Invariant probabilities for discrete time Linear Dynamics via Thermodynamic Formalism

Abstract

In this paper we show existence of invariant ergodic probability measures with full support associated to a suitable weighted shift L : X → X when X is a Banach space, either c 0 (R) or l p (R), 1 ≤ p < ∞. Fixing a Hölder continuous potential A, we associate to the weighted shift L a corresponding Ruelle operator L A and we prove that satisfies a generalized version of the Ruelle Perron Frobenius theorem. The invariant ergodic probability measure for L will be obtained in this way. These results are extended to a wide class of linear operators with non-trivial kernel defined on a general Banach space.

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

“…For the discrete-time dynamical action of a bounded linear operator T : X → X on a Banach space X, the paper [GM14] present results about the existence of T -invariant probability measures with full support in the case that X is reflexive and separable. An extension of this result for a more general setting, including non-reflexive separable Banach spaces, was presented in [LMSV19]. This was obtained via the classical tool in Thermodynamic Formalism known as Ruelle-Perron-Frobenius Theorem.…”

“…When the potential is of Hölder class several nice properties can be derived for its corresponding Gibbs states (see for instance [PP90]). In [LMSV19] the authors show the existence of T -invariant probabilities with full support using the Gibbs state point of view.…”

“…Questions of topological nature as expansivity, shadowing, transitivity, and structural stability in this linear setting were addressed in [BCD + 18, BM20, Gil20]. We refer the reader to the first part of [LMSV19] for a short account of some basic definitions and results concerning the theory of Linear Dynamics on vector spaces of infinite dimension.…”

“…It was assumed that the kernel of the bounded linear operator L is nontrivial and of finite dimension. In order to define this operator, it was necessary in [LMSV19] to fix an a priori probability on the kernel of T with some properties.…”

“…For most of the reasoning of the present paper. we consider similar assumptions as in [LMSV19]. We introduce the concepts of entropy and pressure in the setting of Linear Dynamics on vector spaces of infinite dimension.…”