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

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July 2013, 33(7): 2667-2679. doi: 10.3934/dcds.2013.33.2667

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

James W. Cannon ^1, , Mark H. Meilstrup ^2, and Andreas Zastrow ^3,

1.	Mathematics Department, Brigham Young University, Provo, UT, 84602, United States
2.	Mathematics Department, Southern Utah University, Cedar City, UT, 84720, United States
3.	Institute of Mathematics, University of Gdańsk ul. Wita Stwosza 57, PL-80952 Gdańsk, Poland

Received March 2012 Revised August 2012 Published January 2013

Abstract
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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 $B$.

Keywords: periodic points, aperiodic points, period set, backtracking., Cantor set, Sharkovsky's Theorem.

Mathematics Subject Classification: 37E15, 54H20, 37B1.

Citation: James W. Cannon, Mark H. Meilstrup, Andreas Zastrow. The period set of a map from the Cantor set to itself. Discrete and Continuous Dynamical Systems, 2013, 33 (7) : 2667-2679. doi: 10.3934/dcds.2013.33.2667

References:

[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.

show all references

References:

[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.

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