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On the selection of subaction and measure for a subclass of potentials defined by P. Walters

Published online by Cambridge University Press:  04 July 2012

A. T. BARAVIERA ,
A. O. LOPES  and
J. K. MENGUE
A. T. BARAVIERA
Affiliation:
Instituto de Matemática, Universidade Federal do Rio Grande do Sul, Avenida Bento Gonçalves 9500, Porto Alegre, RS, Brazil (email: [email protected], [email protected], [email protected])
A. O. LOPES
Affiliation:
Instituto de Matemática, Universidade Federal do Rio Grande do Sul, Avenida Bento Gonçalves 9500, Porto Alegre, RS, Brazil (email: [email protected], [email protected], [email protected])
J. K. MENGUE
Affiliation:
Instituto de Matemática, Universidade Federal do Rio Grande do Sul, Avenida Bento Gonçalves 9500, Porto Alegre, RS, Brazil (email: [email protected], [email protected], [email protected])
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Abstract

Suppose $\sigma $ is the shift acting on Bernoulli space $X=\{0,1\}^{\mathbb {N}}$, and consider a fixed function $f:X \to \mathbb {R}$ satisfying the Walters conditions (defined in [P. Walters. A natural space of functions for the Ruelle operator theorem. Ergod. Th. & Dynam. Sys.27 (2007), 1323–1348]). For each real value $t\geq 0$ we consider the Ruelle operator $L_{\mathit {tf}}$. We are interested in the main eigenfunction $h_t$ of $L_{\mathit {tf}}$ and the main eigenmeasure $\nu _t$ for the dual operator $L_{\mathit {tf}}^*$, which we consider normalized in such a way that $h_t(0^\infty )=1$ and $\int h_t \,d\nu _t=1$ for all $t\gt 0$. We denote by $\mu _t= h_t \nu _t$ the Gibbs state for the potential $\mathit {tf}$. By the selection of a subaction $V$, when the temperature goes to zero (or $t\to \infty $), we mean the existence of the limit

\[ V:=\lim _{t\to \infty }\frac {1}{t}\log (h_{t}). \]
By the selection of a measure $\mu $, when the temperature goes to zero (or $t\to \infty $), we mean the existence of the limit (in the weak* sense)
\[\mu :=\lim _{t\to \infty } \mu _t.\]
 We present a large family of non-trivial examples of $f$ where the selection of a measure exists. These $f$ belong to a sub-class of potentials introduced by Walters. In this case, explicit expressions for the selected $V$can be obtained for a certain large family of parameters.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 33 , Issue 5 , October 2013 , pp. 1338 - 1362
Copyright
Copyright © 2012 Cambridge University Press 

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