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A Nonstationary Ruelle-Perron-Frobenius Theorem

Vaughn Climenhaga, Gregory Hemenway

Abstract

The Ruelle-Perron-Frobenius (RPF) theorem is a powerful tool in the study of equilibrium measures and their statistical properties. We prove a nonstationary version of this theorem under general conditions involving an invariant sequence of real convex cones in function space.

A Nonstationary Ruelle-Perron-Frobenius Theorem

Abstract

The Ruelle-Perron-Frobenius (RPF) theorem is a powerful tool in the study of equilibrium measures and their statistical properties. We prove a nonstationary version of this theorem under general conditions involving an invariant sequence of real convex cones in function space.

Paper Structure

This paper contains 11 sections, 25 theorems, 87 equations.

Table of Contents

  1. Introduction
  2. Nonstationary RPF theorems for uniformly expanding maps
  3. Prior work
  4. A general result using convex cones
  5. Convex cones and the Hilbert metric
  6. Abstract cone conditions for an RPF theorem
  7. Deducing cone conditions
  8. The cone of nonnegative functions
  9. The forward cone RPF theorem
  10. A bi-infinite nonstationary RPF theorem
  11. Pseudo-invariant measures

Key Result

Theorem 1.1

Suppose $\{(X_n,T_n,\varphi_n)\}_{n\geq 0}$ satisfy conditions A:pre--A:Hol. Then for $Q$ sufficiently large and $\Lambda_n = \Lambda_n(Q)$ as in eqn:cone, the following are true.

Theorems & Definitions (43)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.3
  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Theorem 2.4: Birkhoff Contraction Theorem gB57
  • Theorem 2.5
  • Theorem 2.6
  • ...and 33 more