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Large deviation principles for periodic points of the Dyck shift

Hiroki Takahasi

Abstract

We investigate periodic points of the Dyck shift from the viewpoint of large deviations. We establish the level-2 Large Deviation Principle with the rate function given in terms of Kolmogorov-Sinai entropies of shift-invariant Borel probability measures. Unlike topologically mixing Markov shifts, the level-2 rate function is non-convex and level-1 rate functions are superpositions of two convex continuous functions. Using the thermodynamic formalism, we show the analyticity of level-1 rate functions in some relevant cases. We display a non-convex level-1 rate function with a non-differentiable point in the interior of its effective domain.

Large deviation principles for periodic points of the Dyck shift

Abstract

We investigate periodic points of the Dyck shift from the viewpoint of large deviations. We establish the level-2 Large Deviation Principle with the rate function given in terms of Kolmogorov-Sinai entropies of shift-invariant Borel probability measures. Unlike topologically mixing Markov shifts, the level-2 rate function is non-convex and level-1 rate functions are superpositions of two convex continuous functions. Using the thermodynamic formalism, we show the analyticity of level-1 rate functions in some relevant cases. We display a non-convex level-1 rate function with a non-differentiable point in the interior of its effective domain.

Paper Structure

This paper contains 21 sections, 26 theorems, 127 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The Dyck shift
  3. Level-2 LDPs
  4. Level-1 LDPs
  5. Generalizations and organization of the paper
  6. Preliminaries
  7. Weak* topology
  8. Words, cylinders, bounded variations of Birkhoff averages
  9. Borel embeddings of full shifts into the Dyck shift
  10. Maximal entropy measures for the Dyck shift
  11. Estimates of numbers of periodic points
  12. The LDP for periodic points of the Dyck shift
  13. The level-2 LDP on two embedded full shifts
  14. Intermediate lower bounds
  15. Intermediate upper bounds
  16. ...and 6 more sections

Key Result

Theorem 1.1

The LDP holds for $(\widetilde{\mu}_n)_{n=1}^\infty$. The rate function $I\colon\mathcal{M}(\Sigma_D)\to[0,\infty]$ is not convex. The minimizers of $I$ are the two ergodic maximal entropy measures.

Figures (1)

  • Figure 1: The graph of the rate function $I_f$ on its effective domain $[0,1]$ for $f$ in \ref{['phi-0']}; (a) $M\leq 3$; (b) $M=4$; (c) $M\geq5$.

Theorems & Definitions (43)

  • Theorem 1.1: the level-2 LDP
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4: the level-1 LDP
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3: Kri74, pp.102--103
  • Lemma 2.4
  • ...and 33 more