Journal of Statistical Physics

, Volume 156, Issue 4, pp 775–799 | Cite as

Toward a Mathematical Holographic Principle

  • Paweł Góra
  • Zhenyang Li
  • Abraham Boyarsky
  • Harald Proppe


In work started in [17] and continued in this paper our objective is to study selectors of multivalued functions which have interesting dynamical properties, such as possessing absolutely continuous invariant measures. We specify the graph of a multivalued function by means of lower and upper boundary maps \(\tau _{1}\) and \(\tau _{2}.\) On these boundary maps we define a position dependent random map \(R_{p}=\{\tau _{1},\tau _{2};p,1-p\},\) which, at each time step, moves the point \(x\) to \(\tau _{1}(x)\) with probability \(p(x)\) and to \(\tau _{2}(x)\) with probability \(1-p(x)\). Under general conditions, for each choice of \(p\), \(R_{p}\) possesses an absolutely continuous invariant measure with invariant density \(f_{p}.\) Let \(\varvec{\tau }\) be a selector which has invariant density function \(f.\) One of our objectives is to study conditions under which \(p(x)\) exists such that \(R_{p}\) has \(f\) as its invariant density function. When this is the case, the long term statistical dynamical behavior of a selector can be represented by the long term statistical behavior of a random map on the boundaries of \(G.\) We refer to such a result as a mathematical holographic principle. We present examples and study the relationship between the invariant densities attainable by classes of selectors and the random maps based on the boundaries and show that, under certain conditions, the extreme points of the invariant densities for selectors are achieved by bang-bang random maps, that is, random maps for which \(p(x)\in \{0,1\}.\)


Multivalued functions Selector Absolutely continuous invariant measure Mathematical holographic principle 

Mathematics Subject Classification (2000)

37A05 37H99 60J05 



We are extremely grateful for the very helpful comments of the anonymous reviewers. Their suggestions and critiques have greatly improved the paper. The research of the authors was supported by NSERC Grants. The research of Z. Li is also supported by NNSF of China (Nos. 11161020 and 11361023)


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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Paweł Góra
    • 1
  • Zhenyang Li
    • 1
  • Abraham Boyarsky
    • 1
  • Harald Proppe
    • 1
  1. 1.Department of Mathematics and StatisticsConcordia UniversityMontrealCanada

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