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Selections and their absolutely continuous invariant measures


Selections and their absolutely continuous invariant measures

Boyarsky, Abraham, Góra, Paweł and Li, Zhenyang (2014) Selections and their absolutely continuous invariant measures. Journal of Mathematical Analysis and Applications, 413 (1). pp. 100-113. ISSN 0022247X

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Selections_November_2013_3.PDF - Accepted Version
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Official URL: http://dx.doi.org/10.1016/j.jmaa.2013.11.043


Let $I=[0,1]$ and let $P$ be a partition of $I$ into a finite number of intervals. Let $\tau _{1},$ $\tau _{2}; I\rightarrow I$ be two piecewise expanding maps on $P.$ Let $G$ $\subset I\times I$ be the region between the
boundaries of the graphs of $\tau _{1}$ and $\tau _{2}.$ Any map $\tau :I\rightarrow I$ that takes values in $G$ is called a selection of the multivalued map defined by $G.$ There are many results devoted to the study of the existence of selections with specified topological properties.
However, there are no results concerning the existence of selection with
measure-theoretic properties. In this paper we prove the existence of
selections which have absolutely continuous invariant measures (acim). By
our assumptions we know that $\tau _{1}$ and $\tau _{2}$ possess acims
preserving the distribution functions $F^{(1)}$ and $F^{(2)}.$
The main result 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 our multivalued maps to random maps.

Divisions:Concordia University > Faculty of Arts and Science > Mathematics and Statistics
Item Type:Article
Authors:Boyarsky, Abraham and Góra, Paweł and Li, Zhenyang
Journal or Publication:Journal of Mathematical Analysis and Applications
Digital Object Identifier (DOI):10.1016/j.jmaa.2013.11.043
ID Code:980102
Deposited By: PAWEL GORA
Deposited On:25 Jun 2015 16:03
Last Modified:18 Jan 2018 17:50
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