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

James W. Cannon; Mark H. Meilstrup; Andreas Zastrow

doi:10.3934/dcds.2013.33.2667

Article Contents

2013, Volume 33, Issue 7: 2667-2679. Doi: 10.3934/dcds.2013.33.2667

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

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

Received: February 29, 2012

Revised: July 31, 2012

Published: June 2013

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Abstract

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 $B$.

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Mathematics Subject Classification: 37E15, 54H20, 37B10.

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