Mixing via families for measure preserving transformations

Rui Kuang, and Xiangdong Ye. "Mixing via families for measure preserving transformations." Colloquium Mathematicae 110.1 (2008): 151-165. <http://eudml.org/doc/286097>.

@article{RuiKuang2008,
abstract = {In topological dynamics a theory of recurrence properties via (Furstenberg) families was established in the recent years. In the current paper we aim to establish a corresponding theory of ergodicity via families in measurable dynamical systems (MDS). For a family ℱ (of subsets of ℤ₊) and a MDS (X,,μ,T), several notions of ergodicity related to ℱ are introduced, and characterized via the weak topology in the induced Hilbert space L²(μ). T is ℱ-convergence ergodic of order k if for any $A₀,...,A_\{k\}$ of positive measure, $0 = e₀ < ⋯ < e_\{k\}$ and ε > 0, $\{n ∈ ℤ₊: |μ(⋂_\{i=0\}^\{k\} T^\{-ne_\{i\}\}A_\{i\}) - ∏_\{i=0\}^\{k\} μ(A_\{i\})| < ε\} ∈ ℱ$. It is proved that the following statements are equivalent: (1) T is Δ*-convergence ergodic of order 1; (2) T is strongly mixing; (3) T is Δ*-convergence ergodic of order 2. Here Δ* is the dual family of the family of difference sets.},
author = {Rui Kuang, Xiangdong Ye},
journal = {Colloquium Mathematicae},
keywords = {family; ergodicity related to a family; weak mixing; mixing; order; set of return times},
language = {eng},
number = {1},
pages = {151-165},
title = {Mixing via families for measure preserving transformations},
url = {http://eudml.org/doc/286097},
volume = {110},
year = {2008},
}

TY - JOUR
AU - Rui Kuang
AU - Xiangdong Ye
TI - Mixing via families for measure preserving transformations
JO - Colloquium Mathematicae
PY - 2008
VL - 110
IS - 1
SP - 151
EP - 165
AB - In topological dynamics a theory of recurrence properties via (Furstenberg) families was established in the recent years. In the current paper we aim to establish a corresponding theory of ergodicity via families in measurable dynamical systems (MDS). For a family ℱ (of subsets of ℤ₊) and a MDS (X,,μ,T), several notions of ergodicity related to ℱ are introduced, and characterized via the weak topology in the induced Hilbert space L²(μ). T is ℱ-convergence ergodic of order k if for any $A₀,...,A_{k}$ of positive measure, $0 = e₀ < ⋯ < e_{k}$ and ε > 0, ${n ∈ ℤ₊: |μ(⋂_{i=0}^{k} T^{-ne_{i}}A_{i}) - ∏_{i=0}^{k} μ(A_{i})| < ε} ∈ ℱ$. It is proved that the following statements are equivalent: (1) T is Δ*-convergence ergodic of order 1; (2) T is strongly mixing; (3) T is Δ*-convergence ergodic of order 2. Here Δ* is the dual family of the family of difference sets.
LA - eng
KW - family; ergodicity related to a family; weak mixing; mixing; order; set of return times
UR - http://eudml.org/doc/286097
ER -