Topological and ergodic properties of symmetric sub-shifts

Rafael Alcaraz Barrera

doi:10.3934/dcds.2014.34.4459

Article Contents

2014, Volume 34, Issue 11: 4459-4486. Doi: 10.3934/dcds.2014.34.4459

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Topological and ergodic properties of symmetric sub-shifts

Rafael Alcaraz Barrera^1,

1.
School of Mathematics, The University of Manchester, Oxford Road, Manchester, M13 9PL

Received: June 30, 2013

Revised: January 31, 2014

Published: October 2014

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Abstract

The family of symmetric one sided sub-shifts in two symbols given by a sequence $a$ is studied. We analyse some of their topological properties such as transitivity, the specification property and intrinsic ergodicity. It is shown that almost every member of this family admits only one measure of maximal entropy. It is shown that the same results hold for attractors of the family of open dynamical systems arising from the doubling map with a centred symmetric hole depending on one parameter, and for the set of points that have unique $\beta$-expansion for $\beta \in (\varphi,2)$ where $\varphi$ is the Golden Ratio.

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Mathematics Subject Classification: Primary: 37B10; Secondary: 28D05, 37C70, 68R15.

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