Article Contents
Article Contents

# The period set of a map from the Cantor set to itself

• In this paper we consider all possible period sets $P(f)$ for self-maps of the Cantor set, $f:C\to C$. We prove that the possible period sets are completely unrestricted provided that, in addition, one allows points that are not periodic. However, if every point is periodic, we show that a surprising finiteness condition is imposed on $P(f)$: namely, there is a finite subset $B$ of $P(f)$ such that every element of $P(f)$ is divisible by at least one element of $B$.
Mathematics Subject Classification: 37E15, 54H20, 37B10.

 Citation:

•  [1] Ll. Alsedà, D. Juher and P. Mumbrú, Sets of periods for piecewise monotone tree maps, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13 (2003), 311-341.doi: 10.1142/S021812740300656X. [2] Lluís Alsedà, Jaume Llibre and Michał Misiurewicz, Periodic orbits of maps of $Y$, Trans. Amer. Math. Soc., 313 (1989), 475-538.doi: 10.2307/2001417. [3] Lluís Alsedà i Soler, Periodic points of continuous mappings of the circle, Publ. Sec. Mat. Univ. Autònoma Barcelona, 24 (1981), 5-71. [4] Stewart Baldwin, An extension of Šarkovskiĭ's theorem to the $n-od$, Ergodic Theory Dynam. Systems, 11 (1991), 249-271.doi: 10.1017/S0143385700006131. [5] Stewart Baldwin, Versions of Sharkovskiĭ's theorem on trees and dendrites, Topology Proc., 18 (1993), 19-31. [6] Louis Block, John Guckenheimer, Michał Misiurewicz and Lai Sang Young, Periodic points and topological entropy of one-dimensional maps, in "Global Theory of Dynamical Systems (Proc. Internat. Conf., Northwestern Univ., Evanston, Ill., 1979)" 819 of Lecture Notes in Math., 18-34. Springer, Berlin, (1980). [7] A. I. Demin, Coexistence of periodic, almost periodic and recurrent points of transformations of $n-od$, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 3 (1996), 84-87. [8] Patrick Gallagher, Approximation by reduced fractions, J. Math. Soc. Japan, 13 (1961), 342-345. [9] Christian Gillot and Jaume Llibre, Periods for maps of the figure-eight space, in "Thirty Years After Sharkovskiĭ's Theorem: New Perspectives (Murcia, 1994)" 8 of World Sci. Ser. Nonlinear Sci. Ser. B Spec. Theme Issues Proc., 95-106. World Sci. Publ., River Edge, NJ, (1995). Reprint of the paper reviewed in MR1361924 (97d:58161). [10] W. T. Ingram, Periodic points for homeomorphisms of hereditarily decomposable chainable continua, Proc. Amer. Math. Soc., 107 (1989), 549-553.doi: 10.2307/2047846. [11] Mark H. Meilstrup, "Wild Low-Dimensional Topology and Dynamics," Ph.D thesis, Brigham Young University, 2010. [12] Michał Misiurewicz, Periodic points of maps of degree one of a circle, Ergodic Theory Dynamical Systems, 2 (1982), 221-227. [13] E. Mochko, V. V. Nekrashevich and V. I. Sushchanskiĭ, Dynamics of triangular transformations of sequences over finite alphabets, Mat. Zametki, 73 (2003), 466-468. Translation in Math. Notes, 73 (2003), 436-439.doi: 10.1023/A:1023234532265. [14] T. Pezda, Polynomial cycles in certain local domains, Acta Arith., 66 (1994), 11-22. [15] A. N. Sharkovskiĭ, Coexistence of cycles of a continuous map of the line into itself, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 5 (1995), 1263-1273. Translated from the Russian Ukrain. Mat. Zh., 16 (1964), 61-71 by J. Tolosa.doi: 10.1142/S0218127495000934. [16] H. W. Siegberg, Chaotic mappings on $S^1$, periods one, two, three imply chaos on $S^1$, in "Numerical Solution of Nonlinear Equations (Bremen, 1980)" 878 of Lecture Notes in Math., 351-370. Springer, Berlin, (1981). [17] V. I. Sushchanski, E. Moćko and V. V. Nekrashevych, Cycles of distance-decreasing mappings in the ring of $n$-adic integers, Colloq. Math., 105 (2006), 197-205.doi: 10.4064/cm105-2-3.