Pointwise convergence along cubes for measure preserving systems

Abstract

Let $(X,\mathcal{B},\mu )$ be a probability measure space and $T_1$, $T_2$, $T_3$ three not necessarily commuting measure preserving transformations on $(X,\mathcal{B},\mu )$. We prove that for all bounded functions $f_1$, $f_2$, $f_3$ the averages $$\frac{1}{N^2}\sum _{n,m=1}^N{f}_1(T_1^{n}x)f_2(T_2^{m}x)f_3(T_3^{n+m}x)$$ converges a.e. Generalizations to averages of $2^k-1$ functions are also given for not necessarily commuting weakly mixing systems.

14 pages

References in corpus (1)

• Pointwise convergence of averages along cubes

Cited by in corpus (4)

• A cubic nonconventional ergodic average with Möbius and Liouville weight
• A cubic nonconventional ergodic average with multiplicative or Mangoldt weights
• Pointwise convergence for cubic and polynomial ergodic averages of non-commuting transformations
• On the pointwise convergence of the cubic average with multiplicative or von Mangoldt weights