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On the spatial mean of the Poincare cycle | | Luis Baez-Duarte
; | | Date: |
28 May 2005 | | Journal: | Bull. Venezuela Acad. Sci. 1964 | | Subject: | Probability | math.PR | | Abstract: | Let $X$ be a measure space and $T:X o X$ a measurable transformation. For any measurable $Esubseteq X$ and $xin E$, the possibly infinite return time is $n_E(x):=inf{n>0: T^n xin E}$. If $T$ is an ergodic tranformation of the probability space $X$, and $mu(E)>0$, then a theorem of M. Kac states that $int_E n_E dmu=1$. We generalize this to any invertible measure preserving transformation $T$ on a finite measure space $X$, by proving independently, and nearly trivially that for any measurable $Esubseteq X$ one has $int_E n_E dmu=mu(I_E)$, where $I_E$ is the smallest invariant set containing $E$. In particular this also provides a simpler proof of Poincaré’s recurrence theorem. | | Source: | arXiv, math.PR/0505625 | | Services: | Forum | Review | PDF | Favorites | |
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