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Cubo (Temuco)

On-line version ISSN 0719-0646

Cubo vol.20 no.3 Temuco Oct. 2018 


The basic ergodic theorems, yet again

Jairo Bochi1 

1Pontificia Universidad Católica de Chile, Facultad de Matemáticas,


A generalization of Rokhlin’s Tower Lemma is presented. The Maximal Ergodic Theorem is then obtained as a corollary. We also use the generalized Rokhlin lemma, this time combined with a subadditive version of Kac’s formula, to deduce a subadditive version of the Maximal Ergodic Theorem due to Silva and Thieullen. In both the additive and subadditive cases, these maximal theorems immediately imply that “heavy” points have positive probability. We use heaviness to prove the pointwise ergodic theorems of Birkhoff and Kingman.

Keywords and Phrases: Maximal ergodic theorem; Birkhoff’s ergodic theorem; Rokhlin lemma; Kingman’s subadditive ergodic theorem.


Se presenta una generalización del Lema de la Torre de Rokhlin. El Teorema Ergódico Maximal se obtiene como corolario. También usamos el lema de Rokhlin generalizado, esta vez combinado con una versión subaditiva de la fórmula de Kac, para deducir una versión subaditiva del Teorema Ergódico Maximal obtenida por Silva y Thieullen. Tanto en el caso aditivo como en el subaditivo, estos teoremas maximales inmediatamente implican que puntos “pesados” tienen probabilidad positiva. Usamos esta pesadez para probar los teoremas ergódicos puntuales de Birkhoff y Kingman.

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