# American Institute of Mathematical Sciences

November  2014, 34(11): 4459-4486. doi: 10.3934/dcds.2014.34.4459

## Topological and ergodic properties of symmetric sub-shifts

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

Received  July 2013 Revised  February 2014 Published  May 2014

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.
Citation: Rafael Alcaraz Barrera. Topological and ergodic properties of symmetric sub-shifts. Discrete & Continuous Dynamical Systems, 2014, 34 (11) : 4459-4486. doi: 10.3934/dcds.2014.34.4459
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