Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The generic isometry and measure preserving homeomorphism are conjugate
  to their powers   It is known that there is a comeagre set of mutually conjugate measure
preserving homeomorphisms of Cantor space equipped with the coinflipping
probability measure, i.e., Haar measure. We show that the generic measure
preserving homeomorphism is moreover conjugate to all of its powers. It follows
that the generic measure preserving homeomorphism extends to an action of
$(\mathbb Q,+)$ by measure preserving homeomorphisms, and, in fact, to an
action of the locally compact ring $\mathfrak A$ of finite ad\`eles.
  Similarly, S. Solecki has proved that there is a comeagre set of mutually
conjugate isometries of the rational Urysohn metric space. We prove that these
are all conjugate with their powers and therefore also embed into $\mathbb
Q$-actions. In fact, we extend these actions to actions of $\mathfrak A$ as in
the case of measure preserving homeomorphisms.
  We also consider a notion of topological similarity in Polish groups and use
this to give simplified proofs of the meagreness of conjugacy classes in the
automorphism group of the standard probability space and in the isometry group
of the Urysohn metric space.

18
The categories of this paper are: econ
19
The categories of this paper are: eess
31
The categories of this paper are: ao-sci
13
The categories of this paper are: quant-ph
27
The categories of this paper are: comp-gas
10
The categories of this paper are: q-bio
20
The categories of this paper are: acc-phys
4
The categories of this paper are: cond-mat
9
The categories of this paper are: nlin
37
The categories of this paper are: supr-con
30
The categories of this paper are: q-alg
17
The categories of this paper are: q-fin
32
The categories of this paper are: atom-ph
16
The categories of this paper are: stat
29
The categories of this paper are: funct-an
12
The categories of this paper are: nucl-th