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Statistical Properties of Endomorphisms and Compact Group Extensions
Author(s) -
Melbourne Ian,
Nicol Matthew
Publication year - 2004
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610704005587
Subject(s) - mathematics , endomorphism , equivariant map , law of the iterated logarithm , compact group , pure mathematics , operator (biology) , ergodicity , lift (data mining) , iterated function , mixing (physics) , group (periodic table) , piecewise , mathematical analysis , logarithm , lie group , biochemistry , chemistry , statistics , physics , organic chemistry , repressor , quantum mechanics , computer science , transcription factor , gene , data mining
The statistical properties of endomorphisms under the assumption that the associated Perron–Frobenius operator is quasicompact are considered. In particular, the central limit theorem, weak invariance principle and law of the iterated logarithm for sufficiently regular observations are examined. The approach clarifies the role of the usual assumptions of ergodicity, weak mixing, and exactness. Sufficient conditions are given for quasicompactness of the Perron–Frobenius operator to lift to the corresponding equivariant operator on a compact group extension of the base. This leads to statistical limit theorems for equivariant observations on compact group extensions. Examples considered include compact group extensions of piecewise uniformly expanding maps (for example Lasota–Yorke maps), and subshifts of finite type, as well as systems that are nonuniformly expanding or nonuniformly hyperbolic.