Invariant measures of full dimension for some expanding maps

1997/02/01 by DIMITRIOS GATZOURAS, D. Gatzouras, Yuval Peres +1 · 6 citations
Mathematics · #Advanced Topology and Set Theory #Mathematical Dynamics and Fractals

paper · doi:10.1017/s0143385797060987

openalex publication_date 1997/02/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/06/11

Abstract

It is an open problem to determine for which maps f, any compact invariant set K carries an ergodic invariant measure of the same Hausdorff dimension as K. If f is conformal and expanding, then it is a known consequence of the thermodynamic formalism that such measures do exist. (We give a proof here under minimal smoothness assumptions.) If f has the form f(x₁,x₂)=(f₁(x₁),f₂(x₂)), where f₁ and f₂ are conformal and expanding maps satisfying inf \vert Df₁\vert≥sup\vert Df₂\vert, then for a large class of invariant sets K, we show that ergodic invariant measures of dimension arbitrarily close to the dimension of K do exist. The proof is based on approximating K by self-affine sets.

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