T-entropy and Variational Principle for the spectral radius of transfer and weighted shift operators

TL;DR

This paper extends the variational principle for spectral radii of transfer and weighted shift operators to arbitrary dynamical systems, introducing t-entropy as a new invariant replacing Kolmogorov--Sinai entropy.

Contribution

It derives a general variational principle involving t-entropy for spectral radii, broadening previous results limited to reversible systems or topological Markov chains.

Findings

Introduces t-entropy as a new invariant of entropy type.

Provides explicit description of the Legendre dual to the spectral potential.

Generalizes variational principles to all dynamical systems.

Abstract

The paper deals with the variational principles for evaluation of the spectral radii of transfer and weighted shift operators associated with a dynamical system. These variational principles have been the matter of numerous investigations and the principal results have been achieved in the situation when the dynamical system is either reversible or it is a topological Markov chain. As the main summands these principles contain the integrals over invariant measures and the Kolmogorov--Sinai entropy. In the article we derive the Variational Principle for an arbitrary dynamical system. It gives the explicit description of the Legendre dual object to the spectral potential. It is shown that in general this principle contains not the Kolmogorov--Sinai entropy but a new invariant of entropy type -- the t-entropy.