**RESEARCH IN PAIRS**

**Convex Real Projective Structures**

**Géométrie projective convexe****17 - 21 June 2019**

1. Constructing a concrete example of deformations in real projective structures: Bulging deformation. An earthquake deformation does not change the domain but bulging deformation will change the domain . This corresponds to changing the vertical twisting parameter in Goldman's coordinates. We want to investigate the dynamical property of this deformation, namely we want to show that the topological entropy remains bounded below but the entropy with respect to Sinai-Ruelle-Bowen measure goes to zero along the bulging deformation.

2. Counting the number of closed geodesics. It is known that the unmarked length spectrum determines the hyperbolic structures up to nite choice. We want to prove a similar theorem for real projective structures with Hilbert metric.

3. We found a mapping class group invariant Kahler metric on the space of convex real projective structures on a closed surface. We want to investigate a curvature properties of this metric. We suspect that it is non-positively curved.

2. Counting the number of closed geodesics. It is known that the unmarked length spectrum determines the hyperbolic structures up to nite choice. We want to prove a similar theorem for real projective structures with Hilbert metric.

3. We found a mapping class group invariant Kahler metric on the space of convex real projective structures on a closed surface. We want to investigate a curvature properties of this metric. We suspect that it is non-positively curved.

**Participants**

Patrick Foulon (CIRM)

Inkang Kim (KIAS)

**Sponsor**