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Marches sur les arbres homogènes suivant une suite substitutive

Zhi-Xiong WenZhi-Ying Wen — 1992

Journal de théorie des nombres de Bordeaux

Ce travail consiste à étudier les comportements des marches sur les arbres homogènes suivant la suite engendrée par une substitution. Dans la première partie, on étudie d’abord les marches sans orientation sur et on détermine complètement, d’après les propriétés combinatoires de la substitution, les conditions assurant que les marches sont bornées, récurrentes ou transientes. Comme corollaire, on obtient le comportement asymptotique des sommes partielles des coefficients de la suite substitutive....

Some properties of packing measure with doubling gauge

Sheng-You WenZhi-Ying Wen — 2004

Studia Mathematica

Let g be a doubling gauge. We consider the packing measure g and the packing premeasure g in a metric space X. We first show that if g ( X ) is finite, then as a function of X, g has a kind of “outer regularity”. Then we prove that if X is complete separable, then λ s u p g ( F ) g ( B ) s u p g ( F ) for every Borel subset B of X, where the supremum is taken over all compact subsets of B having finite g -premeasure, and λ is a positive number depending only on the doubling gauge g. As an application, we show that for every doubling gauge...

Substitutions with Cofinal Fixed Points

Bo TANZhi-Xiong WENJun WUZhi-Ying WEN — 2006

Annales de l’institut Fourier

Let ϕ be a substitution over a 2-letter alphabet, say { a , b } . If ϕ ( a ) and ϕ ( b ) begin with a and b respectively, ϕ has two fixed points beginning with a and b respectively. We characterize substitutions with two cofinal fixed points (i.e., which differ only by prefixes). The proof is a combinatorial one, based on the study of repetitions of words in the fixed points.

Besicovitch subsets of self-similar sets

Ji-Hua MaZhi-Ying WenJun Wu — 2002

Annales de l’institut Fourier

Let E be a self-similar set with similarities ratio r j ( 0 j m - 1 ) and Hausdorff dimension s , let p ( p 0 , p 1 ) ... p m - 1 be a probability vector. The Besicovitch-type subset of E is defined as E ( p ) = x E : lim n 1 n k = 1 n χ j ( x k ) = p j , 0 j m - 1 , where χ j is the indicator function of the set { j } . Let α = dim H ( E ( p ) ) = dim P ( E ( p ) ) = j = 0 m - 1 p j log p j j = 0 m - 1 p i log r j and g be a gauge function, then we prove in this paper:(i) If p = ( r 0 s , r 1 s , , r m - 1 s ) , then s ( E ( p ) ) = s ( E ) , 𝒫 s ( E ( p ) ) = 𝒫 s ( E ) , moreover both of s ( E ) and 𝒫 s ( E ) are finite positive;(ii) If p is a positive probability vector other than ( r 0 s , r 1 s , , r m - 1 s ) , then the gauge functions can be partitioned as follows ...

Hankel determinants of the Thue-Morse sequence

Jean-Paul AlloucheJacques PeyrièreZhi-Xiong WenZhi-Ying Wen — 1998

Annales de l'institut Fourier

Let ϵ = ( ϵ n ) n 0 be the Thue-Morse sequence, i.e., the sequence defined by the recurrence equations: ϵ 0 = 1 , ϵ 2 n = ϵ n , ϵ 2 n + 1 = 1 - ϵ n . We consider { | n p | } n 1 , p 0 , the double sequence of Hankel determinants (modulo 2) associated with the Thue-Morse sequence. Together with three other sequences, it obeys a set of sixteen recurrence equations. It is shown to be automatic. Applications are given, namely to combinatorial properties of the Thue-Morse sequence and to the existence of certain Padé approximants of the power series n 0 ( - 1 ) ϵ n x n .

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