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Invariant and conditionally invariant measures for random open interval maps with countably many branches

Cunyi Nan

Abstract

In this paper, building on previous work, we extend the thermodynamic formalism for random open dynamical systems generated by piecewise monotone interval maps with countably many branches. Under summable and contracting assumptions on the potential, we establish the Ruelle-Perron-Frobenius type theorem for the associated random open operator and prove exponential decay of correlations. In addition, we investigate the escape rate for the hole and conditionally invariant measure for the open system.

Invariant and conditionally invariant measures for random open interval maps with countably many branches

Abstract

In this paper, building on previous work, we extend the thermodynamic formalism for random open dynamical systems generated by piecewise monotone interval maps with countably many branches. Under summable and contracting assumptions on the potential, we establish the Ruelle-Perron-Frobenius type theorem for the associated random open operator and prove exponential decay of correlations. In addition, we investigate the escape rate for the hole and conditionally invariant measure for the open system.
Paper Structure (22 sections, 44 theorems, 256 equations)

This paper contains 22 sections, 44 theorems, 256 equations.

Table of Contents

  1. Introduction
  2. Statement of the main result
  3. Plan of the paper
  4. Piecewise monotone random interval maps with holes
  5. Settings
  6. Transfer operator and potential
  7. Holes and random functionals
  8. Lasota-Yorke type inequality
  9. Birkhoff cones and cone invariance
  10. Birkhoff cones
  11. Good and bad fibers
  12. Auxiliary lemmas
  13. Finite diameter of cones
  14. Conformal and invariant measures
  15. Construction of density functions
  16. ...and 7 more sections

Key Result

Theorem 1.1

(1) There exists a unique random probability measure $\nu\in\mathcal{P}_{\Omega}(\Omega\times I)$ with $\mathop{\mathrm{\mathrm{supp}}}\nolimits(\nu)\subseteq K_{\infty}$ and disintegration $\{\nu_\omega\}_{\omega\in\Omega}$ such that for all $f\in BV(I)$, where $\lambda_\omega:=\nu_{\theta\omega}(\mathcal{L}_{\omega}\mathbbm{1}_\omega)$ is a positive number and $\log\lambda_\omega\in L^1(\mathbb

Theorems & Definitions (88)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Definition 2.2: crauel
  • Lemma 2.3: atnip1
  • Definition 2.4: atnip1
  • Theorem 2.5: atnip1
  • Lemma 2.6
  • proof
  • ...and 78 more