[Paper Review] Equilibrium states of interval maps for hyperbolic potentials

Huaibin Li, Juan Rivera‐Letelier|arXiv (Cornell University)|Oct 25, 2012

Mathematical Dynamics and Fractals18 references3 citations

TL;DR

This paper establishes the existence and uniqueness of equilibrium states for sufficiently regular interval maps under Hölder continuous hyperbolic potentials, proving exponential mixing and real analyticity of the pressure function. Using the Patterson-Sullivan method to construct conformal measures and applying Keller's spectral theory, the authors extend results from complex dynamics to real interval maps, including those with flat critical points.

ABSTRACT

We study the thermodynamic formalism of sufficiently regular interval maps for Holder continuous potentials. We show that for a hyperbolic potential there is a unique equilibrium state, and that this measure is exponentially mixing. Moreover, we show the absence of phase transitions: The pressure function is real analytic at such a potential.

Motivation & Objective

• To extend the thermodynamic formalism of interval maps to Hölder continuous hyperbolic potentials, generalizing results from complex dynamics.
• To establish the existence and uniqueness of equilibrium states for multimodal interval maps with hyperbolic potentials.
• To prove exponential mixing of the equilibrium state and real analyticity of the pressure function.
• To develop a method that avoids bounded distortion assumptions, applicable to maps with flat critical points.
• To provide a foundation for applications in the companion paper [LRL14] on interval dynamics.

Proposed method

• Construct a conformal measure using the Patterson-Sullivan method on the Julia set of the interval map.
• Apply Keller's spectral theory for transfer operators on the space $\operatorname{H}^{\alpha,1}(m)$ of functions with controlled oscillation.
• Use the equivalence between the spectral radius of the transfer operator and the exponential of the pressure to deduce equilibrium state existence.
• Establish analyticity of the pressure function by perturbing the potential and showing analytic dependence of the spectral radius.
• Verify that the hyperbolicity condition $\sup_I \frac{1}{n} S_n(\varphi) < P(f,\varphi)$ ensures the existence of a unique equilibrium state.
• Leverage the fact that the transfer operator acts on functions of bounded variation in a generalized sense, enabling spectral analysis without bounded distortion.

Experimental results

Research questions

• RQ1 Does a unique equilibrium state exist for a multimodal interval map with a Hölder continuous hyperbolic potential?
• RQ2 Is the equilibrium state exponentially mixing under the hyperbolicity condition?
• RQ3 Is the pressure function real analytic in a neighborhood of a hyperbolic potential?
• RQ4 Can the thermodynamic formalism be extended to interval maps with flat critical points without bounded distortion assumptions?
• RQ5 How does the conformal measure construction via the Patterson-Sullivan method facilitate the proof of spectral properties?

Key findings

• For a multimodal interval map $f$ in the class $\mathscr{A}$ and a Hölder continuous potential $\varphi$ that is hyperbolic, there exists a unique equilibrium state.
• The unique equilibrium state is exponentially mixing, implying strong statistical properties such as exponential decay of correlations.
• The pressure function $P(f, \varphi)$ is real analytic in a neighborhood of any hyperbolic potential $\varphi$.
• The method avoids the need for bounded distortion assumptions, allowing application to maps with flat critical points.
• The spectral radius of the transfer operator $\mathscr{L}_t$ equals $\exp(P(f, \varphi_t))$, and this function is real analytic in $t$ near 0.
• The construction of a conformal measure via the Patterson-Sullivan method enables the application of Keller's spectral theory to interval maps, extending techniques from complex dynamics.