Spectral Properties of the Ruelle Operator for Product Type Potentials on Shift Spaces

TL;DR

This paper analyzes the spectral properties of the Ruelle operator for product type potentials on shift spaces, providing explicit formulas for eigenvalues, eigenfunctions, and eigenmeasures, and exploring their uniqueness and existence under various conditions.

Contribution

It derives explicit spectral formulas for the Ruelle operator associated with product type potentials and investigates the existence and uniqueness of eigenfunctions and eigenmeasures.

Findings

Explicit formulas for eigenvalues, eigenfunctions, and eigenmeasures.

Existence of a Bernoulli equilibrium state even without Bowen's condition.

Presence of multiple unbounded eigenfunctions for certain potentials.

Abstract

We study a class of potentials $f$ on one sided full shift spaces over finite or countable alphabets, called potentials of product type. We obtain explicit formulae for the leading eigenvalue, the eigenfunction (which may be discontinuous) and the eigenmeasure of the Ruelle operator. The uniqueness property of these quantities is also discussed and it is shown that there always exists a Bernoulli equilibrium state even if $f$ does not satisfy Bowen's condition. We apply these results to potentials $f:\{-1,1\}^\mathbb{N} \to \mathbb{R}$ of the form $$ f(x_1,x_2,\ldots) = x_1 + 2^{-\gamma} \, x_2 + 3^{-\gamma} \, x_3 + ...+n^{-\gamma} \, x_n + \ldots $$ with $\gamma >1$. For $3/2 < \gamma \leq 2$, we obtain the existence of two different eigenfunctions. Both functions are (locally) unbounded and exist a.s. (but not everywhere) with respect to the eigenmeasure and the measure of maximal entropy, respectively.