A dimension associated with a cutting of the square of a Gibbs measure

Abstract

Given a compact Riemann manifold M. let F:M→M be a diffeomorphism and let μ be an F invariant ergodic measure. In [6] (Ledrappier and Young, 1985), Ledrappier and Young have proved that μ is exact dimensional. We propose to give a direct proof of this result when μ is a Gibbs measure, defined on a symbolic space product ∑r1×∑r2 with 2⩽r1<r2 integers, and invariant by the shift.