{"id":"openai-math/Pointwise-convergence-of-triple-ergodic-averages-for-mixing-transformations","pdf":"https://github.com/openai/math/blob/main/preprints/Pointwise-convergence-of-triple-ergodic-averages-for-mixing-transformations-October-4-2026/pointwise-triple-ergodic-averages-mixing-transformations.pdf","url":"https://github.com/openai/math/tree/main/preprints/Pointwise-convergence-of-triple-ergodic-averages-for-mixing-transformations-October-4-2026","tags":["ergodic-theory","dynamical-systems","measure-theory"],"score":0,"title":"Pointwise convergence of triple ergodic averages for mixing transformations","domain":"Dynamical systems and ergodic theory","family":"154","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.DS","summary":"We prove pointwise convergence of triple ergodic averages for every invertible mixing probability preserving transformation T of an arbitrary probability space $`(X,\\mathcal F,\\mu)`$, with measurable inverse. For every triple of bounded measurable functions $`f_1,f_2,f_3:X\\to\\mathbb C`$, $`\\displaystyle \\frac1N\\sum_{n=1}^N f_1(T^nx)f_2(T^{2n}x)f_3(T^{3n}x) \\longrightarrow \\prod_{j=1}^3\\int_X f_j\\,d\\mu`$ for μ -almost every x as $`N\\to\\infty`$ through all positive integers.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-10-04","reactions":{},"formalized":false,"family_title":"Pointwise multiple ergodic averages for mixing transformations"}