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Unique ergodicity for random noninvertible maps on an interval

Sara Brofferio, Hanna Oppelmayer, Tomasz Szarek

Abstract

In this short note, we investigate non-invertible stochastic dynamical systems on the unit interval $[0, 1]$. We provide a handy condition for unique ergodicity for systems that are injective in mean. On the other hand, we give concrete examples where unique ergodicity fails.

Unique ergodicity for random noninvertible maps on an interval

Abstract

In this short note, we investigate non-invertible stochastic dynamical systems on the unit interval . We provide a handy condition for unique ergodicity for systems that are injective in mean. On the other hand, we give concrete examples where unique ergodicity fails.
Paper Structure (16 sections, 15 theorems, 96 equations)

This paper contains 16 sections, 15 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. Generalized Entropy for piecewise monotone SDS
  3. Pushforward and pushbackward measures
  4. Radon-Nikodym derivative for finite measures on $[0,1]$
  5. Radon--Nikodym derivative of $g^{-1}\nu$
  6. Entropic criterion for characterizing ergodic measures
  7. $\mu$--injectivity and positive entropy
  8. Cardinality of pre-images
  9. Entropy of $\mu$--injective SDS
  10. Practical condition to ensure $h_\mu(\nu)>0$
  11. Examples
  12. A criterion for unique ergodicity
  13. Construction of SDS without unique ergodicity
  14. Appendix
  15. Proof of Proposition \ref{['prop-RN-deri-NC-int']}
  16. ...and 1 more sections

Key Result

Theorem 1.2

Let $(\mathcal{M},\mu)$ be a stochastic dynamical system $\mu$--injective in all but countably many $x\in[0,1]$ and assume that it contracts a neighbourhood of $x_0\in[0,1]$. Let $\nu$ be an atomless, ergodic $\mu$-invariant probability measure with $x_0\in\mathop{\mathrm{supp}}\nolimits(\nu)$. Then

Theorems & Definitions (33)

  • Definition 1.1
  • Theorem 1.2
  • Proposition 2.1
  • Theorem 2.2: Entropic criterion for ergodic measures
  • Proposition 2.3: Contractivity
  • proof : Proof of Theorem \ref{['thm: pos h']}
  • Lemma 3.1
  • proof
  • Definition 3.2
  • Proposition 3.3
  • ...and 23 more