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Equilibrium States for Piecewise Weakly Convex Interval Maps

Nicolás Arévalo-Hurtado

Abstract

We prove the existence of equilibrium states for geometric potentials in a class of piecewise weakly convex interval maps. This class includes systems with indifferent fixed points and non-Markov partitions. Under additional hypotheses we also obtain uniqueness.

Equilibrium States for Piecewise Weakly Convex Interval Maps

Abstract

We prove the existence of equilibrium states for geometric potentials in a class of piecewise weakly convex interval maps. This class includes systems with indifferent fixed points and non-Markov partitions. Under additional hypotheses we also obtain uniqueness.

Paper Structure

This paper contains 5 sections, 16 theorems, 73 equations, 1 figure.

Table of Contents

  1. Introduction
  2. On $a$-convex transformations
  3. Conformal measure for $-s\log|T'|$
  4. Spectral Decomposition of the operator $\overline{F}_{s}$.
  5. Equilibrium states

Key Result

theorem 1.1

Let $T$ be an $a$-convex transformation satisfying conditions $(B)$ and $(C)$ for a given $s>0$. If either then there exists an equilibrium state $\mu_{s}$ for the potential $-s\log|T'|$ which is absolutely continuous with respect to $m_{s}$, and such that $\frac{d\mu_{s}}{dm_{s}}\in \mathcal{J}$. Furthermore, if $\beta=1$ and $m_{s}\neq \delta_{\beta}$, then $\mu_{s}$ is the unique equilibrium s

Figures (1)

  • Figure 1: The graph of $a$-convex transformations from Example \ref{['ExampleParabolicAconvex']} (A) and Example \ref{['ExampleParabolicNonMarkovianAconvex']} (B).

Theorems & Definitions (31)

  • theorem 1.1
  • remark 2.1
  • example 2.2
  • example 2.3
  • proposition 3.1
  • proof
  • example 3.2
  • remark 3.3
  • proposition 3.4
  • proof
  • ...and 21 more