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On the self‐similarity problem for ergodic flows
Author(s) -
Frączek K.,
Lemańczyk M.
Publication year - 2009
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/pdp013
Subject(s) - mathematics , ergodic theory , lebesgue measure , flow (mathematics) , ergodicity , null set , measure (data warehouse) , diophantine equation , pure mathematics , irrational number , mixing (physics) , translation (biology) , similarity (geometry) , interval (graph theory) , zero (linguistics) , lebesgue integration , mathematical analysis , set (abstract data type) , combinatorics , geometry , philosophy , database , linguistics , chemistry , biochemistry , quantum mechanics , programming language , statistics , physics , messenger rna , gene , artificial intelligence , image (mathematics) , computer science
Given an ergodic flow ( T t ) t ∈ ℝ we study the problem of its self‐similarities, that is, we want to describe the set of s ∈ ℝ for which the original flow is isomorphic to the flow ( T st ) t ∈ ℝ . The problem is examined in some classes of special flows over irrational rotations and over interval exchange transformations. In particular, translation flows on translation surfaces are considered: we prove that under the weak mixing condition the set of self‐similarities has Lebesgue measure zero. For von Neumann special flows over irrational rotations given by Diophantine numbers, this set is shown to be equal to {1}, while for horocycle flows a weak convergence in case of some singular (with respect to the volume measure) measures is shown to give rise to some new equidistribution result. The problem of self‐similarity is also studied from the spectral point of view, especially in the class of Gaussian systems.