Ergodic limits on the conformal repellers

Abstract

Let J be the repeller of an expanding, C1+δ-conformal topological mixing map g. Let Φ: J → ℝd be a continuous function and let α(x) = limn→∞1/n∑j=0n-1 Φ(gjx) (when the limit exists) be the ergodic limit. It is known that the possible α(x) are just the values ∫ Φ dμ for all g-invariant measures μ. For any α in the range of the ergodic limits, we prove the following variational formula: ?? where μ is a g-invariant Borel probability measure on J, hg(μ) is the entropy of μ, ∥Dxg∥ is the operator norm of the differential Dxg, and dim is the Hausdorff dimension or the packing dimension. This result gives a substantial extension of the well-known case that Φ is Hölder continuous. We also prove that unless the same ergodic limit exists everywhere, the set of points whose ergodic limit does not exist has the same Hausdorff dimension as the whole space J. © 2002 Elsevier Science (USA).