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

TL;DR

This paper introduces t-entropy as a new invariant for discrete dynamical systems and establishes a variational principle linking it to the spectral radius of weighted shift operators, akin to classical thermodynamic formalism.

Contribution

It defines t-entropy and proves it is the Legendre dual of the spectral radius logarithm, extending variational principles to weighted shift operators.

Findings

T-entropy is a new invariant for dynamical systems.

The variational principle relates t-entropy to spectral radius.

The result parallels classical thermodynamic formalism.

Abstract

In this paper we introduce a new functional invariant of discrete time dynamical systems -- the so-called t-entropy. The main result is that this t-entropy is the Legendre dual functional to the logarithm of the spectral radius of the weighted shift operator on $L^1(X,m)$ generated by the dynamical system. This result is called the Variational principle and is similar to the classical variational principle for the topological pressure.