Entropy, pressure, ground states and calibrated sub-actions for linear dynamics

TL;DR

This paper explores entropy, pressure, ground states, and calibrated sub-actions for linear dynamics on Banach spaces, providing existence results, examples, and explicit constructions under certain regularity conditions.

Contribution

It establishes the existence of ground states and calibrated sub-actions for linear operators on Banach spaces with Schauder bases, including explicit examples and constructions.

Findings

Existence of ground states as temperature approaches zero.

Construction of calibrated sub-actions for H"older potentials.

Examples involving weighted shift operators on Banach spaces.

Abstract

Denote by $X$ a Banach space and by $T : X \to X$ a bounded linear operator with non-trivial kernel satisfying suitable conditions. We consider the concepts of entropy - for $T$-invariant probability measures - and pressure for H\"older continuous potentials. We also prove the existence of ground states (the limit when temperature goes to zero) associated with such class of potentials when the Banach space $X$ is equipped with a Schauder basis. We produce an example concerning weighted shift operators defined on the Banach spaces $c_0(\mathbb{R})$ and $l^p(\mathbb{R})$, $1 \leq p < +\infty$, where our results do apply. In addition, we prove the existence of calibrated sub-actions when the potential satisfies certain regularity conditions using properties of the so-called Ma\~n\'e potential. We also exhibit examples of selection at zero temperature and explicit sub-actions in the class of H\"older continuous potentials.