The triangular metric on projective surfaces

Abstract

In \cite{ThurstonStretch} Thurston introduced stretch paths and stretch maps between hyperbolic surfaces. In this paper we extend similar constructions to the space of convex real projective structures on surfaces. Using Fock-Goncharov-like parameters, we introduce stretch paths and construct explicit stretch maps between the projective surface. We show that stretch maps are $e^t$--Lipschitz with respect to a triangular metric we introduce on the projective surface. This allows us to gain control on several length functions on an open set of the space of convex real projective structures (those where there is no ``local back--tracking"). As an application we obtain pointwise upper bounds for the Hilbert entropy of some convex real projective structures in terms of Fock-Goncharov-like parameters.