On Pesin's entropy formula for dominated splittings without mixed behavior

TL;DR

This paper proves Pesin's entropy formula and Shub's entropy conjecture for certain $C^1$ diffeomorphisms with dominated splittings lacking mixed behavior, advancing understanding of entropy in dynamical systems.

Contribution

It establishes Pesin's entropy formula and confirms Shub's entropy conjecture for $C^1$ diffeomorphisms with specific dominated splittings.

Findings

Pesin's entropy formula holds for invariant measures on topological attractors with dominated splitting without mixed behavior.

Shub's entropy conjecture is verified for diffeomorphisms with such splittings.

The results extend the class of systems where entropy formulas are validated.

Abstract

For $C^1$ diffeomorphisms, we prove that the Pesin's entropy formula holds for some invariant measure supported on any topological attractor that admits a dominated splitting without mixed behavior. We also prove Shub's entropy conjecture for diffeomorphisms having such kind of splittings.