On the spatial mean of the Poincare cycle

Details On the spatial mean of the Poincare cycle On the spatial mean of the Poincare cycle

2005-05-28

arxiv.org/abs/math/0505625v1

Bull. Venezuela Acad. Sci. 1964

Mathematics

Probability

2 pages, Translation into English of a paper by the author generalizing Kac's theorem on the spatial mean of the Poincare cycl

Scientific paper

Let $X$ be a measure space and $T:X\to X$ a measurable transformation. For any measurable $E\subseteq X$ and $x\in E$, the possibly infinite return time is $n_E(x):=\inf\{n>0: T^n x\in E\}$. If $T$ is an ergodic tranformation of the probability space $X$, and $\mu(E)>0$, then a theorem of M. Kac states that $\int_E n_E d\mu=1$. 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 $E\subseteq X$ one has $\int_E n_E d\mu=\mu(I_E)$, where $I_E$ is the smallest invariant set containing $E$. In particular this also provides a simpler proof of Poincar\'{e}'s recurrence theorem.

Affiliated with

Also associated with

No associations

Rating

On the spatial mean of the Poincare cycle does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with On the spatial mean of the Poincare cycle, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On the spatial mean of the Poincare cycle will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-690488