Convergence of Time-Average along Uniformly Behaved in ${\mathbb N}$ Sequences on Every Point

Abstract

We define a uniformly behaved in ${\mathbb N}$ arithmetic sequence ${\bf a}$ and an ${\bf a}$-mean Lyapunov stable dynamical system $f$. We consider the time-average of a continuous function $\phi$ along the ${\bf a}$-orbit of $f$ up to $N$. The main result we prove in the paper is that this partial time-average converges for every point in the space if ${\bf a}$ is uniformly behaved in ${\mathbb N}$ and $f$ is minimal and uniquely ergodic and ${\bf a}$-mean Lyapunov stable. In addition, if ${\bf a}$ is also completely additive, we then prove that the time-average of a continuous function $\phi$ along the square-free ${\bf a}$-orbit of $f$ up to $N$ converges for every point in the space as well. All equicontinuous dynamical systems are ${\bf a}$-mean Lyapunov stable for any sequence ${\bf a}$. When ${\bf a}$ is a subsequence of ${\mathbb N}$ with positive lower density, we give two non-trivial examples of ${\bf a}$-mean Lyapunov stable dynamical systems. We give several examples of uniformly behaved in $\mathbb{N}$ sequences, including the counting function of the prime factors in natural numbers, the subsequence of natural numbers indexed by the Thue-Morse (or Rudin-Shapiro) sequence, and the sequence of even (or odd) prime factor natural numbers. We also show that the sequence of square-free natural numbers (or even (or odd) prime factor square-free natural numbers) is rotationally distributed in ${\mathbb N}$ but not uniformly distributed in ${\mathbb Z}$, thus not uniformly behaved in ${\mathbb N}$. We derive other consequences from the main result relevant to number theory and ergodic theory/dynamical systems.