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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
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show all references

References:
[1]

Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13 (2003), 311-341. doi: 10.1142/S021812740300656X.  Google Scholar

[2]

Trans. Amer. Math. Soc., 313 (1989), 475-538. doi: 10.2307/2001417.  Google Scholar

[3]

Publ. Sec. Mat. Univ. Autònoma Barcelona, 24 (1981), 5-71.  Google Scholar

[4]

Ergodic Theory Dynam. Systems, 11 (1991), 249-271. doi: 10.1017/S0143385700006131.  Google Scholar

[5]

Topology Proc., 18 (1993), 19-31.  Google Scholar

[6]

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]

Vestnik Moskov. Univ. Ser. I Mat. Mekh., 3 (1996), 84-87.  Google Scholar

[8]

J. Math. Soc. Japan, 13 (1961), 342-345.  Google Scholar

[9]

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]

Proc. Amer. Math. Soc., 107 (1989), 549-553. doi: 10.2307/2047846.  Google Scholar

[11]

Ph.D thesis, Brigham Young University, 2010.  Google Scholar

[12]

Ergodic Theory Dynamical Systems, 2 (1982), 221-227.  Google Scholar

[13]

Mat. Zametki, 73 (2003), 466-468. Translation in Math. Notes, 73 (2003), 436-439. doi: 10.1023/A:1023234532265.  Google Scholar

[14]

Acta Arith., 66 (1994), 11-22.  Google Scholar

[15]

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]

in "Numerical Solution of Nonlinear Equations (Bremen, 1980)" 878 of Lecture Notes in Math., 351-370. Springer, Berlin, (1981).  Google Scholar

[17]

Colloq. Math., 105 (2006), 197-205. doi: 10.4064/cm105-2-3.  Google Scholar

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