Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Ergodicity of group actions and spectral gap, applications to random
  walks and Markov shifts   Let $(X, \cal B, \nu)$ be a probability space and let $\Gamma$ be a countable
group of $\nu$-preserving invertible maps of $X$ into itself. To a probability
measure $\mu$ on $\Gamma$ corresponds a random walk on $X$ with Markov operator
$P$ given by $P\psi(x) = \sum_{a} \psi(ax) \, \mu(a)$. A powerful tool is the
spectral gap property for the operator $P$ when it holds. We consider various
examples of ergodic $\Gamma$-actions and random walks and their extensions by a
vector space: groups of automorphisms or affine transformations on compact
nilmanifolds, random walk in random scenery on non amenable groups,
translations on homogeneous spaces of simple Lie groups, random walks on motion
groups. The spectral gap property is applied to obtain limit theorems,
recurrence/transience property and ergodicity for random walks on non compact
extensions of the corresponding dynamical systems.

15
The categories of this paper are: nucl-ex
18
The categories of this paper are: econ
29
The categories of this paper are: funct-an
4
The categories of this paper are: cond-mat
25
The categories of this paper are: patt-sol
9
The categories of this paper are: nlin
10
The categories of this paper are: q-bio
7
The categories of this paper are: hep-th
1
The categories of this paper are: cs
22
The categories of this paper are: chao-dyn
31
The categories of this paper are: ao-sci
30
The categories of this paper are: q-alg
3
The categories of this paper are: physics
14
The categories of this paper are: math-ph
21
The categories of this paper are: adap-org
33
The categories of this paper are: plasm-ph