Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Large deviations for some unbounded observables in dynamical systems

Anselmo Pontes

TL;DR

This work extends large deviation principles to unbounded observables in strongly mixing Markov systems by developing a martingale-coboundary decomposition together with truncation and tail-control arguments. It proves a general large deviation theorem for unbounded observables under regularity-after-truncation, exponential tails, and L^2-control of tails, and derives subexponential deviation bounds in this setting. The framework is then applied to expanding maps and to cocycles with mixed rank, including concrete cases like the tent map, illustrating quantitative LDT bounds for observables with logarithmic singularities and Lyapunov-type quantities. Overall, the paper broadens the scope of LDT from bounded to a wide class of unbounded observables in dynamical systems with strong mixing, offering new quantitative tools for recurrence, entropy, and Lyapunov-analysis problems.

Abstract

In this paper we establish a large deviations type estimate for strongly mixing Markov chains with respect to the Lp norm. As applications we derive such estimates for the iterates of a locally constant random cocycle with mixed rank, as well as for unbounded observables of expanding maps.

Large deviations for some unbounded observables in dynamical systems

TL;DR

This work extends large deviation principles to unbounded observables in strongly mixing Markov systems by developing a martingale-coboundary decomposition together with truncation and tail-control arguments. It proves a general large deviation theorem for unbounded observables under regularity-after-truncation, exponential tails, and L^2-control of tails, and derives subexponential deviation bounds in this setting. The framework is then applied to expanding maps and to cocycles with mixed rank, including concrete cases like the tent map, illustrating quantitative LDT bounds for observables with logarithmic singularities and Lyapunov-type quantities. Overall, the paper broadens the scope of LDT from bounded to a wide class of unbounded observables in dynamical systems with strong mixing, offering new quantitative tools for recurrence, entropy, and Lyapunov-analysis problems.

Abstract

In this paper we establish a large deviations type estimate for strongly mixing Markov chains with respect to the Lp norm. As applications we derive such estimates for the iterates of a locally constant random cocycle with mixed rank, as well as for unbounded observables of expanding maps.
Paper Structure (7 sections, 9 theorems, 47 equations)

This paper contains 7 sections, 9 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Martingale Approach
  4. Large Deviations for some Unbounded Observables
  5. Applications
  6. Large Deviations for the Lyapunov exponent of cocycles with mixed rank
  7. Large Deviations for unbounded observable of expanding maps

Key Result

Theorem 3.1

Let $\{X_i\}_{i=1}^{n}$ be a sequence of martingale differences. If there is $C>0$ such that $\|X_i\|_{\infty} \leq C$, then for all $\varepsilon >0$ we have

Theorems & Definitions (26)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.1
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 3.1
  • Definition 3.2
  • Theorem 3.1: Azuma-Hoeffding
  • Lemma 3.1
  • ...and 16 more