3 - The variational principle for Hausdorff dimension

Summary

Introduction

The variational principle for entropy highlights the significance of measures of maximal entropy. Indeed, these are widely considered to be the most important invariant measures from a purely dynamical perspective. However, other invariant measures can be better adapted to the geometry. This is illustrated by the self-map f of [0, 1), defined by f(x) = 3x/2 for x ∈ [0, 2/3) and f(x) = 3x − 2 for x ∈ [2/3, 1). Since f is conjugate to the 2-shift, its topological entropy is log 2; Lebesgue measure is invariant under f, but does not have maximal entropy.

More generally, given an expanding self-map f of a compact manifold (which by a result of Shub must then be a torus), it is generally accepted that the measure with the greatest geometric significance among f-invariant measures is the one equivalent to Lebesgue measure. (See Krzyzewski and Szlenk (1969) for the existence of such measures.) If instead we consider an expanding map f : K → K of a compact set K of zero Lebesgue measure, the closest analogue to such a measure is an f-invariant measure of full Hausdorff dimension. This survey addresses the existence question for such measures, which is mostly open, and discusses known partial results. We sketch some of the proofs, and give references for the complete arguments (which are somewhat technical).

Consider a smooth map f : U → M, where U ⊂ M is open and M is a Riemannian manifold.