Uniform Convergence of Cesaro Averages for Uniquely Ergodic C*-Dynamical Systems

The Uniform Ergodic Theorem for Dynamical Systems

Necessary and sufficient conditions are given for the uniform convergence over an arbitrary index set in von Neumann’s mean and Birkhoff’s pointwise ergodic theorem. Three different types of conditions already known from probability theory are investigated. Firstly it is shown that the property of being eventually totally bounded in the mean is necessary and sufficient. This condition involves ...

Convergence of Diagonal Ergodic Averages

The case l = 1 is the mean ergodic theorem, and the result can be viewed as a generalization of that theorem. The l = 2 case was proven by Conze and Lesigne [Conze and Lesigne, 1984], and various special cases for higher l have been shown by Zhang [Zhang, 1996], Frantzikinakis and Kra [Frantzikinakis and Kra, 2005], Lesigne [Lesigne, 1993], and Host and Kra [Host and Kra, 2005]. Tao’s argument ...

Existence of Non-uniform Cocycles on Uniquely Ergodic Systems

Computing the speed of convergence of ergodic averages and pseudorandom points in computable dynamical systems

A pseudorandom point in an ergodic dynamical system over a computable metric space is a point which is computable but its dynamics has the same statistical behavior of a typical point of the system. It was proved in [2] that in a system whose dynamics is computable the ergodic averages of computable observables converge effectively. We give an alternative, simpler proof of this result. This imp...

Pointwise Convergence of Some Multiple Ergodic Averages

We show that for every ergodic system (X, μ,T1, . . . ,Td) with commuting transformations, the average 1 Nd+1 ∑ 0≤n1,...,nd≤N−1 ∑ 0≤n≤N−1 f1(T n 1 d ∏ j=1 T n j j x) f2(T n 2 d ∏ j=1 T n j j x) · · · fd(T n d d ∏ j=1 T n j j x). converges for μ-a.e. x ∈ X as N → ∞. If X is distal, we prove that the average 1 N N ∑ i=0 f1(T n 1 x) f2(T n 2 x) · · · fd(T n d x) converges for μ-a.e. x ∈ X as N → ∞...