Results 1  10
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29
The Dimensions of Individual Strings and Sequences
 INFORMATION AND COMPUTATION
, 2003
"... A constructive version of Hausdorff dimension is developed using constructive supergales, which are betting strategies that generalize the constructive supermartingales used in the theory of individual random sequences. This constructive dimension is used to assign every individual (infinite, binary ..."
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Cited by 99 (11 self)
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A constructive version of Hausdorff dimension is developed using constructive supergales, which are betting strategies that generalize the constructive supermartingales used in the theory of individual random sequences. This constructive dimension is used to assign every individual (infinite, binary) sequence S a dimension, which is a real number dim(S) in the interval [0, 1]. Sequences that
On the distribution of longterm time averages on symbolic space
 J. Statist. Phys
"... The pressure was studied in a rather abstract theory as an important notion of the thermodynamic formalism. The present paper gives a more concrete account in the case of symbolic spaces, including subshifts of finite type. We relate the pressure of an interaction function 8 to its longterm time av ..."
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Cited by 27 (12 self)
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The pressure was studied in a rather abstract theory as an important notion of the thermodynamic formalism. The present paper gives a more concrete account in the case of symbolic spaces, including subshifts of finite type. We relate the pressure of an interaction function 8 to its longterm time averages through the Hausdorff and packing dimensions of the subsets on which 8 has prescribed longterm timeaverage values. Functions 8 with values in Rd are considered. For those 8 depending only on finitely many symbols, we get complete results, unifying and completing many partial results.
Generic points in systems of specification and Banach valued Birkhoff ergodic average
 DISCRETE CONTIN DYN. SYST
"... We prove that systems satisfying the specification property are saturated in the sense that the topological entropy of the set of generic points of any invariant measure is equal to the measuretheoretic entropy of the measure. We study Banach valued Birkhoff ergodic averages and obtain a variationa ..."
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Cited by 22 (11 self)
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We prove that systems satisfying the specification property are saturated in the sense that the topological entropy of the set of generic points of any invariant measure is equal to the measuretheoretic entropy of the measure. We study Banach valued Birkhoff ergodic averages and obtain a variational principle for its topological entropy spectrum. As application, we examine a particular example concerning with the set of real numbers for which the frequencies of occurrences in their dyadic expansions of infinitely many words are prescribed. This relies on our explicit determination of a maximal entropy measure.
Distribution of Frequencies of Digits Via Multifractal Analysis
"... We study the Hausdorff dimension of a large class of sets in the real line defined in terms of the distribution of frequencies of digits for the representation in some integer base. In particular, our results unify and extend classical work of Borel, Besicovitch, Eggleston, and Billingsley in severa ..."
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Cited by 14 (4 self)
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We study the Hausdorff dimension of a large class of sets in the real line defined in terms of the distribution of frequencies of digits for the representation in some integer base. In particular, our results unify and extend classical work of Borel, Besicovitch, Eggleston, and Billingsley in several directions. Our methods are based on recent results concerning the multifractal analysis of dynamical systems and often allow us to obtain explicit expressions for the Hausdorff dimension.
ORTHOGONAL EXPONENTIALS, TRANSLATIONS, AND BOHR COMPLETIONS
, 2009
"... We are concerned with an harmonic analysis in Hilbert spaces L²(µ), where µ is a probability measure on R n. The unifying question is the presence of families of orthogonal (complex) exponentials eλ(x) = exp(2πiλx) in L²(µ). This question in turn is connected to the existence of a natural embeddin ..."
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Cited by 14 (10 self)
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We are concerned with an harmonic analysis in Hilbert spaces L²(µ), where µ is a probability measure on R n. The unifying question is the presence of families of orthogonal (complex) exponentials eλ(x) = exp(2πiλx) in L²(µ). This question in turn is connected to the existence of a natural embedding of L²(µ) into an L²space of Bohr almost periodic functions on R n. In particular we explore when L²(µ) contains an orthogonal basis of eλ functions, for λ in a suitable discrete subset in R n; i.e, when the measure µ is spectral. We give a new characterization of finite spectral sets in terms of the existence of a group of local translation. We also consider measures µ that arise as fixed points (in the sense of Hutchinson) of iterated function systems (IFSs), and we specialize to the case when the function system in the IFS consists of affine and contractive mappings in R n. We show in this case that if µ is then assumed spectral then its partitions induced by the IFS at hand have zero overlap measured in µ. This solves part of the ̷LabaWang conjecture. As an application of the new nonoverlap result, we solve the spectralpair problem for Bernoulli convolutions advancing in this way a theorem of KaSing Lau. In addition we present a new perspective on spectral measures and orthogonal Fourier exponentials via the Bohr compactification.
Dimension spectrum for a nonconventional ergodic average
 Real Anal. Ex
"... Abstract. We compute the dimension spectrum of certain nonconventional averages, namely, the Hausdorff dimension of the set of 0, 1 sequences, for which the frequency of the pattern 11 in positions k, 2k equals a given number θ ∈ [0, 1]. 1. Introduction. For a dynamical system (X,T) (say, a continuo ..."
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Cited by 9 (3 self)
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Abstract. We compute the dimension spectrum of certain nonconventional averages, namely, the Hausdorff dimension of the set of 0, 1 sequences, for which the frequency of the pattern 11 in positions k, 2k equals a given number θ ∈ [0, 1]. 1. Introduction. For a dynamical system (X,T) (say, a continuous selfmap of a compact metric space), the dimension spectrum of ordinary Birkhoff averages is defined as the function θ 7 → dimH
A nonconventional strong law of large numbers and fractal dimensions of some multiple recurrence sets
 Stoch. Dynam
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SINGULAR PROBABILITY DISTRIBUTIONS AND FRACTAL PROPERTIES OF SETS OF REAL NUMBERS DEFINED BY THE ASYMPTOTIC FREQUENCIES OF THEIR SADIC DIGITS
, 2006
"... Properties of the set Ts of ”particularly nonnormal numbers ” of the unit interval are studied in details (Ts consists of real numbers x, some of whose sadic digits have the asymptotic frequencies in the nonterminating s − adic expansion of x, and some do not). It is proven that the set Ts is resi ..."
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Cited by 9 (6 self)
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Properties of the set Ts of ”particularly nonnormal numbers ” of the unit interval are studied in details (Ts consists of real numbers x, some of whose sadic digits have the asymptotic frequencies in the nonterminating s − adic expansion of x, and some do not). It is proven that the set Ts is residual in the topological sense (i.e., it is of the first Baire category) and it is generic in the sense of fractal geometry (Ts is a superfractal set, i.e., its HausdorffBesicovitch dimension is equal to 1). A topological and fractal classification of sets of real numbers via analysis of asymptotic frequencies of digits in their sadic expansions is presented.
Ubiquity and large intersections properties under digit frequencies constraints
, 2009
"... digit frequencies constraints ..."
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Hyperbolicity And Recurrence In Dynamical Systems: A Survey Of Recent Results
, 2002
"... We discuss selected topics of current research interest in the theory of dynamical systems, with emphasis on dimension theory, multifractal analysis, and quantitative recurrence. The topics include the quantitative versus the qualitative behavior of Poincare recurrence, the product structure of inva ..."
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Cited by 8 (4 self)
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We discuss selected topics of current research interest in the theory of dynamical systems, with emphasis on dimension theory, multifractal analysis, and quantitative recurrence. The topics include the quantitative versus the qualitative behavior of Poincare recurrence, the product structure of invariant measures and return times, the dimension of invariant sets and invariant measures, the complexity of the level sets of local quantities from the point of view of Hausdorff dimension, and the conditional variational principles as well as their applications to problems in number theory.