Pointwise convergence of bilinear averages for two commuting transformations
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For commuting measure-preserving maps T,S of a probability space, do N⁻¹∑ₙ₌₀ᴺ⁻¹ f(Tⁿx)g(Sⁿx) converge almost everywhere when f∈Lᵖ, g∈Lᑫ and 1/p+1/q≤1? No mixing or invertibility is assumed.
Why it matters and what counts as progress
Why it matters: Pointwise convergence is stronger than norm convergence and goes beyond known distal or single-transformation settings.
What counts as progress: An almost-everywhere theorem for the full exponent and map range, or a valid counterexample, without adding stronger dynamical assumptions.
How the problem developed
- 2018 · Partial result [1]
Sebastián Donoso and Wenbo Sun Proved pointwise convergence for commuting transformations in distal systems, a restricted setting for bilinear averages.
Floris van Doorn, Polona Durcik, Lenka Slavíková, Joris Roos and Christoph Thiele Reported almost-everywhere convergence for two commuting measure-preserving maps and f∈Lᵖ, g∈Lᑫ when 1/p+1/q≤1, without mixing or invertibility assumptions.
AI contribution
Pointwise convergence of ergodic averages for two commuting transformations
For two commuting measure-preserving maps T and S of a probability space, the bilinear averages N^-1 ∑_{n<N} f(T^n x)g(S^n x) converge almost everywhere when f∈Lp, g∈Lq, p,q∈[1,∞] and 1/p+1/q≤1. No mixing or invertibility assumption is imposed.
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Pointwise convergence of ergodic averages for two commuting transformations