Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Selections and their Absolutely Continuous Invariant Measures   Let $I=[0,1]$ and consider disjoint closed regions $G_{1},....,G_{n}$ in $%
I\times I$ and subintervals $I_{1},......,I_{n},$ such that $G_{i}$ projects
onto $I_{i.}$ We define the lower and upper maps $\tau_{1},$ $\tau_{2}$ by the
lower and upper boundaries of $G_{i},i=1,....,n,$ respectively. We assume
$\tau_{1}$, $\tau_{2}$ to be piecewise monotonic and preserving continuous
invariant measures $\mu_{1}$ and $\mu_{2}$, respectively. Let $% F^{(1)}$ and
$F^{(2)}$ be the distribution functions of $\mu_{1}$ and $\mu_{2}.$ The main
results shows that for any convex combination $F$ of $% F^{(1)} $ and $F^{(2)}$
we can find a map $\eta $ with values between the graphs of $\tau_{1}$ and
$\tau_{2}$ (that is, a selection) such that $F$ is the $\eta $-invariant
distribution function. Examples are presented. We also study the relationship
of the dynamics of multi-valued maps to random maps.

5
The categories of this paper are: gr-qc
13
The categories of this paper are: quant-ph
22
The categories of this paper are: chao-dyn
3
The categories of this paper are: physics
29
The categories of this paper are: funct-an
7
The categories of this paper are: hep-th
14
The categories of this paper are: math-ph
8
The categories of this paper are: hep-ex
34
The categories of this paper are: chem-ph
25
The categories of this paper are: patt-sol
37
The categories of this paper are: supr-con
19
The categories of this paper are: eess
20
The categories of this paper are: acc-phys
15
The categories of this paper are: nucl-ex
18
The categories of this paper are: econ
23
The categories of this paper are: solv-int