Maximal transverse measures of expanding foliations

TL;DR

This paper introduces maximal transverse measures for expanding foliations in dynamical systems, demonstrating their convergence and relation to entropy measures, providing insights into the statistical behavior of foliation intersections.

Contribution

It constructs and analyzes maximal transverse measures for expanding foliations, establishing their convergence and connection to maximal u-entropy measures, which was previously unexplored.

Findings

Maximal transverse measures are well-defined and finite.

Convergence of the averaging process is exponential.

Maximal measures relate to maximal u-entropy measures.

Abstract

For an expanding (unstable) foliation of a diffeomorphism, we use a natural dynamical averaging to construct transverse measures, which we call \emph{maximal}, describing the statistics of how the iterates of a given leaf intersect the cross-sections to the foliation. For a suitable class of diffeomorphisms, we prove that this averaging converges, even exponentially fast, and the limit measures have finite ergodic decompositions. These results are obtained through relating the maximal transverse measures to the maximal $u$-entropy measures of the diffeomorphism.