Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the spatial mean of the Poincare cycle   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.

29
The categories of this paper are: funct-an
9
The categories of this paper are: nlin
8
The categories of this paper are: hep-ex
34
The categories of this paper are: chem-ph
0
The categories of this paper are: hep-ph
3
The categories of this paper are: physics
7
The categories of this paper are: hep-th
14
The categories of this paper are: math-ph
32
The categories of this paper are: atom-ph
12
The categories of this paper are: nucl-th
28
The categories of this paper are: alg-geom
13
The categories of this paper are: quant-ph
24
The categories of this paper are: dg-ga
37
The categories of this paper are: supr-con
35
The categories of this paper are: mtrl-th
17
The categories of this paper are: q-fin