Ergodic Limits on the Conformal Repellers
By De-Jun Feng, Ka-Sing Lau, Jun Wu. 2002
122 citations recorded by the source corpus.
Abstract Let J be the repeller of an expanding, C1+δ-conformal topological mixing map g. Let Φ:J→ R d be a continuous function and let α(x)=limn→∞ 1 n ∑n−1j=0Φ(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: dim x∈J:α(x)=α= max μ h g (μ) ∫log∥D x g∥dμ(x) :∫Φ dμ=α , 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 Holder 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.
Subfields: Analytical Chemistry, Cell Biology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/2da29f53b38b33d2f2244d7b7e9e770cb3fbf79e
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Analytical Chemistry
In Cell Biology
In Mathematical Physics
Citations in the corpus
- Multifractal analysis of Birkhoff averages for countable Markov maps (2010, background): “Feng et al [9], then studied the case of continuous potentials for conformal expanding maps.”
- Lyapunov exponents for products of matrices and multifractal analysis. Part II: General matrices (2003, background): “Theorem A extends the corresponding results for the Birkho averages of scalar functions, for which one is referred to, e.g., [1, 2, 3, 9, 10, 16 , 35, 38, 40, 41, 47].”
- The limited Rademacher functions and Bernoulli convolutions associated with Pisot numbers (2005, background): “For some applications of homogeneous Moran sets in the dimension theory of dynamical systems, see [10,11,17,50].”
- Lyapunov Spectrum of Asymptotically Sub-additive Potentials (2009, background): “average of f. In this case, the topological entropy (or the Hausdor dimension) of the level sets of Birkho averages has been extensively studied in the recent two decades (see, e.g., [13, 7, 40, 9, 20, 21, 33, 27 , 8, 41, 46, 23, 16, 22, 1, 2, 28] and references therein).”