Pointwise convergence along cubes for measure preserving systems

2003/11/17 by Idris Assani, Assani, Idris
Mathematics · #37A30 #Dynamical Systems (math.DS) #FOS: Mathematics #math.DS #msc:37A30

paper · pdf · doi:10.48550/arxiv.math/0311274

14 pages

arxiv created 2003/11/17 · arxiv updated 2009/12/01

Abstract

Let (X, B, μ) be a probability measure space and T₁, T₂, T₃ three not necessarily commuting measure preserving transformations on (X, B, μ). We prove that for all bounded functions f₁, f₂, f₃ the averages (1)/(N²)∑_{n, m =1}^N f₁(T₁ⁿx)f₂(T₂^mx)f₃(T₃^n+mx) converges a.e. Generalizations to averages of 2^k -1 functions are also given for not necessarily commuting weakly mixing systems.

Related

• Averages along cubes for not necessarily commuting measure preserving transformations
• Pointwise convergence of averages along cubes
• Pointwise convergence of averages along cubes II
• Pointwise multiple averages for systems with two commuting transformations
• Pointwise recurrence for commuting measure preserving transformations