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Metric, Fractal Dimensional and Baire Results on the Distribution of Subsequences
Author(s) -
Goldstern Martin,
Schmeling Jörg,
Winkler Reinhard
Publication year - 2000
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/1522-2616(200011)219:1<97::aid-mana97>3.0.co;2-6
Subject(s) - mathematics , subsequence , unit interval , baire category theorem , baire measure , combinatorics , hausdorff dimension , sequence (biology) , baire space , metric space , lebesgue measure , distribution (mathematics) , borel set , compact space , discrete mathematics , hausdorff distance , lebesgue integration , pure mathematics , mathematical analysis , biology , bounded function , genetics
Abstract Let X be a locally compact metric space. One important object connected with the distribution behavior of an arbitrary sequence x on X is the set M ( x ) of limit measures of x . It is defined as the set of accumulation points of the sequence of the discrete measures induced by x . Using binary representation of reals one gets a natural bijective correspondence between infinite subsets of the set ℕ of positive integers and numbers in the unit interval I = 〈0, 1]. Hence to each sequence x = ( x n ) n ∈ℕ ∈ X ℕ and every a I there corresponds a subsequence denoted by a x . We investigate the set M ( a x ) for given x with emphasis on the behavior for “typical” a in the sense of Baire category, Lebesgue measure and Hausdorff dimension.