounts to just
9006:(2009). "Evolution Profiles and Functional Equations".
8688:. Vol. I (new ed.). Boston, USA. p. 203.
8641:) has been widely used and is now the prevailing one.
8525: V=∫ V, we may write similarly sin.
5470:{\displaystyle {\frac {a}{c}}+{\frac {bc-ad}{c}}\left}
7918:
7747:
7655:
7583:
7505:
7304:
7202:
6961:
6776:
6597:
6436:
6363:
6280:
6239:
6131:
6032:
5989:
5951:
5883:
5815:
5732:
5658:
5571:
5488:
5259:
5204:
5111:
5023:
4939:
4887:
4825:
4747:
4672:
4623:
4551:
4503:
4471:
4444:
4405:
4375:
3529:
3234:
3128:
2955:
2680:
2435:
2180:
1960:
1716:
633:
438:
329:
262:
87:
27:
Result of repeatedly applying a mathematical function
8563:
Persistence of rival notations for inverse function.
8461:, but what is usually written thus, arc (cos.=
7991:{\displaystyle f_{t}(f_{\tau }(x))=f_{t+\tau }(x)~.}
4153:
If the function is linear and can be described by a
3523:, expand around the fixed point 1 to get the series
1176:
Fractional iterates and flows, and negative iterates
1133:
Other limiting behaviors are possible; for example,
9123:
Evolution surfaces and Schröder functional methods.
8927:Berkolaiko, G.; Rabinovich, S.; Havlin, S. (1998).
8706:
199:The formal definition of an iterated function on a
9223:Journal of Physics A: Mathematical and Theoretical
8797:
8754:
8597:. The prevailing notation at present is sin
7990:
7890:
7723:
7635:
7563:
7389:
7276:
7163:Shift operator § Functions of a real variable
7145:
6932:
6756:
6486:
6412:
6346:
6265:
6222:
6116:
6015:
5974:
5934:
5863:
5798:
5717:
5639:
5556:
5469:
5239:
5185:
5092:
5007:{\displaystyle ax^{2}+bx+{\frac {b^{2}-2b-8}{4a}}}
5006:
4920:
4868:
4808:
4730:
4657:
4606:
4536:
4486:
4456:
4427:
4390:
3758:
3429:
3167:
3106:
2893:
2659:
2415:
2146:
1944:
721:This is structurally identical to the property of
710:
444:
397:
315:
163:
36:Iterated transformations of the object on the left
8863:
6215:
6147:
6079:
6048:
5791:
5748:
5710:
5674:
4731:{\displaystyle a^{\frac {b^{n}-1}{b-1}}x^{b^{n}}}
3179:times (and possibly the interpolated values when
9312:
8209:"On a Remarkable Application of Cotes's Theorem"
7636:{\displaystyle h(x)=\int {\frac {1}{v(x)}}\,dx.}
7156:
6560:, iterated functions occur as a special case of
4869:{\displaystyle {\frac {2\alpha ^{2^{n}}-b}{2a}}}
4809:{\displaystyle ax^{2}+bx+{\frac {b^{2}-2b}{4a}}}
164:{\displaystyle L=F(K),\ M=F\circ F(K)=F^{2}(K).}
9220:
8998:
8795:
8753:Kuczma, M., Choczewski B., and Ger, R. (1990).
8465:)." He admits that some authors use cos.
5935:{\displaystyle g^{-1}{\bigl (}g(x)+nb{\bigr )}}
4082:in this expression serves as a plain exponent:
3786:are two iterated functions, and there exists a
1074:applied to an iterated fixed point is known as
8724:(in French). Vol. I. p. 195. Part I.
8712:
8457:must not be understood to signify 1/cos.
5864:{\displaystyle g^{-1}{\bigl (}g(x)+b{\bigr )}}
4607:{\displaystyle a^{n}x+{\frac {a^{n}-1}{a-1}}b}
1514:, a sort of continuous "time" of a continuous
761:. On a logarithmic scale, this reduces to the
38:On top is a clockwise rotation by 90°. It has
8324:
8322:
8320:
5927:
5899:
5856:
5831:
5784:
5765:
5703:
5684:
1521:In such cases, one refers to the system as a
518:to denote the compositional meaning, writing
9195:
8781:: CS1 maint: multiple names: authors list (
8255:
8197:
6927:
6915:
6889:
6859:
6853:
6841:
6751:
6739:
6713:
6680:
6674:
6662:
4348:Most functions do not have explicit general
4173:. Well-known iterated functions include the
9277:(Dover Books on Mathematics, 2006), Ch. 6,
8069:Infinite compositions of analytic functions
6540:Iterated functions can be studied with the
6487:{\displaystyle T_{mn}=\cos(m^{n}\arccos x)}
4187:, in 1870, worked out special cases of the
4113:), in the first half-period. Half-iterate (
3479:, the series computes the inverse function
1417:can be written using the index notation as
1128:infinite compositions of analytic functions
8847:"Finding f such that f(f(x))=g(x) given g"
8698:: CS1 maint: location missing publisher (
8674:
8653:
8317:
6576:Definitions in terms of iterated functions
4921:{\displaystyle \alpha ={\frac {2ax+b}{2}}}
2949:, so the above formula terminates to just
1558:is the inverse composed with itself, i.e.
9234:
9213:
9019:
8944:
8903:
8877:
8740:Functional equations in a single variable
8299:
8230:
7724:{\displaystyle f^{n}(x)=h^{-1}(h(x)+n)~.}
7623:
6413:{\displaystyle T_{m}(x)=\cos(m\arccos x)}
1272:must be used with care when the equation
9136:
8796:Carleson, L.; Gamelin, T. D. W. (1993).
8450:
8289:
8261:
8203:
8120:
6943:
4125:) and ending with the 64th iterate. The
4104:
1179:
612:Abelian property and iteration sequences
31:
9132:
9130:
8313:(in French). Vol. IV. p. 229.
8161:, introduced by Hans Maurer (1901) and
7390:{\displaystyle g(f(x))=\exp \leftg(x).}
478:. This notation has been traced to and
14:
9313:
8752:
8734:
8680:
8660:(3rd ed.). CRC Press. p. 2.
8328:
6551:
4008:locally conjugate to a mere dilation,
1645:Some formulas for fractional iteration
1078:, and produces quadratic convergence.
9140:(1870). "Ueber iterirte Functionen".
9096:. This solution is also the infinite
8517:for log. log. log.
8305:
7400:For example, for rigid advection, if
3766:which is simply the Taylor series of
1122:, according to the behavior of small
1081:
1018:(that is, the period of the orbit of
514:), some mathematicians choose to use
9298:
9127:
8446:Philosophical Transactions of London
8443:, etc., was published by him in the
8288:(NB. Inhere, Herschel refers to his
7564:{\displaystyle f(x)=h^{-1}(h(x)+1),}
5240:{\displaystyle {\frac {ax+b}{cx+d}}}
1197:is a trivial functional 5th root of
1140:
9267:
9183:, September 1955, 489–492.
8826:Fixed Point Theory, An Introduction
8364:We note here the symbolism used by
8342:. pp. 108, 176–179, 336, 346.
8335:A History of Mathematical Notations
984:problem in computer science is the
24:
8657:Encounters with Chaos and Fractals
8654:Gulick, Denny; Ford, Jeff (2024).
8579:Powers of trigonometric functions.
8433:'s notation for inverse functions,
7773:
7762:
7358:
7354:
7249:
7225:
6266:{\displaystyle {\sqrt {ax^{2}+b}}}
6016:{\displaystyle {\sqrt {x^{2}+bn}}}
4360:, as well as non-negative integer
3848:is topologically conjugate to the
1439:is the function defined such that
365:
362:
359:
292:
289:
286:
175:Iterated functions are studied in
66:is a function that is obtained by
25:
9357:
9292:
9202:The Michigan Mathematical Journal
6535:
6499:Note: these two special cases of
5975:{\displaystyle {\sqrt {x^{2}+b}}}
4658:{\displaystyle ax^{b}\ (b\neq 1)}
1346:are solutions; so the expression
1062:There are several techniques for
8263:Herschel, John Frederick William
8205:Herschel, John Frederick William
7646:This is evident by noting that
6568:, or narrower ones, such as the
5249:fractional linear transformation
4148:
510:(the latter is commonly used in
9196:Berkson, E.; Porta, H. (1978).
9189:
9168:
9115:
9052:
8992:
8967:
8953:
8920:
8857:
8839:
8818:
8789:
8746:
8728:
8613:; second, sin (sin
8505:, we ought to write sin.
8277:from the original on 2020-08-04
7907:translation functional equation
7734:For continuous iteration index
4537:{\displaystyle ax+b\ (a\neq 1)}
4109:Iterates of the sine function (
991:
751:translation functional equation
480:John Frederick William Herschel
256:is a non-negative integer, by:
9253:10.1088/1751-8113/43/44/445101
9038:10.1088/1751-8113/42/48/485208
8761:. Cambridge University Press.
8757:Iterative Functional Equations
8647:
8449:, for the year 1813. He says (
8097:
7979:
7973:
7951:
7948:
7942:
7929:
7882:
7879:
7873:
7860:
7851:
7848:
7839:
7833:
7827:
7811:
7802:
7796:
7785:
7779:
7712:
7703:
7697:
7691:
7672:
7666:
7617:
7611:
7593:
7587:
7555:
7546:
7540:
7534:
7515:
7509:
7470:Conversely, one may specify
7381:
7375:
7349:
7343:
7323:
7320:
7314:
7308:
7244:
7238:
7212:
7206:
7140:
7131:
7125:
7106:
7094:
7088:
7077:
7071:
7046:
7043:
7037:
7018:
7001:
6995:
6984:
6978:
6886:
6880:
6856:
6824:
6818:
6710:
6704:
6677:
6645:
6639:
6481:
6459:
6407:
6392:
6380:
6374:
6171:
6165:
6111:
6087:
6068:
6062:
5913:
5907:
5845:
5839:
5779:
5773:
5698:
5692:
5608:
5595:
5525:
5512:
5447:
5426:
5410:
5389:
5368:
5347:
5325:
5304:
5160:
5141:
4652:
4640:
4531:
4519:
4422:
4416:
4385:
4379:
3997:for a function Ψ, which makes
3977:, a form known as the
3735:
3722:
3719:
3700:
3697:
3678:
3641:
3628:
3625:
3606:
3580:
3568:
3546:
3540:
3469:causes the series to diverge.
3458:. Using the other fixed point
3415:
3397:
3389:
3374:
3361:
3358:
3343:
3330:
3315:
3302:
3290:
3284:
2972:
2966:
2877:
2871:
2848:
2841:
2830:
2818:
2785:
2772:
2763:
2760:
2754:
2740:
2725:
2712:
2697:
2691:
2639:
2633:
2608:
2601:
2587:
2572:
2565:
2554:
2548:
2537:
2522:
2509:
2494:
2487:
2476:
2464:
2452:
2446:
2387:
2380:
2360:
2354:
2332:
2317:
2310:
2299:
2293:
2282:
2267:
2254:
2239:
2232:
2221:
2209:
2197:
2191:
2135:
2132:
2126:
2107:
2093:
2090:
2084:
2071:
2060:
2057:
2051:
2045:
2034:
2028:
2017:
2005:
1999:
1993:
1977:
1971:
1916:
1910:
1851:
1838:
1812:
1806:
1773:
1761:
1755:
1749:
1733:
1727:
1615:, or, equivalently, such that
1526:
749:, this relation is called the
503:exponentiation of the function
155:
149:
133:
127:
103:
97:
13:
1:
9177:American Mathematical Monthly
8453:): "This notation cos.
8340:Open court publishing company
8190:
8179:pre-superscript notation for
7157:Lie's data transport equation
4366:
3993:(0) = 0, one may often solve
2667:There is a special case when
1101:of the orbit is known as the
1066:of the sequences produced by
960:. The smallest such value of
194:
9306:. Colorado State University.
9110:) − 2)/(ln 2)
8824:Istratescu, Vasile (1981).
8686:Curves, Functions and Forces
6767:and the equivalent product:
4098:) has generalized to a full
4094:of the Picard sequence (cf.
4086:Here, however, the exponent
3920:, for any function
3852:. As a special case, taking
3773:
3504:
3183:is not an integer). We have
3117:
2911:
533:-th iterate of the function
7:
8001:
4191:, such as the chaotic case
4164:
3867:, one has the iteration of
3114:which is trivial to check.
1477:may be defined as equal to
1160:can both be interpreted as
1057:Brouwer fixed point theorem
880:The sequence of functions
482:in 1813. Herschel credited
10:
9362:
9336:Fixed points (mathematics)
9090:) = ln 2 Ψ(
8509:for sin. sin.
8148:must not be confused with
7901:The initial flow velocity
7160:
1361:can always be obtained if
1126:under iteration. Also see
1053:Banach fixed point theorem
1047:. There exist a number of
53:, which both have order 3.
8629:and log (log
6542:Artin–Mazur zeta function
4179:iterated function systems
1543:is the normal inverse of
8828:, D. Reidel, Holland.
8171:David Patterson Ellerman
8150:Rudolf von Bitter Rucker
8090:
8014:Iterated function system
7291:iterate of the function
4428:{\displaystyle f^{n}(x)}
4096:transformation semigroup
3928:Making the substitution
1170:subshifts of finite type
1064:convergence acceleration
956:, the orbit is called a
574:. For the same purpose,
8311:Formulaire mathématique
8223:Royal Society of London
7191:beta function (physics)
4350:closed-form expressions
3821:topologically conjugate
1702:around the fixed point
9341:Functions and mappings
9215:10.1307/mmj/1029002009
8946:10.1006/jmaa.1998.5986
8232:10.1098/rstl.1813.0005
8163:Reuben Louis Goodstein
8049:Functional square root
7992:
7892:
7725:
7637:
7565:
7492:, through the generic
7391:
7278:
7147:
6934:
6814:
6758:
6635:
6572:of computer programs.
6570:denotational semantics
6488:
6414:
6348:
6267:
6224:
6118:
6017:
5976:
5936:
5865:
5800:
5719:
5641:
5558:
5471:
5241:
5187:
5094:
5008:
4922:
4870:
4810:
4732:
4659:
4608:
4538:
4488:
4458:
4429:
4392:
4134:
3760:
3441:is positive. Also see
3431:
3169:
3108:
2934:gives the fixed point
2895:
2661:
2417:
2148:
1946:
1263:
1088:attractive fixed point
712:
544:, as in, for example,
446:
445:{\displaystyle \circ }
399:
317:
165:
55:
8585:, namely, (sin
8294:Hans Heinrich Bürmann
8146:function compositions
7993:
7893:
7726:
7638:
7566:
7392:
7279:
7148:
6950:functional derivative
6944:Functional derivative
6935:
6794:
6759:
6615:
6489:
6415:
6349:
6268:
6225:
6119:
6018:
5977:
5937:
5866:
5801:
5720:
5642:
5559:
5472:
5242:
5188:
5095:
5009:
4923:
4871:
4811:
4733:
4660:
4609:
4539:
4489:
4459:
4430:
4393:
4108:
3761:
3432:
3170:
3109:
2916:For example, setting
2896:
2662:
2427:geometric progression
2418:
2149:
1947:
1592:is defined such that
1224:. The computation of
1183:
1068:fixed point iteration
767:Chebyshev polynomials
713:
489:Because the notation
484:Hans Heinrich Bürmann
447:
400:
318:
189:renormalization group
166:
49:Below that are their
35:
9346:Functional equations
9331:Sequences and series
9067:) = Ψ((ln 2) Ψ(
9008:Journal of Physics A
8029:Sarkovskii's theorem
7916:
7745:
7653:
7581:
7503:
7463:, action by a plain
7302:
7200:
6959:
6774:
6595:
6434:
6422:Chebyshev polynomial
6361:
6278:
6237:
6129:
6030:
5987:
5949:
5881:
5813:
5730:
5656:
5569:
5486:
5257:
5202:
5109:
5021:
4937:
4885:
4823:
4745:
4670:
4621:
4549:
4501:
4487:{\displaystyle x+nb}
4469:
4442:
4403:
4391:{\displaystyle f(x)}
4373:
3527:
3232:
3126:
2953:
2678:
2433:
2178:
1958:
1714:
1512:continuous parameter
1092:unstable fixed point
1049:fixed-point theorems
892:Charles Émile Picard
631:
476:function composition
436:
327:
260:
85:
46:with infinite order.
9245:2010JPhA...43R5101C
9079:Schröder's equation
9030:2009JPhA...42V5208C
8976:Funkcialaj Ekvacioj
8933:J. Math. Anal. Appl
8888:1998JMP....39.5324A
8521:. Just as we write
8362:Iterated logarithms
8133:'s (1907) notation
8084:Functional equation
8064:Böttcher's equation
8059:Schröder's equation
8044:Schröder's equation
8039:Recurrence relation
8034:Fractional calculus
8009:Irrational rotation
7481:given an arbitrary
6562:recursive functions
6552:In computer science
4457:{\displaystyle x+b}
3995:Schröder's equation
3770:expanded around 1.
3200:. A fixed point is
3175:where this is done
2429:to simplify terms,
1099:accumulation points
1076:Steffensen's method
1070:. For example, the
976:itself is called a
970:period of the orbit
755:Schröder's equation
9273:Aczel, J. (2006),
9154:10.1007/BF01443992
8982:2012-04-26 at the
8714:Pringsheim, Alfred
8625:⋅ log
8621:, where log
8609:⋅ sin
8221:(Part 1). London:
8074:Flow (mathematics)
7988:
7888:
7721:
7633:
7561:
7432:. Consequently,
7387:
7274:
7187:iteration velocity
7143:
6930:
6754:
6546:transfer operators
6484:
6410:
6344:
6263:
6220:
6114:
6013:
5972:
5932:
5861:
5796:
5715:
5637:
5554:
5467:
5237:
5183:
5090:
5004:
4918:
4866:
4806:
4728:
4655:
4604:
4534:
4484:
4454:
4425:
4388:
4135:
3756:
3509:With the function
3427:
3165:
3122:Find the value of
3104:
2891:
2657:
2413:
2156:Substitute in for
2144:
1942:
1293:Babbage's equation
1264:
1082:Limiting behaviour
708:
442:
395:
313:
187:, mathematics and
161:
56:
18:Iterative function
9321:Dynamical systems
9121:Curtright, T. L.
8529:=arc (sin.=
8349:978-1-60206-714-1
8127:Alfred Pringsheim
8109:is taken for the
7984:
7789:
7770:
7767:
7758:
7717:
7621:
7496:discussed above,
7365:
7256:
7098:
7005:
6497:
6496:
6342:
6337:
6261:
6208:
6086:
6058:
6011:
5970:
5635:
5629:
5552:
5546:
5461:
5295:
5268:
5235:
5181:
5175:
5088:
5002:
4916:
4864:
4804:
4708:
4639:
4599:
4518:
4236:) = sin(2 arcsin(
4171:many chaotic maps
4155:stochastic matrix
3752:
3666:
3594:
3419:
3259:
3250:
3241:
3153:
3144:
3135:
3100:
3093:
3026:
2994:
2813:
2798:
2738:
2649:
2535:
2280:
1898:
1864:
1794:
1151:transfer operator
1147:invariant measure
1141:Invariant measure
952:for some integer
704:
591:Alfred Pringsheim
418:identity function
375:
370:
348:
302:
297:
275:
185:dynamical systems
111:
64:iterated function
54:
16:(Redirected from
9353:
9307:
9302:(January 2017).
9286:
9271:
9265:
9264:
9238:
9219:
9217:
9193:
9187:
9172:
9166:
9165:
9134:
9125:
9119:
9113:
9111:
9095:
9089:
9088:
9072:
9056:
9050:
9049:
9023:
9000:Curtright, T. L.
8996:
8990:
8971:
8965:
8964:
8957:
8951:
8950:
8948:
8924:
8918:
8917:
8907:
8896:10.1063/1.532574
8881:
8861:
8855:
8854:
8843:
8837:
8822:
8816:
8815:
8803:
8800:Complex dynamics
8793:
8787:
8786:
8780:
8772:
8760:
8750:
8744:
8743:
8732:
8726:
8725:
8710:
8704:
8703:
8697:
8689:
8682:Peirce, Benjamin
8678:
8672:
8671:
8651:
8645:
8643:
8469:for (cos.
8357:
8356:
8326:
8315:
8314:
8303:
8297:
8285:
8283:
8282:
8259:
8253:
8252:
8234:
8201:
8184:
8178:
8160:
8143:
8124:
8118:
8115:
8108:
8101:
8019:Iterative method
7997:
7995:
7994:
7989:
7982:
7972:
7971:
7941:
7940:
7928:
7927:
7904:
7897:
7895:
7894:
7889:
7872:
7871:
7826:
7825:
7792:
7791:
7790:
7788:
7771:
7768:
7765:
7760:
7756:
7737:
7730:
7728:
7727:
7722:
7715:
7690:
7689:
7665:
7664:
7642:
7640:
7639:
7634:
7622:
7620:
7603:
7570:
7568:
7567:
7562:
7533:
7532:
7491:
7480:
7462:
7431:
7417:
7396:
7394:
7393:
7388:
7371:
7367:
7366:
7364:
7353:
7294:
7290:
7283:
7281:
7280:
7275:
7273:
7272:
7261:
7257:
7255:
7247:
7237:
7236:
7223:
7181:
7152:
7150:
7149:
7144:
7124:
7123:
7099:
7097:
7080:
7070:
7069:
7050:
7036:
7035:
7017:
7006:
7004:
6987:
6977:
6976:
6963:
6939:
6937:
6936:
6931:
6914:
6913:
6896:
6892:
6831:
6827:
6813:
6808:
6763:
6761:
6760:
6755:
6738:
6737:
6720:
6716:
6652:
6648:
6634:
6629:
6558:computer science
6512:
6493:
6491:
6490:
6485:
6471:
6470:
6449:
6448:
6419:
6417:
6416:
6411:
6373:
6372:
6353:
6351:
6350:
6345:
6343:
6338:
6336:
6325:
6318:
6317:
6307:
6302:
6301:
6292:
6291:
6282:
6272:
6270:
6269:
6264:
6262:
6254:
6253:
6241:
6229:
6227:
6226:
6221:
6219:
6218:
6209:
6207:
6196:
6189:
6188:
6178:
6161:
6160:
6151:
6150:
6144:
6143:
6123:
6121:
6120:
6115:
6084:
6083:
6082:
6056:
6052:
6051:
6045:
6044:
6022:
6020:
6019:
6014:
6012:
6001:
6000:
5991:
5981:
5979:
5978:
5973:
5971:
5963:
5962:
5953:
5941:
5939:
5938:
5933:
5931:
5930:
5903:
5902:
5896:
5895:
5871: (generic
5870:
5868:
5867:
5862:
5860:
5859:
5835:
5834:
5828:
5827:
5805:
5803:
5802:
5797:
5795:
5794:
5788:
5787:
5769:
5768:
5762:
5761:
5752:
5751:
5745:
5744:
5724:
5722:
5721:
5716:
5714:
5713:
5707:
5706:
5688:
5687:
5678:
5677:
5671:
5670:
5646:
5644:
5643:
5638:
5636:
5631:
5630:
5616:
5615:
5594:
5579:
5563:
5561:
5560:
5555:
5553:
5548:
5547:
5533:
5532:
5511:
5496:
5476:
5474:
5473:
5468:
5466:
5462:
5460:
5459:
5458:
5422:
5421:
5387:
5386:
5385:
5343:
5342:
5302:
5296:
5291:
5274:
5269:
5261:
5246:
5244:
5243:
5238:
5236:
5234:
5220:
5206:
5192:
5190:
5189:
5184:
5182:
5177:
5176:
5168:
5167:
5140:
5119:
5099:
5097:
5096:
5091:
5089:
5087:
5079:
5072:
5071:
5070:
5069:
5046:
5045:
5044:
5043:
5025:
5013:
5011:
5010:
5005:
5003:
5001:
4993:
4977:
4976:
4966:
4952:
4951:
4927:
4925:
4924:
4919:
4917:
4912:
4895:
4875:
4873:
4872:
4867:
4865:
4863:
4855:
4848:
4847:
4846:
4845:
4827:
4815:
4813:
4812:
4807:
4805:
4803:
4795:
4785:
4784:
4774:
4760:
4759:
4737:
4735:
4734:
4729:
4727:
4726:
4725:
4724:
4710:
4709:
4707:
4696:
4689:
4688:
4678:
4664:
4662:
4661:
4656:
4637:
4636:
4635:
4613:
4611:
4610:
4605:
4600:
4598:
4587:
4580:
4579:
4569:
4561:
4560:
4543:
4541:
4540:
4535:
4516:
4493:
4491:
4490:
4485:
4463:
4461:
4460:
4455:
4434:
4432:
4431:
4426:
4415:
4414:
4397:
4395:
4394:
4389:
4367:
4336:
4327:
4321:
4319:
4318:
4315:
4312:
4296:
4290:
4288:
4287:
4284:
4281:
4268:
4246:
4244:
4243:
4226:
4224:
4223:
4209:
4128:
4124:
4120:
4116:
4112:
4100:continuous group
4089:
4081:
4073:
4058:
4047:
4025:
4007:
3992:
3988:
3976:
3950:
3923:
3919:
3889:
3888:) + 1)
3866:
3843:
3818:
3814:
3810:
3792:
3785:
3781:
3765:
3763:
3762:
3757:
3750:
3743:
3742:
3712:
3711:
3690:
3689:
3677:
3676:
3667:
3665:
3654:
3649:
3648:
3618:
3617:
3605:
3604:
3595:
3587:
3567:
3566:
3539:
3538:
3522:
3500:
3498:
3497:
3494:
3491:
3484:
3478:
3468:
3457:
3456:
3455:
3436:
3434:
3433:
3428:
3420:
3418:
3392:
3382:
3381:
3357:
3356:
3328:
3323:
3322:
3283:
3282:
3270:
3269:
3268:
3267:
3266:
3265:
3260:
3255:
3251:
3246:
3242:
3237:
3227:
3220:
3210:
3199:
3198:
3197:
3174:
3172:
3171:
3166:
3164:
3163:
3162:
3161:
3160:
3159:
3154:
3149:
3145:
3140:
3136:
3131:
3113:
3111:
3110:
3105:
3098:
3094:
3092:
3081:
3080:
3079:
3063:
3055:
3054:
3042:
3041:
3032:
3028:
3027:
3025:
3011:
2995:
2993:
2979:
2965:
2964:
2948:
2933:
2900:
2898:
2897:
2892:
2884:
2880:
2870:
2856:
2855:
2840:
2814:
2806:
2799:
2794:
2793:
2792:
2770:
2753:
2739:
2734:
2733:
2732:
2710:
2690:
2689:
2673:
2666:
2664:
2663:
2658:
2650:
2648:
2632:
2623:
2616:
2615:
2600:
2591:
2586:
2585:
2564:
2547:
2536:
2531:
2530:
2529:
2507:
2502:
2501:
2486:
2445:
2444:
2425:Make use of the
2422:
2420:
2419:
2414:
2406:
2402:
2401:
2400:
2379:
2353:
2331:
2330:
2309:
2292:
2281:
2276:
2275:
2274:
2252:
2247:
2246:
2231:
2190:
2189:
2169:
2153:
2151:
2150:
2145:
2125:
2124:
2106:
2083:
2082:
2070:
2044:
2027:
1992:
1991:
1970:
1969:
1951:
1949:
1948:
1943:
1935:
1934:
1923:
1919:
1909:
1908:
1899:
1897:
1896:
1895:
1882:
1881:
1872:
1865:
1860:
1859:
1858:
1836:
1831:
1830:
1819:
1815:
1805:
1804:
1795:
1793:
1782:
1748:
1747:
1726:
1725:
1701:
1683:
1666:
1640:
1614:
1591:
1580:
1557:
1546:
1542:
1525:(cf. section on
1509:
1505:
1491:
1476:
1465:
1438:
1427:
1416:
1406:. This function
1405:
1382:
1378:
1367:
1356:
1345:
1330:
1316:
1301:
1290:
1271:
1261:
1260:
1256:
1247:
1246:
1242:
1237:
1236:
1232:
1223:
1196:
1168:. The theory of
1158:Koopman operator
1135:wandering points
1046:
1029:
1025:
1021:
1017:
1013:
1009:
975:
967:
963:
955:
951:
929:
919:
904:
900:
885:
876:
861:
830:
806:
763:nesting property
748:
744:
737:
717:
715:
714:
709:
702:
701:
700:
682:
681:
669:
668:
656:
655:
643:
642:
623:
619:
607:
584:
573:
554:
543:
532:
528:
517:
509:
500:
494:
473:
452:
451:
449:
448:
443:
425:
415:
404:
402:
401:
396:
391:
390:
373:
372:
371:
369:
368:
356:
351:
346:
345:
344:
322:
320:
319:
314:
312:
311:
300:
299:
298:
296:
295:
283:
278:
273:
272:
271:
251:
241:
228:
214:
177:computer science
171:
170:
168:
167:
162:
148:
147:
109:
48:
21:
9361:
9360:
9356:
9355:
9354:
9352:
9351:
9350:
9311:
9310:
9295:
9290:
9289:
9272:
9268:
9194:
9190:
9173:
9169:
9138:Schröder, Ernst
9135:
9128:
9120:
9116:
9101:
9086:
9084:
9082:
9059:
9057:
9053:
8997:
8993:
8984:Wayback Machine
8972:
8968:
8961:"Tetration.org"
8959:
8958:
8954:
8925:
8921:
8879:physics/9712026
8862:
8858:
8845:
8844:
8840:
8823:
8819:
8812:
8794:
8790:
8774:
8773:
8769:
8751:
8747:
8733:
8729:
8711:
8707:
8691:
8690:
8679:
8675:
8668:
8652:
8648:
8637:for (sin
8567:Benjamin Peirce
8423:
8417:
8407:
8397:
8391:
8381:
8372:in their joint
8354:
8352:
8350:
8330:Cajori, Florian
8327:
8318:
8307:Peano, Giuseppe
8304:
8300:
8296:'s older work.)
8280:
8278:
8260:
8256:
8202:
8198:
8193:
8188:
8187:
8174:
8156:
8134:
8125:
8121:
8111:
8104:
8102:
8098:
8093:
8088:
8024:Rotation number
8004:
7961:
7957:
7936:
7932:
7923:
7919:
7917:
7914:
7913:
7902:
7867:
7863:
7818:
7814:
7772:
7761:
7759:
7752:
7748:
7746:
7743:
7742:
7735:
7682:
7678:
7660:
7656:
7654:
7651:
7650:
7607:
7602:
7582:
7579:
7578:
7525:
7521:
7504:
7501:
7500:
7482:
7471:
7433:
7419:
7401:
7357:
7352:
7339:
7335:
7303:
7300:
7299:
7292:
7288:
7262:
7248:
7232:
7228:
7224:
7222:
7219:
7218:
7201:
7198:
7197:
7168:
7165:
7159:
7113:
7109:
7081:
7059:
7055:
7051:
7049:
7025:
7021:
7010:
6988:
6972:
6968:
6964:
6962:
6960:
6957:
6956:
6946:
6897:
6840:
6836:
6835:
6809:
6798:
6781:
6777:
6775:
6772:
6771:
6721:
6661:
6657:
6656:
6630:
6619:
6602:
6598:
6596:
6593:
6592:
6578:
6566:lambda calculus
6554:
6538:
6500:
6466:
6462:
6441:
6437:
6435:
6432:
6431:
6368:
6364:
6362:
6359:
6358:
6326:
6313:
6309:
6308:
6306:
6297:
6293:
6287:
6283:
6281:
6279:
6276:
6275:
6249:
6245:
6240:
6238:
6235:
6234:
6214:
6213:
6197:
6184:
6180:
6179:
6177:
6156:
6152:
6146:
6145:
6136:
6132:
6130:
6127:
6126:
6078:
6077:
6047:
6046:
6037:
6033:
6031:
6028:
6027:
5996:
5992:
5990:
5988:
5985:
5984:
5958:
5954:
5952:
5950:
5947:
5946:
5926:
5925:
5898:
5897:
5888:
5884:
5882:
5879:
5878:
5855:
5854:
5830:
5829:
5820:
5816:
5814:
5811:
5810:
5790:
5789:
5783:
5782:
5764:
5763:
5757:
5753:
5747:
5746:
5737:
5733:
5731:
5728:
5727:
5709:
5708:
5702:
5701:
5683:
5682:
5673:
5672:
5663:
5659:
5657:
5654:
5653:
5611:
5607:
5593:
5580:
5578:
5570:
5567:
5566:
5528:
5524:
5510:
5497:
5495:
5487:
5484:
5483:
5477:
5454:
5450:
5417:
5413:
5388:
5375:
5371:
5332:
5328:
5303:
5301:
5297:
5275:
5273:
5260:
5258:
5255:
5254:
5221:
5207:
5205:
5203:
5200:
5199:
5163:
5159:
5139:
5120:
5118:
5110:
5107:
5106:
5100:
5080:
5065:
5061:
5057:
5053:
5039:
5035:
5034:
5030:
5026:
5024:
5022:
5019:
5018:
5015:
4994:
4972:
4968:
4967:
4965:
4947:
4943:
4938:
4935:
4934:
4896:
4894:
4886:
4883:
4882:
4876:
4856:
4841:
4837:
4836:
4832:
4828:
4826:
4824:
4821:
4820:
4817:
4796:
4780:
4776:
4775:
4773:
4755:
4751:
4746:
4743:
4742:
4720:
4716:
4715:
4711:
4697:
4684:
4680:
4679:
4677:
4673:
4671:
4668:
4667:
4631:
4627:
4622:
4619:
4618:
4588:
4575:
4571:
4570:
4568:
4556:
4552:
4550:
4547:
4546:
4502:
4499:
4498:
4470:
4467:
4466:
4443:
4440:
4439:
4410:
4406:
4404:
4401:
4400:
4374:
4371:
4370:
4332:
4316:
4313:
4310:
4309:
4307:
4298:
4285:
4282:
4279:
4278:
4276:
4270:
4251:
4239:
4237:
4228:
4219:
4217:
4211:
4192:
4167:
4151:
4143:Carleman matrix
4126:
4122:
4118:
4114:
4110:
4087:
4079:
4063:
4053:
4030:
4009:
3998:
3990:
3986:
3955:
3929:
3921:
3894:
3868:
3853:
3827:
3819:are said to be
3816:
3812:
3794:
3790:
3783:
3779:
3776:
3738:
3734:
3707:
3703:
3685:
3681:
3672:
3668:
3658:
3653:
3644:
3640:
3613:
3609:
3600:
3596:
3586:
3562:
3558:
3534:
3530:
3528:
3525:
3524:
3510:
3507:
3495:
3492:
3486:
3485:
3482:
3480:
3473:
3459:
3453:
3451:
3446:
3393:
3377:
3373:
3346:
3342:
3329:
3327:
3318:
3314:
3278:
3274:
3261:
3254:
3253:
3252:
3245:
3244:
3243:
3236:
3235:
3233:
3230:
3229:
3222:
3215:
3201:
3195:
3193:
3184:
3155:
3148:
3147:
3146:
3139:
3138:
3137:
3130:
3129:
3127:
3124:
3123:
3120:
3082:
3075:
3071:
3064:
3062:
3050:
3046:
3037:
3033:
3015:
3010:
3003:
2999:
2983:
2978:
2960:
2956:
2954:
2951:
2950:
2935:
2917:
2914:
2863:
2851:
2847:
2833:
2805:
2804:
2800:
2788:
2784:
2771:
2769:
2746:
2728:
2724:
2711:
2709:
2685:
2681:
2679:
2676:
2675:
2668:
2625:
2624:
2611:
2607:
2593:
2592:
2590:
2575:
2571:
2557:
2540:
2525:
2521:
2508:
2506:
2497:
2493:
2479:
2440:
2436:
2434:
2431:
2430:
2390:
2386:
2372:
2346:
2339:
2335:
2320:
2316:
2302:
2285:
2270:
2266:
2253:
2251:
2242:
2238:
2224:
2185:
2181:
2179:
2176:
2175:
2157:
2114:
2110:
2099:
2078:
2074:
2063:
2037:
2020:
1987:
1983:
1965:
1961:
1959:
1956:
1955:
1924:
1904:
1900:
1891:
1887:
1883:
1877:
1873:
1871:
1870:
1867:
1866:
1854:
1850:
1837:
1835:
1820:
1800:
1796:
1786:
1781:
1780:
1777:
1776:
1743:
1739:
1721:
1717:
1715:
1712:
1711:
1692:
1671:
1654:
1647:
1616:
1593:
1582:
1559:
1548:
1544:
1533:
1507:
1493:
1478:
1467:
1440:
1429:
1418:
1407:
1384:
1380:
1376:
1365:
1347:
1332:
1318:
1303:
1296:
1273:
1267:
1258:
1254:
1253:
1244:
1240:
1239:
1234:
1230:
1229:
1198:
1184:
1178:
1162:shift operators
1143:
1084:
1037:
1027:
1023:
1019:
1015:
1011:
997:
994:
982:cycle detection
973:
965:
961:
953:
934:
927:
910:
902:
898:
888:Picard sequence
881:
863:
835:
816:
808:
800:
787:
778:
770:
746:
742:
726:
690:
686:
677:
673:
664:
660:
651:
647:
638:
634:
632:
629:
628:
621:
617:
614:
598:
587:Benjamin Peirce
575:
556:
545:
534:
530:
519:
515:
505:
496:
490:
437:
434:
433:
432:
427:
421:
414:
408:
386:
382:
358:
357:
352:
350:
349:
334:
330:
328:
325:
324:
307:
303:
285:
284:
279:
277:
276:
267:
263:
261:
258:
257:
247:
246:-th iterate of
237:
216:
210:
197:
143:
139:
86:
83:
82:
81:
47:
37:
28:
23:
22:
15:
12:
11:
5:
9359:
9349:
9348:
9343:
9338:
9333:
9328:
9323:
9309:
9308:
9294:
9293:External links
9291:
9288:
9287:
9283:978-0486445236
9266:
9229:(44): 445101.
9188:
9167:
9148:(2): 296–322.
9126:
9114:
9051:
9014:(48): 485208.
8991:
8966:
8952:
8919:
8856:
8838:
8817:
8810:
8788:
8767:
8745:
8727:
8705:
8673:
8666:
8646:
8573:," "log
8553:), sin.
8533:), log.
8419:
8413:
8403:
8393:
8387:
8377:
8348:
8316:
8298:
8254:
8195:
8194:
8192:
8189:
8186:
8185:
8119:
8095:
8094:
8092:
8089:
8087:
8086:
8081:
8076:
8071:
8066:
8061:
8056:
8051:
8046:
8041:
8036:
8031:
8026:
8021:
8016:
8011:
8005:
8003:
8000:
7999:
7998:
7987:
7981:
7978:
7975:
7970:
7967:
7964:
7960:
7956:
7953:
7950:
7947:
7944:
7939:
7935:
7931:
7926:
7922:
7899:
7898:
7887:
7884:
7881:
7878:
7875:
7870:
7866:
7862:
7859:
7856:
7853:
7850:
7847:
7844:
7841:
7838:
7835:
7832:
7829:
7824:
7821:
7817:
7813:
7810:
7807:
7804:
7801:
7798:
7795:
7787:
7784:
7781:
7778:
7775:
7764:
7755:
7751:
7732:
7731:
7720:
7714:
7711:
7708:
7705:
7702:
7699:
7696:
7693:
7688:
7685:
7681:
7677:
7674:
7671:
7668:
7663:
7659:
7644:
7643:
7632:
7629:
7626:
7619:
7616:
7613:
7610:
7606:
7601:
7598:
7595:
7592:
7589:
7586:
7572:
7571:
7560:
7557:
7554:
7551:
7548:
7545:
7542:
7539:
7536:
7531:
7528:
7524:
7520:
7517:
7514:
7511:
7508:
7465:shift operator
7398:
7397:
7386:
7383:
7380:
7377:
7374:
7370:
7363:
7360:
7356:
7351:
7348:
7345:
7342:
7338:
7334:
7331:
7328:
7325:
7322:
7319:
7316:
7313:
7310:
7307:
7285:
7284:
7271:
7268:
7265:
7260:
7254:
7251:
7246:
7243:
7240:
7235:
7231:
7227:
7221:
7217:
7214:
7211:
7208:
7205:
7158:
7155:
7154:
7153:
7142:
7139:
7136:
7133:
7130:
7127:
7122:
7119:
7116:
7112:
7108:
7105:
7102:
7096:
7093:
7090:
7087:
7084:
7079:
7076:
7073:
7068:
7065:
7062:
7058:
7054:
7048:
7045:
7042:
7039:
7034:
7031:
7028:
7024:
7020:
7016:
7013:
7009:
7003:
7000:
6997:
6994:
6991:
6986:
6983:
6980:
6975:
6971:
6967:
6945:
6942:
6941:
6940:
6929:
6926:
6923:
6920:
6917:
6912:
6909:
6906:
6903:
6900:
6895:
6891:
6888:
6885:
6882:
6879:
6876:
6873:
6870:
6867:
6864:
6861:
6858:
6855:
6852:
6849:
6846:
6843:
6839:
6834:
6830:
6826:
6823:
6820:
6817:
6812:
6807:
6804:
6801:
6797:
6793:
6790:
6787:
6784:
6780:
6765:
6764:
6753:
6750:
6747:
6744:
6741:
6736:
6733:
6730:
6727:
6724:
6719:
6715:
6712:
6709:
6706:
6703:
6700:
6697:
6694:
6691:
6688:
6685:
6682:
6679:
6676:
6673:
6670:
6667:
6664:
6660:
6655:
6651:
6647:
6644:
6641:
6638:
6633:
6628:
6625:
6622:
6618:
6614:
6611:
6608:
6605:
6601:
6580:Two important
6577:
6574:
6553:
6550:
6537:
6536:Means of study
6534:
6495:
6494:
6483:
6480:
6477:
6474:
6469:
6465:
6461:
6458:
6455:
6452:
6447:
6444:
6440:
6429:
6409:
6406:
6403:
6400:
6397:
6394:
6391:
6388:
6385:
6382:
6379:
6376:
6371:
6367:
6355:
6354:
6341:
6335:
6332:
6329:
6324:
6321:
6316:
6312:
6305:
6300:
6296:
6290:
6286:
6273:
6260:
6257:
6252:
6248:
6244:
6231:
6230:
6217:
6212:
6206:
6203:
6200:
6195:
6192:
6187:
6183:
6176:
6173:
6170:
6167:
6164:
6159:
6155:
6149:
6142:
6139:
6135:
6124:
6113:
6110:
6107:
6104:
6101:
6098:
6095:
6092:
6089:
6081:
6076:
6073:
6070:
6067:
6064:
6061:
6055:
6050:
6043:
6040:
6036:
6024:
6023:
6010:
6007:
6004:
5999:
5995:
5982:
5969:
5966:
5961:
5957:
5943:
5942:
5929:
5924:
5921:
5918:
5915:
5912:
5909:
5906:
5901:
5894:
5891:
5887:
5876:
5858:
5853:
5850:
5847:
5844:
5841:
5838:
5833:
5826:
5823:
5819:
5807:
5806:
5793:
5786:
5781:
5778:
5775:
5772:
5767:
5760:
5756:
5750:
5743:
5740:
5736:
5725:
5712:
5705:
5700:
5697:
5694:
5691:
5686:
5681:
5676:
5669:
5666:
5662:
5650:
5649:
5648:
5647:
5634:
5628:
5625:
5622:
5619:
5614:
5610:
5606:
5603:
5600:
5597:
5592:
5589:
5586:
5583:
5577:
5574:
5564:
5551:
5545:
5542:
5539:
5536:
5531:
5527:
5523:
5520:
5517:
5514:
5509:
5506:
5503:
5500:
5494:
5491:
5465:
5457:
5453:
5449:
5446:
5443:
5440:
5437:
5434:
5431:
5428:
5425:
5420:
5416:
5412:
5409:
5406:
5403:
5400:
5397:
5394:
5391:
5384:
5381:
5378:
5374:
5370:
5367:
5364:
5361:
5358:
5355:
5352:
5349:
5346:
5341:
5338:
5335:
5331:
5327:
5324:
5321:
5318:
5315:
5312:
5309:
5306:
5300:
5294:
5290:
5287:
5284:
5281:
5278:
5272:
5267:
5264:
5252:
5233:
5230:
5227:
5224:
5219:
5216:
5213:
5210:
5196:
5195:
5194:
5193:
5180:
5174:
5171:
5166:
5162:
5158:
5155:
5152:
5149:
5146:
5143:
5138:
5135:
5132:
5129:
5126:
5123:
5117:
5114:
5086:
5083:
5078:
5075:
5068:
5064:
5060:
5056:
5052:
5049:
5042:
5038:
5033:
5029:
5016:
5000:
4997:
4992:
4989:
4986:
4983:
4980:
4975:
4971:
4964:
4961:
4958:
4955:
4950:
4946:
4942:
4931:
4930:
4929:
4928:
4915:
4911:
4908:
4905:
4902:
4899:
4893:
4890:
4862:
4859:
4854:
4851:
4844:
4840:
4835:
4831:
4818:
4802:
4799:
4794:
4791:
4788:
4783:
4779:
4772:
4769:
4766:
4763:
4758:
4754:
4750:
4739:
4738:
4723:
4719:
4714:
4706:
4703:
4700:
4695:
4692:
4687:
4683:
4676:
4665:
4654:
4651:
4648:
4645:
4642:
4634:
4630:
4626:
4615:
4614:
4603:
4597:
4594:
4591:
4586:
4583:
4578:
4574:
4567:
4564:
4559:
4555:
4544:
4533:
4530:
4527:
4524:
4521:
4515:
4512:
4509:
4506:
4495:
4494:
4483:
4480:
4477:
4474:
4464:
4453:
4450:
4447:
4436:
4435:
4424:
4421:
4418:
4413:
4409:
4398:
4387:
4384:
4381:
4378:
4185:Ernst Schröder
4175:Mandelbrot set
4166:
4163:
4150:
4147:
4076:
4075:
4050:
4049:
3983:
3982:
3926:
3925:
3914:) +
3865: + 1
3775:
3772:
3755:
3749:
3746:
3741:
3737:
3733:
3730:
3727:
3724:
3721:
3718:
3715:
3710:
3706:
3702:
3699:
3696:
3693:
3688:
3684:
3680:
3675:
3671:
3664:
3661:
3657:
3652:
3647:
3643:
3639:
3636:
3633:
3630:
3627:
3624:
3621:
3616:
3612:
3608:
3603:
3599:
3593:
3590:
3585:
3582:
3579:
3576:
3573:
3570:
3565:
3561:
3557:
3554:
3551:
3548:
3545:
3542:
3537:
3533:
3506:
3503:
3426:
3423:
3417:
3414:
3411:
3408:
3405:
3402:
3399:
3396:
3391:
3388:
3385:
3380:
3376:
3372:
3369:
3366:
3363:
3360:
3355:
3352:
3349:
3345:
3341:
3338:
3335:
3332:
3326:
3321:
3317:
3313:
3310:
3307:
3304:
3301:
3298:
3295:
3292:
3289:
3286:
3281:
3277:
3273:
3264:
3258:
3249:
3240:
3158:
3152:
3143:
3134:
3119:
3116:
3103:
3097:
3091:
3088:
3085:
3078:
3074:
3070:
3067:
3061:
3058:
3053:
3049:
3045:
3040:
3036:
3031:
3024:
3021:
3018:
3014:
3009:
3006:
3002:
2998:
2992:
2989:
2986:
2982:
2977:
2974:
2971:
2968:
2963:
2959:
2913:
2910:
2902:
2901:
2890:
2887:
2883:
2879:
2876:
2873:
2869:
2866:
2862:
2859:
2854:
2850:
2846:
2843:
2839:
2836:
2832:
2829:
2826:
2823:
2820:
2817:
2812:
2809:
2803:
2797:
2791:
2787:
2783:
2780:
2777:
2774:
2768:
2765:
2762:
2759:
2756:
2752:
2749:
2745:
2742:
2737:
2731:
2727:
2723:
2720:
2717:
2714:
2708:
2705:
2702:
2699:
2696:
2693:
2688:
2684:
2656:
2653:
2647:
2644:
2641:
2638:
2635:
2631:
2628:
2622:
2619:
2614:
2610:
2606:
2603:
2599:
2596:
2589:
2584:
2581:
2578:
2574:
2570:
2567:
2563:
2560:
2556:
2553:
2550:
2546:
2543:
2539:
2534:
2528:
2524:
2520:
2517:
2514:
2511:
2505:
2500:
2496:
2492:
2489:
2485:
2482:
2478:
2475:
2472:
2469:
2466:
2463:
2460:
2457:
2454:
2451:
2448:
2443:
2439:
2423:
2412:
2409:
2405:
2399:
2396:
2393:
2389:
2385:
2382:
2378:
2375:
2371:
2368:
2365:
2362:
2359:
2356:
2352:
2349:
2345:
2342:
2338:
2334:
2329:
2326:
2323:
2319:
2315:
2312:
2308:
2305:
2301:
2298:
2295:
2291:
2288:
2284:
2279:
2273:
2269:
2265:
2262:
2259:
2256:
2250:
2245:
2241:
2237:
2234:
2230:
2227:
2223:
2220:
2217:
2214:
2211:
2208:
2205:
2202:
2199:
2196:
2193:
2188:
2184:
2154:
2143:
2140:
2137:
2134:
2131:
2128:
2123:
2120:
2117:
2113:
2109:
2105:
2102:
2098:
2095:
2092:
2089:
2086:
2081:
2077:
2073:
2069:
2066:
2062:
2059:
2056:
2053:
2050:
2047:
2043:
2040:
2036:
2033:
2030:
2026:
2023:
2019:
2016:
2013:
2010:
2007:
2004:
2001:
1998:
1995:
1990:
1986:
1982:
1979:
1976:
1973:
1968:
1964:
1952:
1941:
1938:
1933:
1930:
1927:
1922:
1918:
1915:
1912:
1907:
1903:
1894:
1890:
1886:
1880:
1876:
1869:
1863:
1857:
1853:
1849:
1846:
1843:
1840:
1834:
1829:
1826:
1823:
1818:
1814:
1811:
1808:
1803:
1799:
1792:
1789:
1785:
1779:
1775:
1772:
1769:
1766:
1763:
1760:
1757:
1754:
1751:
1746:
1742:
1738:
1735:
1732:
1729:
1724:
1720:
1689:
1668:
1646:
1643:
1379:is a function
1375:of a function
1177:
1174:
1142:
1139:
1083:
1080:
993:
990:
978:periodic point
968:is called the
958:periodic orbit
920:is called the
890:, named after
834:The relation
812:
796:
783:
774:
723:exponentiation
719:
718:
707:
699:
696:
693:
689:
685:
680:
676:
672:
667:
663:
659:
654:
650:
646:
641:
637:
613:
610:
441:
410:
394:
389:
385:
381:
378:
367:
364:
361:
355:
343:
340:
337:
333:
310:
306:
294:
291:
288:
282:
270:
266:
196:
193:
173:
172:
160:
157:
154:
151:
146:
142:
138:
135:
132:
129:
126:
123:
120:
117:
114:
108:
105:
102:
99:
96:
93:
90:
26:
9:
6:
4:
3:
2:
9358:
9347:
9344:
9342:
9339:
9337:
9334:
9332:
9329:
9327:
9324:
9322:
9319:
9318:
9316:
9305:
9301:
9297:
9296:
9284:
9280:
9276:
9270:
9262:
9258:
9254:
9250:
9246:
9242:
9237:
9232:
9228:
9224:
9216:
9211:
9207:
9203:
9199:
9192:
9186:
9182:
9179:
9178:
9171:
9163:
9159:
9155:
9151:
9147:
9143:
9139:
9133:
9131:
9124:
9118:
9109:
9105:
9099:
9093:
9080:
9076:
9070:
9066:
9062:
9055:
9047:
9043:
9039:
9035:
9031:
9027:
9022:
9017:
9013:
9009:
9005:
9004:Zachos, C. K.
9001:
8995:
8988:
8985:
8981:
8978:
8977:
8970:
8962:
8956:
8947:
8942:
8938:
8934:
8930:
8923:
8915:
8911:
8906:
8901:
8897:
8893:
8889:
8885:
8880:
8875:
8871:
8867:
8866:J. Math. Phys
8860:
8852:
8848:
8842:
8835:
8834:90-277-1224-7
8831:
8827:
8821:
8813:
8811:0-387-97942-5
8807:
8802:
8801:
8792:
8784:
8778:
8770:
8768:0-521-35561-3
8764:
8759:
8758:
8749:
8741:
8737:
8736:Kuczma, Marek
8731:
8723:
8719:
8715:
8709:
8701:
8695:
8687:
8683:
8677:
8669:
8667:9781003835776
8663:
8659:
8658:
8650:
8642:
8640:
8636:
8632:
8628:
8624:
8620:
8616:
8612:
8608:
8604:
8600:
8596:
8592:
8589:), sin
8588:
8584:
8580:
8576:
8572:
8568:
8564:
8560:
8556:
8552:
8548:
8544:
8540:
8536:
8532:
8528:
8524:
8520:
8516:
8513:, log.
8512:
8508:
8504:
8500:
8496:
8492:
8488:
8484:
8480:
8476:
8472:
8468:
8464:
8460:
8456:
8452:
8448:
8447:
8442:
8438:
8434:
8432:
8431:John Herschel
8427:
8422:
8416:
8411:
8406:
8401:
8396:
8390:
8385:
8380:
8376:article: "log
8375:
8371:
8367:
8363:
8351:
8345:
8341:
8337:
8336:
8331:
8325:
8323:
8321:
8312:
8308:
8302:
8295:
8292:and mentions
8291:
8287:
8276:
8272:
8268:
8264:
8258:
8250:
8246:
8242:
8238:
8233:
8228:
8224:
8220:
8216:
8215:
8210:
8206:
8200:
8196:
8182:
8177:
8172:
8168:
8164:
8159:
8155:
8151:
8147:
8141:
8137:
8132:
8128:
8123:
8117:
8116:th derivative
8114:
8107:
8100:
8096:
8085:
8082:
8080:
8077:
8075:
8072:
8070:
8067:
8065:
8062:
8060:
8057:
8055:
8054:Abel function
8052:
8050:
8047:
8045:
8042:
8040:
8037:
8035:
8032:
8030:
8027:
8025:
8022:
8020:
8017:
8015:
8012:
8010:
8007:
8006:
7985:
7976:
7968:
7965:
7962:
7958:
7954:
7945:
7937:
7933:
7924:
7920:
7912:
7911:
7910:
7908:
7885:
7876:
7868:
7864:
7857:
7854:
7845:
7842:
7836:
7830:
7822:
7819:
7815:
7808:
7805:
7799:
7793:
7782:
7776:
7753:
7749:
7741:
7740:
7739:
7718:
7709:
7706:
7700:
7694:
7686:
7683:
7679:
7675:
7669:
7661:
7657:
7649:
7648:
7647:
7630:
7627:
7624:
7614:
7608:
7604:
7599:
7596:
7590:
7584:
7577:
7576:
7575:
7558:
7552:
7549:
7543:
7537:
7529:
7526:
7522:
7518:
7512:
7506:
7499:
7498:
7497:
7495:
7494:Abel equation
7489:
7485:
7478:
7474:
7468:
7466:
7460:
7456:
7452:
7448:
7444:
7440:
7436:
7430:
7426:
7422:
7416:
7412:
7408:
7404:
7384:
7378:
7372:
7368:
7361:
7346:
7340:
7336:
7332:
7329:
7326:
7317:
7311:
7305:
7298:
7297:
7296:
7269:
7266:
7263:
7258:
7252:
7241:
7233:
7229:
7215:
7209:
7203:
7196:
7195:
7194:
7192:
7188:
7183:
7179:
7175:
7171:
7164:
7137:
7134:
7128:
7120:
7117:
7114:
7110:
7103:
7100:
7091:
7085:
7082:
7074:
7066:
7063:
7060:
7056:
7052:
7040:
7032:
7029:
7026:
7022:
7014:
7011:
7007:
6998:
6992:
6989:
6981:
6973:
6969:
6965:
6955:
6954:
6953:
6951:
6924:
6921:
6918:
6910:
6907:
6904:
6901:
6898:
6893:
6883:
6877:
6874:
6871:
6868:
6865:
6862:
6850:
6847:
6844:
6837:
6832:
6828:
6821:
6815:
6810:
6805:
6802:
6799:
6795:
6791:
6788:
6785:
6782:
6778:
6770:
6769:
6768:
6748:
6745:
6742:
6734:
6731:
6728:
6725:
6722:
6717:
6707:
6701:
6698:
6695:
6692:
6689:
6686:
6683:
6671:
6668:
6665:
6658:
6653:
6649:
6642:
6636:
6631:
6626:
6623:
6620:
6616:
6612:
6609:
6606:
6603:
6599:
6591:
6590:
6589:
6587:
6583:
6573:
6571:
6567:
6563:
6559:
6549:
6547:
6543:
6533:
6530:
6528:
6524:
6520:
6516:
6511:
6507:
6503:
6478:
6475:
6472:
6467:
6463:
6456:
6453:
6450:
6445:
6442:
6438:
6430:
6427:
6423:
6404:
6401:
6398:
6395:
6389:
6386:
6383:
6377:
6369:
6365:
6357:
6356:
6339:
6333:
6330:
6327:
6322:
6319:
6314:
6310:
6303:
6298:
6294:
6288:
6284:
6274:
6258:
6255:
6250:
6246:
6242:
6233:
6232:
6210:
6204:
6201:
6198:
6193:
6190:
6185:
6181:
6174:
6168:
6162:
6157:
6153:
6140:
6137:
6133:
6125:
6108:
6105:
6102:
6099:
6096:
6093:
6090:
6074:
6071:
6065:
6059:
6053:
6041:
6038:
6034:
6026:
6025:
6008:
6005:
6002:
5997:
5993:
5983:
5967:
5964:
5959:
5955:
5945:
5944:
5922:
5919:
5916:
5910:
5904:
5892:
5889:
5885:
5877:
5874:
5873:Abel equation
5851:
5848:
5842:
5836:
5824:
5821:
5817:
5809:
5808:
5776:
5770:
5758:
5754:
5741:
5738:
5734:
5726:
5695:
5689:
5679:
5667:
5664:
5660:
5652:
5651:
5632:
5626:
5623:
5620:
5617:
5612:
5604:
5601:
5598:
5590:
5587:
5584:
5581:
5575:
5572:
5565:
5549:
5543:
5540:
5537:
5534:
5529:
5521:
5518:
5515:
5507:
5504:
5501:
5498:
5492:
5489:
5482:
5481:
5480:
5463:
5455:
5451:
5444:
5441:
5438:
5435:
5432:
5429:
5423:
5418:
5414:
5407:
5404:
5401:
5398:
5395:
5392:
5382:
5379:
5376:
5372:
5365:
5362:
5359:
5356:
5353:
5350:
5344:
5339:
5336:
5333:
5329:
5322:
5319:
5316:
5313:
5310:
5307:
5298:
5292:
5288:
5285:
5282:
5279:
5276:
5270:
5265:
5262:
5253:
5250:
5231:
5228:
5225:
5222:
5217:
5214:
5211:
5208:
5198:
5197:
5178:
5172:
5169:
5164:
5156:
5153:
5150:
5147:
5144:
5136:
5133:
5130:
5127:
5124:
5121:
5115:
5112:
5105:
5104:
5103:
5084:
5081:
5076:
5073:
5066:
5062:
5058:
5054:
5050:
5047:
5040:
5036:
5031:
5027:
5017:
4998:
4995:
4990:
4987:
4984:
4981:
4978:
4973:
4969:
4962:
4959:
4956:
4953:
4948:
4944:
4940:
4933:
4932:
4913:
4909:
4906:
4903:
4900:
4897:
4891:
4888:
4881:
4880:
4879:
4860:
4857:
4852:
4849:
4842:
4838:
4833:
4829:
4819:
4800:
4797:
4792:
4789:
4786:
4781:
4777:
4770:
4767:
4764:
4761:
4756:
4752:
4748:
4741:
4740:
4721:
4717:
4712:
4704:
4701:
4698:
4693:
4690:
4685:
4681:
4674:
4666:
4649:
4646:
4643:
4632:
4628:
4624:
4617:
4616:
4601:
4595:
4592:
4589:
4584:
4581:
4576:
4572:
4565:
4562:
4557:
4553:
4545:
4528:
4525:
4522:
4513:
4510:
4507:
4504:
4497:
4496:
4481:
4478:
4475:
4472:
4465:
4451:
4448:
4445:
4438:
4437:
4419:
4411:
4407:
4399:
4382:
4376:
4369:
4368:
4365:
4363:
4359:
4355:
4351:
4346:
4344:
4340:
4335:
4329:
4325:
4305:
4301:
4297:, and hence
4294:
4274:
4266:
4262:
4258:
4254:
4248:
4242:
4235:
4231:
4222:
4215:
4207:
4203:
4199:
4195:
4190:
4186:
4182:
4180:
4176:
4172:
4162:
4160:
4156:
4149:Markov chains
4146:
4144:
4140:
4139:eigenfunction
4132:
4107:
4103:
4101:
4097:
4093:
4085:
4071:
4067:
4062:
4061:
4060:
4056:
4045:
4041:
4037:
4033:
4029:
4028:
4027:
4024:
4020:
4016:
4012:
4005:
4001:
3996:
3980:
3979:Abel equation
3974:
3970:
3966:
3962:
3958:
3954:
3953:
3952:
3948:
3944:
3940:
3936:
3932:
3917:
3913:
3909:
3905:
3901:
3897:
3893:
3892:
3891:
3887:
3883:
3879:
3875:
3871:
3864:
3860:
3856:
3851:
3847:
3842:
3838:
3835: ○
3834:
3831: =
3830:
3824:
3822:
3809:
3805:
3801:
3797:
3789:
3788:homeomorphism
3771:
3769:
3753:
3747:
3744:
3739:
3731:
3728:
3725:
3716:
3713:
3708:
3704:
3694:
3691:
3686:
3682:
3673:
3669:
3662:
3659:
3655:
3650:
3645:
3637:
3634:
3631:
3622:
3619:
3614:
3610:
3601:
3597:
3591:
3588:
3583:
3577:
3574:
3571:
3563:
3559:
3555:
3552:
3549:
3543:
3535:
3531:
3521:
3517:
3513:
3502:
3490:
3476:
3470:
3466:
3462:
3449:
3444:
3440:
3424:
3421:
3412:
3409:
3406:
3403:
3400:
3394:
3386:
3383:
3378:
3370:
3367:
3364:
3353:
3350:
3347:
3339:
3336:
3333:
3324:
3319:
3311:
3308:
3305:
3299:
3296:
3293:
3287:
3279:
3275:
3271:
3262:
3256:
3247:
3238:
3225:
3218:
3212:
3208:
3204:
3191:
3187:
3182:
3178:
3156:
3150:
3141:
3132:
3115:
3101:
3095:
3089:
3086:
3083:
3076:
3072:
3068:
3065:
3059:
3056:
3051:
3047:
3043:
3038:
3034:
3029:
3022:
3019:
3016:
3012:
3007:
3004:
3000:
2996:
2990:
2987:
2984:
2980:
2975:
2969:
2961:
2957:
2946:
2942:
2938:
2932:
2928:
2924:
2920:
2909:
2907:
2888:
2885:
2881:
2874:
2867:
2864:
2860:
2857:
2852:
2844:
2837:
2834:
2827:
2824:
2821:
2815:
2810:
2807:
2801:
2795:
2789:
2781:
2778:
2775:
2766:
2757:
2750:
2747:
2743:
2735:
2729:
2721:
2718:
2715:
2706:
2703:
2700:
2694:
2686:
2682:
2671:
2654:
2651:
2645:
2642:
2636:
2629:
2626:
2620:
2617:
2612:
2604:
2597:
2594:
2582:
2579:
2576:
2568:
2561:
2558:
2551:
2544:
2541:
2532:
2526:
2518:
2515:
2512:
2503:
2498:
2490:
2483:
2480:
2473:
2470:
2467:
2461:
2458:
2455:
2449:
2441:
2437:
2428:
2424:
2410:
2407:
2403:
2397:
2394:
2391:
2383:
2376:
2373:
2369:
2366:
2363:
2357:
2350:
2347:
2343:
2340:
2336:
2327:
2324:
2321:
2313:
2306:
2303:
2296:
2289:
2286:
2277:
2271:
2263:
2260:
2257:
2248:
2243:
2235:
2228:
2225:
2218:
2215:
2212:
2206:
2203:
2200:
2194:
2186:
2182:
2173:
2168:
2164:
2160:
2155:
2141:
2138:
2129:
2121:
2118:
2115:
2111:
2103:
2100:
2096:
2087:
2079:
2075:
2067:
2064:
2054:
2048:
2041:
2038:
2031:
2024:
2021:
2014:
2011:
2008:
2002:
1996:
1988:
1984:
1980:
1974:
1966:
1962:
1953:
1939:
1936:
1931:
1928:
1925:
1920:
1913:
1905:
1901:
1892:
1888:
1884:
1878:
1874:
1861:
1855:
1847:
1844:
1841:
1832:
1827:
1824:
1821:
1816:
1809:
1801:
1797:
1790:
1787:
1783:
1770:
1767:
1764:
1758:
1752:
1744:
1740:
1736:
1730:
1722:
1718:
1709:
1708:Taylor series
1705:
1699:
1695:
1690:
1687:
1682:
1678:
1674:
1669:
1665:
1661:
1657:
1652:
1651:
1650:
1642:
1639:
1635:
1631:
1627:
1623:
1619:
1612:
1608:
1604:
1600:
1596:
1589:
1585:
1578:
1574:
1570:
1566:
1562:
1555:
1551:
1540:
1536:
1530:
1528:
1524:
1519:
1517:
1513:
1504:
1500:
1496:
1489:
1485:
1481:
1474:
1470:
1463:
1459:
1455:
1451:
1447:
1443:
1436:
1432:
1428:. Similarly,
1425:
1421:
1414:
1410:
1403:
1399:
1395:
1391:
1387:
1374:
1369:
1364:
1360:
1354:
1350:
1343:
1339:
1335:
1329:
1325:
1321:
1314:
1310:
1306:
1299:
1294:
1288:
1284:
1280:
1276:
1270:
1251:
1227:
1221:
1217:
1213:
1209:
1205:
1201:
1195:
1191:
1187:
1182:
1173:
1171:
1167:
1163:
1159:
1154:
1152:
1148:
1138:
1136:
1131:
1129:
1125:
1124:neighborhoods
1121:
1120:unstable sets
1117:
1112:
1110:
1106:
1105:
1100:
1095:
1093:
1089:
1079:
1077:
1073:
1072:Aitken method
1069:
1065:
1060:
1058:
1054:
1050:
1044:
1040:
1035:
1034:
1007:
1003:
999:
989:
987:
983:
979:
971:
959:
949:
945:
941:
937:
931:
925:
924:
917:
913:
908:
895:
893:
889:
884:
878:
875:
871:
867:
859:
855:
851:
847:
843:
839:
832:
828:
824:
820:
815:
811:
804:
799:
795:
791:
786:
782:
777:
773:
768:
764:
760:
759:Abel equation
756:
752:
739:
736:
732:
729:
724:
705:
697:
694:
691:
687:
683:
678:
674:
670:
665:
661:
657:
652:
648:
644:
639:
635:
627:
626:
625:
609:
605:
601:
596:
592:
588:
582:
578:
571:
567:
563:
559:
552:
548:
541:
537:
526:
522:
513:
508:
504:
499:
493:
487:
485:
481:
477:
471:
467:
463:
459:
455:
439:
431:
424:
419:
413:
405:
392:
387:
383:
379:
376:
353:
341:
338:
335:
331:
308:
304:
280:
268:
264:
255:
250:
245:
240:
234:
232:
227:
223:
219:
215:be a set and
213:
207:
205:
202:
192:
190:
186:
182:
178:
158:
152:
144:
140:
136:
130:
124:
121:
118:
115:
112:
106:
100:
94:
91:
88:
80:
79:
78:
75:
73:
69:
65:
61:
52:
45:
44:shear mapping
41:
34:
30:
19:
9274:
9269:
9226:
9222:
9205:
9201:
9191:
9180:
9175:
9170:
9145:
9141:
9117:
9107:
9103:
9097:
9091:
9074:
9068:
9064:
9060:
9054:
9011:
9007:
8994:
8986:
8975:
8969:
8955:
8936:
8932:
8922:
8872:(10): 5324.
8869:
8865:
8859:
8851:MathOverflow
8850:
8841:
8825:
8820:
8799:
8791:
8756:
8748:
8739:
8730:
8721:
8708:
8685:
8676:
8656:
8649:
8638:
8634:
8630:
8626:
8622:
8618:
8614:
8610:
8606:
8602:
8598:
8594:
8593:, sin
8590:
8586:
8582:
8578:
8574:
8570:
8562:
8554:
8550:
8546:
8542:
8538:
8534:
8530:
8526:
8522:
8518:
8514:
8510:
8506:
8502:
8498:
8497:, ΔΔΔ
8494:
8490:
8486:
8482:
8478:
8474:
8470:
8466:
8462:
8458:
8454:
8444:
8440:
8439:, tan
8436:
8429:
8425:
8420:
8414:
8409:
8404:
8399:
8394:
8388:
8383:
8378:
8374:Encyclopédie
8373:
8361:
8359:
8353:. Retrieved
8334:
8310:
8301:
8279:. Retrieved
8270:
8257:
8218:
8212:
8199:
8175:
8157:
8139:
8135:
8122:
8112:
8105:
8099:
7906:
7900:
7733:
7645:
7573:
7487:
7483:
7476:
7472:
7469:
7458:
7454:
7450:
7446:
7442:
7438:
7434:
7428:
7424:
7420:
7414:
7410:
7406:
7402:
7399:
7286:
7184:
7177:
7173:
7169:
7166:
6947:
6766:
6579:
6555:
6539:
6531:
6526:
6522:
6518:
6514:
6509:
6505:
6501:
6498:
6425:
6424:for integer
5478:
5101:
4877:
4361:
4357:
4353:
4347:
4333:
4330:
4323:
4303:
4299:
4292:
4272:
4264:
4260:
4256:
4252:
4249:
4240:
4233:
4229:
4220:
4213:
4205:
4201:
4197:
4193:
4189:logistic map
4183:
4168:
4159:Markov chain
4152:
4136:
4083:
4077:
4069:
4065:
4054:
4051:
4043:
4039:
4035:
4031:
4026:, that is
4022:
4018:
4014:
4010:
4003:
3999:
3984:
3972:
3968:
3964:
3960:
3956:
3946:
3942:
3938:
3934:
3930:
3927:
3915:
3911:
3907:
3903:
3899:
3895:
3885:
3881:
3877:
3873:
3869:
3862:
3858:
3854:
3850:logistic map
3840:
3836:
3832:
3828:
3825:
3807:
3803:
3799:
3795:
3777:
3767:
3519:
3515:
3511:
3508:
3488:
3474:
3471:
3464:
3460:
3447:
3438:
3223:
3216:
3213:
3206:
3202:
3189:
3185:
3180:
3176:
3121:
2944:
2940:
2936:
2930:
2926:
2922:
2918:
2915:
2905:
2903:
2669:
2171:
2166:
2162:
2158:
1703:
1697:
1693:
1685:
1680:
1676:
1672:
1663:
1659:
1655:
1648:
1637:
1633:
1629:
1625:
1621:
1617:
1610:
1606:
1602:
1598:
1594:
1587:
1583:
1576:
1572:
1568:
1564:
1560:
1553:
1549:
1538:
1534:
1531:
1520:
1511:
1502:
1498:
1494:
1487:
1483:
1479:
1472:
1468:
1461:
1457:
1453:
1449:
1445:
1441:
1434:
1430:
1423:
1419:
1412:
1408:
1401:
1397:
1393:
1389:
1385:
1373:half iterate
1370:
1362:
1358:
1352:
1348:
1341:
1337:
1333:
1327:
1323:
1319:
1312:
1308:
1304:
1297:
1286:
1282:
1278:
1274:
1268:
1265:
1249:
1225:
1219:
1215:
1211:
1207:
1203:
1199:
1193:
1189:
1185:
1164:action on a
1155:
1144:
1132:
1113:
1108:
1102:
1096:
1085:
1061:
1042:
1038:
1031:
1030:is called a
1005:
1001:
998:
995:
992:Fixed points
972:. The point
969:
964:for a given
957:
947:
943:
939:
935:
932:
921:
915:
911:
897:For a given
896:
887:
886:is called a
882:
879:
873:
869:
865:
857:
853:
849:
845:
841:
837:
833:
826:
822:
818:
813:
809:
802:
797:
793:
789:
784:
780:
775:
771:
762:
750:
740:
734:
730:
727:
720:
615:
603:
599:
585:was used by
580:
576:
569:
565:
561:
557:
550:
546:
539:
535:
524:
520:
512:trigonometry
506:
497:
491:
488:
469:
465:
461:
457:
453:
429:
422:
411:
406:
253:
248:
243:
238:
235:
225:
221:
217:
211:
208:
203:
198:
174:
76:
63:
57:
51:compositions
29:
9208:: 101–115.
8905:11449/65519
8718:Molk, Jules
8501:, ΣΣ
8428:)." §533.
8165:(1947) for
6582:functionals
4216:) = arcsin(
3793:such that
1954:Expand out
1266:The notion
1262:) is shown.
1166:shift space
1116:stable sets
1109:ω-limit set
1033:fixed point
986:algorithmic
909:of values
60:mathematics
9315:Categories
9300:Gill, John
8989:, 197-238.
8577:." §537.
8485:, Σ
8481:, Δ
8451:p. 10
8435:sin
8366:Pringsheim
8355:2016-01-18
8281:2020-08-04
8191:References
8173:'s (1995)
8169:, or with
8152:'s (1982)
8144:to denote
8131:Jules Molk
7295:, we have
7185:Given the
7161:See also:
5014:(see note)
4816:(see note)
4343:free group
4269:, yielded
4210:, so that
4169:There are
2170:, for any
1510:becomes a
1383:such that
597:suggested
595:Jules Molk
195:Definition
9261:115176169
9236:1002.0104
9162:116998358
9142:Math. Ann
9100:limit of
9046:115173476
9021:0909.2424
8939:: 81–90.
8914:119675869
8777:cite book
8694:cite book
8402:), …, log
8290:1813 work
8249:118124706
8207:(1813) .
8167:tetration
8079:Tetration
7969:τ
7938:τ
7820:−
7774:∂
7763:∂
7684:−
7600:∫
7527:−
7359:∂
7355:∂
7333:
7250:∂
7226:∂
7135:−
7118:−
7104:δ
7083:δ
7064:−
7053:δ
7030:−
6990:δ
6966:δ
6902:−
6857:→
6833:≡
6796:∏
6726:−
6678:→
6654:≡
6617:∑
6586:summation
6544:and with
6476:
6457:
6402:
6390:
6331:−
6320:−
6202:−
6191:−
6138:−
6100:∨
6094:≠
6039:−
5890:−
5822:−
5739:−
5665:−
5602:−
5591:−
5573:β
5519:−
5490:α
5452:β
5445:β
5436:−
5424:−
5415:α
5408:α
5399:−
5380:−
5373:β
5366:β
5357:−
5345:−
5337:−
5330:α
5323:α
5314:−
5283:−
5170:−
5137:±
5113:α
5074:−
5059:−
5055:α
5032:α
4988:−
4979:−
4889:α
4850:−
4834:α
4787:−
4702:−
4691:−
4647:≠
4593:−
4582:−
4526:≠
3774:Conjugacy
3748:⋯
3729:−
3714:−
3692:−
3635:−
3620:−
3575:−
3505:Example 3
3443:Tetration
3425:⋯
3422:−
3410:−
3404:
3384:−
3368:
3337:
3309:
3300:−
3263:⋯
3157:⋯
3118:Example 2
3087:−
3069:−
3020:−
3008:−
2988:−
2912:Example 1
2906:Conjugacy
2889:⋯
2825:−
2779:−
2719:−
2655:⋯
2643:−
2618:−
2580:−
2516:−
2471:−
2411:⋯
2395:−
2367:⋯
2325:−
2261:−
2216:−
2142:⋯
2119:−
2097:⋯
2012:−
1940:⋯
1845:−
1768:−
1527:conjugacy
1326:) = 6 − 2
1104:limit set
1010:for some
807:, since
671:∘
645:∘
608:instead.
440:∘
380:∘
236:Defining
206:follows.
191:physics.
122:∘
72:iteration
68:composing
9326:Fractals
8980:Archived
8738:(1968).
8720:(1907).
8684:(1852).
8309:(1903).
8275:Archived
8265:(1820).
8154:notation
8002:See also
7445:) = exp(
7287:for the
7015:′
5247: (
4352:for the
4291:ln(1 − 2
4227:, hence
4165:Examples
4057:'(0) ≠ 1
3951:yields
3846:tent map
3811:, then
3481:2
2868:‴
2838:″
2751:″
2672:'(a) = 1
2630:′
2598:′
2562:′
2545:″
2484:′
2377:′
2351:′
2307:′
2290:″
2229:′
2104:′
2068:′
2042:′
2025:′
1684:for all
1547:, while
1529:below.)
1466:, while
1218:) = sin(
1055:and the
1026:), then
954:m > 0
907:sequence
821:) = cos(
589:whereas
555:meaning
529:for the
474:denotes
252:, where
231:function
181:fractals
9241:Bibcode
9085:√
9026:Bibcode
8884:Bibcode
8559:Burmann
8493:
8477:
8424:
8408:
8398:
8382:
8129:'s and
7418:, then
6525:= 4 = –
6517:= 2 = –
5479:where:
5102:where:
4878:where:
4337:is the
4322:((1 − 2
4320:
4308:
4289:
4277:
4238:√
4218:√
4141:Ψ, cf.
4068:'(0) Ψ(
4042:'(0) Ψ(
4038:) = Ψ(
3499:
3467:(4) = 4
3452:√
3214:So set
3209:(2) = 2
3194:√
1691:Expand
1670:Define
1317:, both
1257:⁄
1243:⁄
1233:⁄
1107:or the
825:arccos(
416:is the
242:as the
9281:
9259:
9185:online
9160:
9073:, for
9044:
8912:
8832:
8808:
8765:
8664:
8360:§473.
8346:
8247:
8241:107384
8239:
8103:while
7983:
7769:
7766:
7757:
7716:
7574:where
6473:arccos
6399:arccos
6085:
6057:
4638:
4517:
4339:action
4326:) − 1)
4115:orange
4092:monoid
4078:where
3751:
3450:(1) =
3099:
2943:/(1 −
1456:))) =
980:. The
905:, the
753:, cf.
703:
407:where
374:
347:
301:
274:
110:
9257:S2CID
9231:arXiv
9158:S2CID
9075:any n
9042:S2CID
9016:arXiv
8910:S2CID
8874:arXiv
8489:mean
8412:= log
8386:= log
8245:S2CID
8237:JSTOR
8181:roots
8091:Notes
7189:, or
4306:) = −
4275:) = −
4263:(1 −
4259:) = 2
4204:(1 −
4200:) = 4
4127:green
4021:'(0)
3989:= 0,
3967:)) =
1706:as a
1628:)) =
1605:)) =
1516:orbit
1396:)) =
1366:'
1340:) = 2
1311:) = 4
942:) =
923:orbit
868:) = (
844:) = (
792:)) =
725:that
229:be a
62:, an
40:order
9279:ISBN
8830:ISBN
8806:ISBN
8783:link
8763:ISBN
8700:link
8662:ISBN
8418:(log
8392:(log
8370:Molk
8368:and
8344:ISBN
7909:,
7427:) =
7409:) =
7193:,
6948:The
6521:and
4177:and
4119:blue
4111:blue
4017:) =
3941:) =
3902:) =
3890:as
3876:) =
3861:) =
3815:and
3782:and
3518:) =
3496:ln 2
3477:= −1
3472:For
3221:and
3192:) =
2925:) =
2165:) =
1679:) =
1662:) =
1636:) =
1567:) =
1523:flow
1331:and
1302:and
1281:) =
1238:) =
1118:and
933:If
872:) =
852:) =
757:and
745:and
620:and
593:and
460:) =
426:and
323:and
209:Let
9249:doi
9210:doi
9150:doi
9034:doi
8941:doi
8937:224
8900:hdl
8892:doi
8545:),
8227:doi
8219:103
7453:)
7449:∂/∂
7330:exp
6556:In
6454:cos
6387:cos
4331:If
4133:.)
4123:red
3975:+1)
3798:=
3778:If
3487:ln
3226:(1)
3219:= 1
1344:− 2
1315:− 6
1300:= 2
1039:Fix
1022:is
1014:in
1002:= f
996:If
926:of
901:in
798:m n
765:of
572:)))
501:or
420:on
220::
201:set
58:In
9317::
9255:.
9247:.
9239:.
9227:43
9225:.
9206:25
9204:.
9200:.
9181:62
9156:.
9144:.
9129:^
9083:Ψ(
9081:,
9071:))
9040:.
9032:.
9024:.
9012:42
9010:.
9002:;
8987:14
8935:.
8931:.
8908:.
8898:.
8890:.
8882:.
8870:39
8868:.
8849:.
8779:}}
8775:{{
8716:;
8696:}}
8692:{{
8491:dd
8358:.
8319:^
8269:.
8243:.
8235:.
8217:.
8211:.
7467:.
7441:+
7413:+
7182:.
7180:))
6588::
6548:.
6508:+
6506:bx
6504:+
6502:ax
6428:)
5875:)
5251:)
5173:16
4364:.
4345:.
4328:.
4271:Ψ(
4247:.
4245:))
4212:Ψ(
4181:.
4161:.
4102:.
4072:))
4064:Ψ(
4046:))
3933:=
3839:○
3823:.
3806:○
3802:○
3501:.
3463:=
3445::
3401:ln
3365:ln
3334:ln
3306:ln
3211:.
3205:=
2939:=
2929:+
2927:Cx
2908:.
2674:,
2174:,
1710:,
1641:.
1579:))
1518:.
1501:=
1497:○
1490:))
1248:=
1210:,
1202::
1188::
1130:.
1111:.
1094:.
1059:.
930:.
894:.
877:.
848:)(
840:)(
831:.
829:))
769:,
738:.
733:=
624:,
472:))
456:)(
409:id
305:id
233:.
224:→
183:,
179:,
9285:.
9263:.
9251::
9243::
9233::
9218:.
9212::
9164:.
9152::
9146:3
9112:.
9108:x
9106:(
9104:f
9102:(
9098:m
9094:)
9092:x
9087:2
9069:x
9065:x
9063:(
9061:f
9048:.
9036::
9028::
9018::
8963:.
8949:.
8943::
8916:.
8902::
8894::
8886::
8876::
8853:.
8836:.
8814:.
8785:)
8771:.
8702:)
8670:.
8639:x
8635:x
8631:x
8627:x
8623:x
8619:x
8615:x
8611:x
8607:x
8603:x
8599:x
8595:x
8591:x
8587:x
8583:x
8575:x
8571:x
8555:x
8551:x
8549:(
8547:f
8543:x
8541:(
8539:f
8535:x
8531:x
8527:x
8523:d
8519:x
8515:x
8511:x
8507:x
8503:x
8499:x
8495:x
8487:x
8483:x
8479:x
8475:d
8471:A
8467:A
8463:e
8459:e
8455:e
8441:x
8437:x
8426:a
8421:b
8415:b
8410:a
8405:b
8400:a
8395:b
8389:b
8384:a
8379:b
8284:.
8251:.
8229::
8183:.
8176:x
8158:x
8142:)
8140:x
8138:(
8136:f
8113:n
8106:f
7986:.
7980:)
7977:x
7974:(
7966:+
7963:t
7959:f
7955:=
7952:)
7949:)
7946:x
7943:(
7934:f
7930:(
7925:t
7921:f
7903:v
7886:.
7883:)
7880:)
7877:x
7874:(
7869:t
7865:f
7861:(
7858:g
7855:=
7852:)
7849:)
7846:t
7843:+
7840:)
7837:x
7834:(
7831:h
7828:(
7823:1
7816:h
7812:(
7809:g
7806:=
7803:)
7800:x
7797:(
7794:g
7786:)
7783:x
7780:(
7777:h
7754:t
7750:e
7736:t
7719:.
7713:)
7710:n
7707:+
7704:)
7701:x
7698:(
7695:h
7692:(
7687:1
7680:h
7676:=
7673:)
7670:x
7667:(
7662:n
7658:f
7631:.
7628:x
7625:d
7618:)
7615:x
7612:(
7609:v
7605:1
7597:=
7594:)
7591:x
7588:(
7585:h
7559:,
7556:)
7553:1
7550:+
7547:)
7544:x
7541:(
7538:h
7535:(
7530:1
7523:h
7519:=
7516:)
7513:x
7510:(
7507:f
7490:)
7488:x
7486:(
7484:v
7479:)
7477:x
7475:(
7473:f
7461:)
7459:x
7457:(
7455:g
7451:x
7447:t
7443:t
7439:x
7437:(
7435:g
7429:t
7425:x
7423:(
7421:v
7415:t
7411:x
7407:x
7405:(
7403:f
7385:.
7382:)
7379:x
7376:(
7373:g
7369:]
7362:x
7350:)
7347:x
7344:(
7341:v
7337:[
7327:=
7324:)
7321:)
7318:x
7315:(
7312:f
7309:(
7306:g
7293:f
7289:n
7270:0
7267:=
7264:n
7259:|
7253:n
7245:)
7242:x
7239:(
7234:n
7230:f
7216:=
7213:)
7210:x
7207:(
7204:v
7178:x
7176:(
7174:f
7172:(
7170:g
7141:)
7138:y
7132:)
7129:x
7126:(
7121:1
7115:N
7111:f
7107:(
7101:+
7095:)
7092:y
7089:(
7086:f
7078:)
7075:x
7072:(
7067:1
7061:N
7057:f
7047:)
7044:)
7041:x
7038:(
7033:1
7027:N
7023:f
7019:(
7012:f
7008:=
7002:)
6999:y
6996:(
6993:f
6985:)
6982:x
6979:(
6974:N
6970:f
6928:}
6925:1
6922:,
6919:a
6916:{
6911:1
6908:+
6905:a
6899:b
6894:)
6890:}
6887:)
6884:i
6881:(
6878:g
6875:x
6872:,
6869:1
6866:+
6863:i
6860:{
6854:}
6851:x
6848:,
6845:i
6842:{
6838:(
6829:}
6825:)
6822:i
6819:(
6816:g
6811:b
6806:a
6803:=
6800:i
6792:,
6789:1
6786:+
6783:b
6779:{
6752:}
6749:0
6746:,
6743:a
6740:{
6735:1
6732:+
6729:a
6723:b
6718:)
6714:}
6711:)
6708:i
6705:(
6702:g
6699:+
6696:x
6693:,
6690:1
6687:+
6684:i
6681:{
6675:}
6672:x
6669:,
6666:i
6663:{
6659:(
6650:}
6646:)
6643:i
6640:(
6637:g
6632:b
6627:a
6624:=
6621:i
6613:,
6610:1
6607:+
6604:b
6600:{
6527:a
6523:b
6519:a
6515:b
6510:c
6482:)
6479:x
6468:n
6464:m
6460:(
6451:=
6446:n
6443:m
6439:T
6426:m
6420:(
6408:)
6405:x
6396:m
6393:(
6384:=
6381:)
6378:x
6375:(
6370:m
6366:T
6340:b
6334:1
6328:a
6323:1
6315:n
6311:a
6304:+
6299:2
6295:x
6289:n
6285:a
6259:b
6256:+
6251:2
6247:x
6243:a
6216:)
6211:b
6205:1
6199:a
6194:1
6186:n
6182:a
6175:+
6172:)
6169:x
6166:(
6163:g
6158:n
6154:a
6148:(
6141:1
6134:g
6112:)
6109:0
6106:=
6103:b
6097:1
6091:a
6088:(
6080:)
6075:b
6072:+
6069:)
6066:x
6063:(
6060:g
6054:a
6049:(
6042:1
6035:g
6009:n
6006:b
6003:+
5998:2
5994:x
5968:b
5965:+
5960:2
5956:x
5928:)
5923:b
5920:n
5917:+
5914:)
5911:x
5908:(
5905:g
5900:(
5893:1
5886:g
5857:)
5852:b
5849:+
5846:)
5843:x
5840:(
5837:g
5832:(
5825:1
5818:g
5792:)
5785:)
5780:)
5777:x
5774:(
5771:g
5766:(
5759:n
5755:h
5749:(
5742:1
5735:g
5711:)
5704:)
5699:)
5696:x
5693:(
5690:g
5685:(
5680:h
5675:(
5668:1
5661:g
5633:2
5627:c
5624:b
5621:4
5618:+
5613:2
5609:)
5605:d
5599:a
5596:(
5588:d
5585:+
5582:a
5576:=
5550:2
5544:c
5541:b
5538:4
5535:+
5530:2
5526:)
5522:d
5516:a
5513:(
5508:+
5505:d
5502:+
5499:a
5493:=
5464:]
5456:n
5448:)
5442:+
5439:a
5433:x
5430:c
5427:(
5419:n
5411:)
5405:+
5402:a
5396:x
5393:c
5390:(
5383:1
5377:n
5369:)
5363:+
5360:a
5354:x
5351:c
5348:(
5340:1
5334:n
5326:)
5320:+
5317:a
5311:x
5308:c
5305:(
5299:[
5293:c
5289:d
5286:a
5280:c
5277:b
5271:+
5266:c
5263:a
5232:d
5229:+
5226:x
5223:c
5218:b
5215:+
5212:x
5209:a
5179:4
5165:2
5161:)
5157:b
5154:+
5151:x
5148:a
5145:2
5142:(
5134:b
5131:+
5128:x
5125:a
5122:2
5116:=
5085:a
5082:2
5077:b
5067:n
5063:2
5051:2
5048:+
5041:n
5037:2
5028:2
4999:a
4996:4
4991:8
4985:b
4982:2
4974:2
4970:b
4963:+
4960:x
4957:b
4954:+
4949:2
4945:x
4941:a
4914:2
4910:b
4907:+
4904:x
4901:a
4898:2
4892:=
4861:a
4858:2
4853:b
4843:n
4839:2
4830:2
4801:a
4798:4
4793:b
4790:2
4782:2
4778:b
4771:+
4768:x
4765:b
4762:+
4757:2
4753:x
4749:a
4722:n
4718:b
4713:x
4705:1
4699:b
4694:1
4686:n
4682:b
4675:a
4653:)
4650:1
4644:b
4641:(
4633:b
4629:x
4625:a
4602:b
4596:1
4590:a
4585:1
4577:n
4573:a
4566:+
4563:x
4558:n
4554:a
4532:)
4529:1
4523:a
4520:(
4514:b
4511:+
4508:x
4505:a
4482:b
4479:n
4476:+
4473:x
4452:b
4449:+
4446:x
4423:)
4420:x
4417:(
4412:n
4408:f
4386:)
4383:x
4380:(
4377:f
4362:n
4358:n
4354:n
4334:f
4324:x
4317:2
4314:/
4311:1
4304:x
4302:(
4300:f
4295:)
4293:x
4286:2
4283:/
4280:1
4273:x
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