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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

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 & 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.  Google Scholar

[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.  Google Scholar

[3]

Lluís Alsedà i Soler, Periodic points of continuous mappings of the circle, Publ. Sec. Mat. Univ. Autònoma Barcelona, 24 (1981), 5-71.  Google Scholar

[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.  Google Scholar

[5]

Stewart Baldwin, Versions of Sharkovskiĭ's theorem on trees and dendrites, Topology Proc., 18 (1993), 19-31.  Google Scholar

[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).  Google Scholar

[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.  Google Scholar

[8]

Patrick Gallagher, Approximation by reduced fractions, J. Math. Soc. Japan, 13 (1961), 342-345.  Google Scholar

[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).  Google Scholar

[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.  Google Scholar

[11]

Mark H. Meilstrup, "Wild Low-Dimensional Topology and Dynamics," Ph.D thesis, Brigham Young University, 2010.  Google Scholar

[12]

Michał Misiurewicz, Periodic points of maps of degree one of a circle, Ergodic Theory Dynamical Systems, 2 (1982), 221-227.  Google Scholar

[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.  Google Scholar

[14]

T. Pezda, Polynomial cycles in certain local domains, Acta Arith., 66 (1994), 11-22.  Google Scholar

[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.  Google Scholar

[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).  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

Lluís Alsedà i Soler, Periodic points of continuous mappings of the circle, Publ. Sec. Mat. Univ. Autònoma Barcelona, 24 (1981), 5-71.  Google Scholar

[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.  Google Scholar

[5]

Stewart Baldwin, Versions of Sharkovskiĭ's theorem on trees and dendrites, Topology Proc., 18 (1993), 19-31.  Google Scholar

[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).  Google Scholar

[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.  Google Scholar

[8]

Patrick Gallagher, Approximation by reduced fractions, J. Math. Soc. Japan, 13 (1961), 342-345.  Google Scholar

[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).  Google Scholar

[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.  Google Scholar

[11]

Mark H. Meilstrup, "Wild Low-Dimensional Topology and Dynamics," Ph.D thesis, Brigham Young University, 2010.  Google Scholar

[12]

Michał Misiurewicz, Periodic points of maps of degree one of a circle, Ergodic Theory Dynamical Systems, 2 (1982), 221-227.  Google Scholar

[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.  Google Scholar

[14]

T. Pezda, Polynomial cycles in certain local domains, Acta Arith., 66 (1994), 11-22.  Google Scholar

[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.  Google Scholar

[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).  Google Scholar

[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.  Google Scholar

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