From: Douglas Zare Subject: Re: Cycles of continuous functions Date: Sun, 12 Sep 1999 17:35:44 -0400 Newsgroups: sci.math Keywords: Sarkovskii's Theorem on cycles for functions with f^n=identity Seraph-sama wrote: > Has any research been done on finding all the cycles of a continuous function > f:R -> R or f:C -> C? If I do any research, will I just be digging up old > results? lol, thanks. This would be one of the questions that the field of dynamical systems asks. There is a lot of interesting stuff that might be accessible, and you might try reading, e.g., Devaney's _An Introduction to Chaotic Dynamical Systems_. I don't think this is an area where one now expects results that don't build on any previous work. There might be decent expositions on the web of Sarkovskii's [multiple spellings] Theorem, which implies among other things that if f is continuous and f(x) is not x but f^5(x)=x, then f has cycles of all orders except possibly 3; a cycle of length 3 implies cycles of all lengths. Douglas Zare ============================================================================== From: "G. A. Edgar" Subject: Re: Cycles of continuous functions Date: Mon, 13 Sep 1999 07:35:25 -0400 Newsgroups: sci.math In article <37DC1CAE.6DCAEBB3@math.columbia.edu>, Douglas Zare wrote: > Seraph-sama wrote: > > > Has any research been done on finding all the cycles of a continuous > > function > > f:R -> R or f:C -> C? If I do any research, will I just be digging up old > > results? lol, thanks. > > This would be one of the questions that the field of dynamical systems asks. > There is a lot of interesting stuff that might be accessible, and you might > try > reading, e.g., Devaney's _An Introduction to Chaotic Dynamical Systems_. I > don't > think this is an area where one now expects results that don't build on any > previous work. > > There might be decent expositions on the web of Sarkovskii's [multiple > spellings] Theorem, which implies among other things that if f is continuous > and > f(x) is not x but f^5(x)=x, then f has cycles of all orders except possibly > 3; a > cycle of length 3 implies cycles of all lengths. > > Douglas Zare Sarkovskii is for functions R -> R but not for C -> C, right? -- Gerald A. Edgar edgar@math.ohio-state.edu Department of Mathematics telephone: 614-292-0395 (Office) The Ohio State University 614-292-4975 (Math. Dept.) Columbus, OH 43210 614-292-1479 (Dept. Fax) ============================================================================== From: "G. A. Edgar" Subject: Re: Cycles of continuous functions Date: Wed, 15 Sep 1999 07:45:16 -0400 Newsgroups: sci.math In article <19990914173616.02162.00000145@ng-fd1.aol.com>, Seraph-sama wrote: >That is interesting. It is fairly obvious that a 3-cycle implies a 2-cycle. Is >it considered trivial to prove that a 3-cycle implies all n-cycles? (Sorry for >the ridiculous question.) Not trivial, but well-known. There is a famous Monthly article: Li, Tien Yien; Yorke, James A. "Period three implies chaos. " Amer. Math. Monthly 82 (1975), no. 10, 985--992. It's famous because it is the first time the word "chaos" was used in the now-common mathematical sense. -- Gerald A. Edgar edgar@math.ohio-state.edu Department of Mathematics telephone: 614-292-0395 (Office) The Ohio State University 614-292-4975 (Math. Dept.) Columbus, OH 43210 614-292-1479 (Dept. Fax) ============================================================================== From: Douglas Zare Subject: Re: Cycles of continuous functions Date: Fri, 17 Sep 1999 18:13:38 -0400 Newsgroups: sci.math Seraph-sama wrote: > Douglas Zare wrote: > >[...] > >There might be decent expositions on the web of Sarkovskii's [multiple > >spellings] Theorem, which implies among other things that if f is continuous > >and > >f(x) is not x but f^5(x)=x, then f has cycles of all orders except possibly > >3; a > >cycle of length 3 implies cycles of all lengths. > > That is interesting. It is fairly obvious that a 3-cycle implies a 2-cycle. Is > it considered trivial to prove that a 3-cycle implies all n-cycles? (Sorry for > the ridiculous question.) I wouldn't say that it is trivial, but it is not hard, and a sketch follows. It is also not the full theorem. I believe that Sarkovskii's theorem would be that there is an ordering 3>5>7>9>11>... \/ 6>10>14>18>... \/ 12>20>... \/ ... ...>64>32>16>8>4>2>1 so that a>b implies that a continuous real (thanks, GAE) function with an a-cycle has a b-cycle. To prove this seems much harder to me than to prove the 3>n property. Sketch of proof of 3>n: Lemma 1: One might as well assume that f is x+1/2 on [0,1/2] and 2-2x on [1/2,1]. One can construct fixed points for any function from the fixed points of this one. Lemma 2: The above function has F(n+2)-F(n-2) points of order dividing n, where F(n) is the nth Fibonacci number (F(0)=0). This is clear from the graph, which is made of F(n+2) line segments, F(n-2) of which are only between 1/2 and 1 in either order and are over the first half of the domain; the rest will intersect the line y=x. Lemma 3: F(n+2)-F(n-2) is much larger than the sum of mu(n/m)(F(m+2)-F(m-2)) which counts the points of orders properly dividing n, where mu(n)=+-1 is the Mobius function. F(n) grows exponentially; one can check the first few terms and then use the estimate that mu<=1 and proper factors are at most n/2. (The sum was inclusion-exclusion to eliminate the points of order dividing but not equal to n.) In particular, of the at least 123 fixed points of f^10, at least 110=123-11-3+1 of these are actually of order 10, arranged in at least 11 cycles. So any real continuous function with a 3-cycle has at least 11 10-cycles. Devaney's book has a different proof for the 3>n which generalizes more easily to prove the rest of the ordering. I haven't yet read the _Monthly_ article. Does anyone know the current status of 2-d generalizations of the Sarkovskii ordering? I remember seeing a talk on that, but I forget most of the content. Douglas Zare ============================================================================== From: Douglas Zare Subject: Re: Cycles of continuous functions Date: Tue, 21 Sep 1999 19:36:11 -0400 Newsgroups: sci.math Seraph-sama wrote: > Douglas Zare wrote: > >Does anyone know the current status of 2-d generalizations of the Sarkovskii > >ordering? I remember seeing a talk on that, but I forget most of the content. > > Are you talking about complex continuous functions f:C -> C? Let n be a > positive integer and let f:z -> rotation of z about 2pi/n radians. Then f is > continuous and has ONLY n-cycles and a fixed point 0. If there were a > generalization, it would have to be much weaker, placing extra conditions on > the function or the cycles/classes. There are a number of possible types of descriptions of the dynamics on subsets, and there are many statements one can make about the implications between them. For example, that some topological circle in the plane is mapped into itself implies that there is a fixed point. Of course, if there is an interval which is mapped into itself with a point of order 3, then there is an interval which is mapped into itself with periodic points of all orders; all of the complexity of real dynamics is still there. I don't remember the types of systems the speaker considered, but if the talk was repeated elsewhere perhaps someone on sci.math would know. A search on mathscinet for Sarkovskii anywhere returns at least two articles which might have some organized partial results, but they were about a decade ago. 82i:58052 Collet, P.; Eckmann, J.-P.; Koch, H. Period doubling bifurcations for families of maps on ${R}\sp{n}$. J. Statist. Phys. 25 (1981), no. 1, 1--14. (Reviewer: Zbigniew Nitecki) 58F15 (26B15 58F14 58F20 76F99 82A05) 89k:58214 Harikrishnan, K. P.; Nandakumaran, V. M. Evidence for the existence of Sarkovski\u\i ordering in a two-dimensional map. Phys. Lett. A 133 (1988), no. 6, 305--308. 58F14 (26A18 58F13) 91f:58072 Gambaudo, Jean-Marc; van Strien, Sebastian; Tresser, Charles Vers un ordre de Sarkovski\u\i pour les plongements du disque préservant l'orientation. (French) [Toward a Sharkovskii order for $C\sp 1$ orientation-preserving embeddings of the disk] C. R. Acad. Sci. Paris Sér. I Math. 310 (1990), no. 5, 291--294. (Reviewer: Benjamin D. Mestel) 58F20 (58F08) The case of differentiable maps from R^2 to itself might be a natural restriction, as differentiability allows the description of more phenomena. Incidentally, one usually does not refer to maps from C to itself unless these are complex-differentiable or reasonably close (either meromorphic or quasiconformal). Douglas Zare