Joint ergodicity of piecewise monotone interval maps

Medientyp: E-Artikel

Titel: Joint ergodicity of piecewise monotone interval maps

Beteiligte: Bergelson, Vitaly; Son, Younghwan

Erschienen: IOP Publishing, 2023

Sprache: Nicht zu entscheiden

DOI: 10.1088/1361-6544/acd29a

ISSN: 0951-7715; 1361-6544

Schlagwörter: Applied Mathematics ; General Physics and Astronomy ; Mathematical Physics ; Statistical and Nonlinear Physics

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Beschreibung: Abstract For i = 0 , 1 , 2 , … , k , let µ i be a Borel probability measure on [ 0 , 1 ] which is equivalent to the Lebesgue measure λ and let T i : [ 0 , 1 ] → [ 0 , 1 ] be µ i -preserving ergodic transformations. We say that transformations T 0 , T 1 , … , T k are uniformly jointly ergodic with respect to ( λ ; μ 0 , μ 1 , … , μ k ) if for any f 0 , f 1 , … , f k ∈ L ∞ , lim N − M → ∞ 1 N − M ∑ n = M N − 1 f 0 ( T 0 n x ) ⋅ f 1 ( T 1 n x ) ⋯ f k ( T k n x ) = ∏ i = 0 k ∫ f i d μ i in L 2 ( λ ) . We establish convenient criteria for uniform joint ergodicity and obtain numerous applications, most of which deal with interval maps. Here is a description of one such application. Let T G denote the Gauss map, T G ( x ) = 1 x ( m o d 1 ) , and, for β > 1, let T β denote the β-transformation defined by T β x = β x ( m o d 1 ) . Let T 0 be an ergodic interval exchange transformation. Let β 1 , … , β k be distinct real numbers with β i > 1 and assume that log β i ≠ π 2 6 log 2 for all i = 1 , 2 , … , k . Then for any f 0 , f 1 , f 2 , … , f k + 1 ∈ L ∞ ( λ ) , lim N − M → ∞ 1 N − M ∑ n = M N − 1 f 0 ( T 0 n x ) ⋅ f 1 ( T β 1 n x ) ⋯ f k ( T β k n x ) ⋅ f k + 1 ( T G n x ) = ∫ f 0 d λ ⋅ ∏ i = 1 k ∫ f i d μ β i ⋅ ∫ f k + 1 d μ G in L 2 ( λ ) . We also study the phenomenon of joint mixing. Among other things we establish joint mixing for skew tent maps and for restrictions of finite Blaschke products to the unit circle.

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