Multifractal analysis of ergodic averages: A generalization of eggleston's theorem

A. A. Tempelman

Research output: Contribution to journalArticle

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Abstract

Let V be a finite set, S be an infinite countable commutative semigroup, {τs, s ∈ S} be the semigroup of translations in the function space X = VS, A = {An} be a sequence of finite sets in S, f be a continuous function on X with values in a separable real Banach space B, and let α ∈ B. We introduce in X a "scale metric" generating the product topology. Under some assumptions on f and A, we evaluate the Hausdorff dimension of the set Xα,f,A defined by the following formula: Xα,f,A = {x : x ∈ X, limn→∞ 1/|An| ∑s∈A(n) f(τsx) = α}. It turns out that this dimension does not depend on the choice of a Følner "pointwise averaging" sequence A and is completely specified by the "scale index" of the metric in X. This general model includes the important special cases where S = ℤd or ℤ+d, d ≥ 1, and the sets An are infinitely increasing cubes; if B = ℝm then f(x) = (f1(x), ..., fm(x)), α = (α1, ..., αm), and Xα,f,A = {x : x ∈ X, limn→∞ 1/|An| ∑s∈A(n) f1sx) = = α1, ..., lim n→∞ 1/|An| ∑s∈A(n) fmsx) = αm}. Thus the multifractal analysis of the ergodic averages of several continuous functions is a special case of our results; in particular, in Examples 4 and 5 we generalize the well-known theorems due to Eggleston [3] and Billingsley [1].

Original languageEnglish (US)
Pages (from-to)535-551
Number of pages17
JournalJournal of Dynamical and Control Systems
Volume7
Issue number4
DOIs
StatePublished - Oct 1 2001

All Science Journal Classification (ASJC) codes

  • Control and Systems Engineering
  • Algebra and Number Theory
  • Numerical Analysis
  • Control and Optimization

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