Geodesic flow and decay of traces on hyperbolic surfaces

Abstract

We study pseudodifferential operators on a hyperbolic surface using `Zelditch quantization'. We motivate and study the trace of $A_2^* A_1(t)$, where $A_2$ is a fixed operator and the Zelditch symbol of $A_1(t)$ evolves by geodesic flow. We find conditions under which the trace decays exponentially as $t \to \pm \infty$.

Figures (2)

• Figure 1: By identifying $g$ with a point in $S\mathbb{D}$, the geodesic flow, stable horocyclic flow, unstable horocyclic flow and rotation in the fibres are represented by the right action on $g$ by $\mathbf{a}_t$, $\mathbf{n}_u$, $\overline{\mathbf{n}_v}$ and $\mathbf{k}_{\theta}$ respectively. The images of the two horocyclic flows trace out horocycles in the base point $z$, which (in the disk model) are circles tangent to the conformal boundary at the forward and backward endpoints ($b$ and $b'$ respectively) of the geodesic determined by $g$.
• Figure 2: On the left, a horocycle through $z$ and $b$ and a horocycle through $w$ and $b$ on the hyperbolic disk. The so-called Busemann function, $\langle z, b\rangle$, is the distance from the origin $o$ to this horocycle. $\langle z, b\rangle-\langle w, b\rangle$ represents the signed distance between the given horocycles. On the right, a family of horocycles through a single point $b\in B$. Any two horocycles through the same point $b$ are equidistant to each other.

Theorems & Definitions (20)

• Definition : Anosov flow
• Lemma 1: geodesic flow on $S\mathbb{H}^2$ is an Anosov flow
• proof
• Definition : Busemann function
• Theorem 2: Helgason's non-Euclidean Fourier transform, Theorem 4.2 in helg.1
• Theorem 3: Boundary distribution of Laplacian eigenfunctions, Theorem 4.3 in helg.1
• Corollary 4
• Theorem 5: Theorem 4 in non.zwo.1
• Theorem 6
• Lemma 7