# Research Projects

• ### Entropy and self-intersection number of geodesic currents. (In preparation)

abstract:  We know that the measure-theoretic entropy of measured laminations on a compact hyperbolic surface is zero. On the other
hand, measured laminations are exactly the geodesic currents with zero self-intersection number.
This argument inspires us to ask: Does small self-intersection number imply small measure-theoretic entropy? Let $$i(-,-), h_C^X$$ refer to intersection number and measure-theoretic entropy. Let C be an ergodic geodesic current with length 1.  We show:

#### $$h_C^X \leq \frac{c_g}{sys(X)}i(C,C)\log{i(C,C)}$$,

where $$c_g<0$$ is a constant only depends on the genus and $$sys(X)$$ is the systole of X.
Therefore,  $$h_C^X \to 0$$ as i(C, C) → 0, when C is ergodic.

• ### Intersection points of the closed geodesics on finite volume hyperbolic surfaces. (In preparation)(Youtube part 1 and 2)

abstract:  Let X be a finite volume hyperbolic surface. It is not hard to see that the intersection points of the closed geodesics are dense on X. We show that the points are also equidistributed on X. In other words, if we consider all of the intersection points between the pairs of closed geodesics with length < T (including self-intersection points) then they are equidistributed, with respect
to the area measure, as T → ∞. Note that X may have cusps and part of the project is to
show that there is no escape of mass to the cusps in the limit.

• ### Intersection number, length and systole on compact hyperbolic surfaces. (pdf)

abstract: Assume that X is a compact hyperbolic surface with genus g. We define:

#### $$I(X) := \sup \limits_{\gamma_1,\gamma_2} \frac{i(\gamma_1,\gamma_2)}{\ell(\gamma_1)\ell(\gamma_2)},$$

and refer to it as the interaction strength of X because it controls the upper bound on $$i(\gamma_1,\gamma_2)$$
in terms of $$\ell(\gamma_1)\ell(\gamma_2)$$. Let $$\mathcal{M}_g$$ be the moduli space of genus g compact hyperbolic surfaces.
The main result establishes:

#### $$I(X)\sim \frac{1}{2sys(X)\log(1/sys(X))},$$

as X → ∞ in  $$\mathcal{M}_g$$, or equivalently, as sys(X) → 0, where sys(X) denotes the length of the shortest closed geodesic on X. Here, the notation $$\sim$$ means that their ratio tends to 1.

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• ### Geodesic planes in a geometrically finite end and halo of measured laminations ( with Y. Zhang) (pdf)

Recent works have shed light on the topological behavior of geodesic planes in the convex core of a geometrically finite hyperbolic 3-manifolds $$M$$of infinite volume. In this paper, we focus on the remaining case of geodesic planes outside the convex core of $$M$$, giving a complete classification of their closures in $$M$$. Behavior of the geodesic planes are strongly realated to the  bending  laminations of the ends, on conves core boundaries. We show that the classification depends on whether a special class of roof, exotic roofs, exist or not. More precisely, we  show a necessary condition for the existence of exotic roofs is the existence of a special geodesic ray, called exotic ray, for the bending lamination. We also show that the existence of geodesic rays with a stronger condition than being exotic -- phrased only in terms of the bending lamination -- is sufficient for the existence of exotic roofs. As a result, we show that quasifuchsian manifolds with exotic roofs exist in every genus. Moreover, in genus 1, when the quasifuchsian manifold is homotopic to a punctured torus, a generic one (in the sense of Baire category) contains uncountably many exotic roofs.