[Paper Review] Rates of mixing for the Weil-Petersson geodesic flow II: exponential mixing in exceptional moduli spaces

Keith Burns, Howard Masur+2 more|arXiv (Cornell University)|May 29, 2016

Mathematical Dynamics and Fractals11 references3 citations

TL;DR

This paper establishes exponential mixing for the Weil-Petersson geodesic flow on the moduli spaces $\mathcal{M}_{1,1}$ and $\mathcal{M}_{0,4}$ by analyzing singular, negatively curved surfaces with cusp-like singularities modeled on $y = x^r$ for $r > 2$. The authors introduce a novel metric rescaling technique to handle singularities and prove that the flow exhibits exponential decay of correlations, contrasting sharply with slower mixing in higher-dimensional moduli spaces.

ABSTRACT

We establish exponential mixing for the geodesic flow $φ_t\colon T^1S o T^1S$ of an incomplete, negatively curved surface $S$ with cusp-like singularities of a prescribed order. As a consequence, we obtain that the Weil-Petersson flows for the moduli spaces ${\mathcal M}_{1,1}$ and ${\mathcal M}_{0,4}$ are exponentially mixing, in sharp contrast to the flows for ${\mathcal M}_{g,n}$ with $3g-3+n>1$, which fail to be rapidly mixing. In the proof, we present a new method of analyzing invariant foliations for hyperbolic flows with singularities, based on changing the Riemannian metric on the phase space $T^1S$ and rescaling the flow $φ_t$.

Motivation & Objective

• To establish exponential mixing for the Weil-Petersson geodesic flow on exceptional moduli spaces $\mathcal{M}_{1,1}$ and $\mathcal{M}_{0,4}$, which are known to have unique dynamical behavior.
• To resolve the contrast between rapid mixing in low-complexity moduli spaces and polynomial mixing in higher-dimensional cases ($3g-3+n > 1$).
• To develop a new method for analyzing hyperbolic flows with singularities by rescaling the Riemannian metric on the unit tangent bundle $T^1S$ and adjusting the flow dynamics.
• To prove that the stable and unstable foliations are uniformly $C^{1+\alpha}$, enabling the construction of a Young tower with exponential decay of correlations.

Proposed method

• The authors introduce a modified distance function $\overline{\delta}$ to control the geometry near cusp singularities, replacing the standard distance $\delta$ to the cusp point.
• They rescale the Riemannian metric on $T^1S$ to create a complete, adapted metric that allows for uniform control of the flow’s derivative along unstable directions.
• A time change is applied to convert the singular geodesic flow into an Anosov flow, enabling the use of standard exponential mixing machinery.
• The construction of a Young tower relies on defining return maps via stable holonomy and a carefully chosen return time function $R(v)$, which is shown to have exponential tails.
• The key technical innovation lies in analyzing the derivative of the return time function $R \circ h_j$ using bounded distortion and $C^{1+\alpha}$ regularity of the center-stable foliation.
• The proof leverages the fact that the flow preserves a contact 1-form, ensuring the non-integrability of the stable and unstable foliations, which is essential for the UNI condition in the Young tower framework.

Experimental results

Research questions

• RQ1 Why do the Weil-Petersson geodesic flows on $\mathcal{M}_{1,1}$ and $\mathcal{M}_{0,4}$ exhibit exponential mixing, while those on higher-dimensional moduli spaces fail to be rapidly mixing?
• RQ2 Can a new geometric method be developed to handle singularities in hyperbolic flows, particularly in incomplete, negatively curved surfaces with cusp-like ends?
• RQ3 How can the invariant foliations of a singular geodesic flow be analyzed when standard $C^1$ regularity fails near the singular set?
• RQ4 What conditions on curvature and higher-order derivatives ensure exponential decay of correlations in such systems?
• RQ5 Is it possible to construct a Young tower with exponential tails for the Weil-Petersson flow in exceptional moduli spaces using metric rescaling and holonomy techniques?

Key findings

• The Weil-Petersson geodesic flow on $\mathcal{M}_{1,1}$ and $\mathcal{M}_{0,4}$ is exponentially mixing, with correlation decay bounded by $Ce^{-ct}\|u_1\|_{C^1}\|u_2\|_{C^1}$ for all $t > 0$.
• The flow exhibits exponential decay of correlations despite the presence of cusp-like singularities, which are modeled on surfaces of revolution $y = x^r$ with $r > 2$.
• The authors construct a Young tower with exponential tails for the return time function $R(v)$, satisfying $\left|\{v : R(v) \geq k\}\right| \leq C\lambda^k$ for some $\lambda < 1$.
• The stable and unstable foliations are uniformly $C^{1+\alpha}$, and the flow preserves a contact 1-form, ensuring the non-integrability required for the UNI condition.
• The return map $h_j$ from $\Delta_0$ to $\Delta_j$ is uniformly $C^{1+\alpha}$, and its derivative satisfies $|h_j'| \asymp \|D^u \varphi_{R_0(h_j(v))}\|^{-1}$, enabling distortion control.
• The method of metric rescaling and flow time change allows the authors to overcome the lack of global $C^1$ regularity near cusps, enabling the application of exponential mixing techniques.