Weak* and entropy approximation of nonhyperbolic measures: a geometrical approach

TL;DR

This paper investigates nonhyperbolic measures in certain robustly transitive, partially hyperbolic dynamical systems, showing they can be approximated by hyperbolic measures with positive or negative central exponents, and analyzing entropy variation.

Contribution

It introduces a geometrical approach to approximate nonhyperbolic measures by hyperbolic ones in $C^1$-robust systems with a one-dimensional central bundle.

Findings

Nonhyperbolic measures can be approximated by hyperbolic measures with positive or negative central exponents.

Entropy varies continuously across measures with central Lyapunov exponents near zero.

Nonhyperbolic ergodic measures lie in the convex hull of hyperbolic measures with opposite signs of central exponents.

Abstract

We study $C^1$-robustly transitive and nonhyperbolic diffeomorphisms having a partially hyperbolic splitting with one-dimensional central bundle whose strong un-/stable foliations are both minimal. {In dimension $3$, an important class of examples of such systems is given by those with a simple closed periodic curve tangent to the central bundle.} We prove that there is a $C^1$-open and dense subset of such diffeomorphisms such that every nonhyperbolic ergodic measure (i.e. with zero central exponent) can be approximated in the weak$\ast$ topology and in entropy by measures supported in basic sets with positive (negative) central Lyapunov exponent. Our method also allows to show how entropy changes across measures with central Lyapunov exponent close to zero. We also prove that any nonhyperbolic ergodic measure is in the intersection of the convex hulls of the measures with positive central exponent and with negative central exponent.