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

A. A. Tempelman

Research output: Contribution to journalArticle

9 Citations (Scopus)

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

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Ergodic Averages
Multifractal Analysis
Finite Set
Continuous Function
Semigroup
Product topology
Metric
Hausdorff Dimension
Theorem
Function Space
Regular hexahedron
Averaging
Countable
Banach spaces
Banach space
Generalise
Evaluate
Topology
Generalization
Model

All Science Journal Classification (ASJC) codes

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

Cite this

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title = "Multifractal analysis of ergodic averages: A generalization of eggleston's theorem",
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 F{\o}lner {"}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) f1(τsx) = = α1, ..., lim n→∞ 1/|An| ∑s∈A(n) fm(τsx) = 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].",
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Multifractal analysis of ergodic averages : A generalization of eggleston's theorem. / Tempelman, A. A.

In: Journal of Dynamical and Control Systems, Vol. 7, No. 4, 01.10.2001, p. 535-551.

Research output: Contribution to journalArticle

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