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Existence of Large deviations rate function for any $S$-unimodal map

Hiroki Takahasi, Masato Tsujii

Abstract

For an arbitrary negative Schwarzian unimodal map with non-flat critical point, we establish the level-2 Large Deviation Principle (LDP) for empirical distributions. We also give an example of a multimodal map for which the level-2 LDP does not hold.

Existence of Large deviations rate function for any $S$-unimodal map

Abstract

For an arbitrary negative Schwarzian unimodal map with non-flat critical point, we establish the level-2 Large Deviation Principle (LDP) for empirical distributions. We also give an example of a multimodal map for which the level-2 LDP does not hold.

Paper Structure

This paper contains 24 sections, 19 theorems, 186 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Classification of attracting periodic orbits
  4. Renormalization of $S$-unimodal maps
  5. Structures associated with renormalization
  6. Symbolic dynamics on each cycle
  7. Distortion estimate on each cycle
  8. Topological results
  9. Notation
  10. Infinitely renormalizable maps
  11. Large deviations lower bound
  12. Key lower bound
  13. Approximation of a measure on each cycle
  14. Proof of Proposition \ref{['low']}
  15. Large deviations upper bound
  16. ...and 9 more sections

Key Result

Lemma 2.1

Let $f\colon X\to X$ be a renormalizable $S$-unimodal map, and let $J$ be a restrictive interval with period $p$ containing $c$ and not contained in any other restrictive interval (with period smaller than $p$). If $z\in\partial J$ is periodic and two-sided attracting, then $z$ is a fixed point of $

Figures (2)

  • Figure 1: The graphs of the renormalized unimodal maps $f^{p_{m}}|_{J_{m}}\colon J_m\to J_m$.
  • Figure 2: The graph of the bimodal map $f_a$

Theorems & Definitions (41)

  • Lemma 2.1
  • proof
  • Remark 2.2
  • Remark 2.3
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • proof
  • Lemma 2.6
  • proof
  • ...and 31 more