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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
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
\[ V:=\lim _{t\to \infty }\frac {1}{t}\log (h_{t}). \]
$\mu $, when the temperature goes to zero (or
$t\to \infty $), we mean the existence of the limit (in the weak* sense)
\[\mu :=\lim _{t\to \infty } \mu _t.\]
$f$ 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.
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- Research Article
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- Copyright © 2012 Cambridge University Press
References
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