Entropy formula for systems with inducing schemes

TL;DR

This paper derives entropy formulas for SRB measures in dynamical systems with inducing schemes, extending classical results to systems with singularities and providing new examples where traditional inequalities fail.

Contribution

It establishes Pesin entropy formulas for noninvertible and invertible systems with SRB measures via inducing schemes, including systems with singularities.

Findings

Derived entropy formulas for SRB measures using inducing schemes.

Extended Pesin entropy formula to systems with singular sets.

Provided examples where Ruelle inequality does not hold.

Abstract

We obtain entropy formulas for SRB measures with finite entropy given by inducing schemes. In the first part of the work, we obtain Pesin entropy formula for the class of noninvertible systems whose SRB measures are given by Gibbs-Markov induced maps. In the second part, we obtain Pesin entropy formula for invertible maps whose SRB measures given by Young sets, taking into account a classical compression technique along the stable direction that allows a reduction of the return map associated with a Young set to a Gibbs-Markov map. In both cases, we give applications of our main results to several classes of dynamical systems with singular sets, where the classical results by Ruelle and Pesin cannot be applied. We also present examples of systems with SRB measures given by inducing schemes for which Ruelle inequality does not hold.