Invariant probabilities for discrete time Linear Dynamics via Thermodynamic Formalism

Artur O. Lopes, Ali Messaoudi, Victor Vargas

Published 2019-10-10Version 1

We denote by $X$ a real vector space that can be either $ l^p(\mathbb{R}),\; 1 \leq p < \infty,$ or $c_0(\mathbb{R})$. We consider a certain class of linear maps $L : X \to X$ (called weighted shift operators). We fix a suitable {\it a priori} probability measure $m$ on the kernel of $L$ in order to introduce the Ruelle operator $\mathcal{L}_A$ associated to a bounded H\"older continuous potential $A: X \to \mathbb{R}$. We adapt the Thermodynamic Formalism for generalized $XY$ models (in the case the alphabet is non-compact) to the present setting. We are able to show the existence of eigenvalues, eigenfunctions and eigenprobabilities for $\mathcal{L}_A$. Moreover, we show the existence of $L$-invariant ergodic probabilities with full support in a similar manner as in Classical Thermodynamic Formalism. Finally, these results are extended to a large class of linear operators defined on Banach spaces.