A horseshoe with a discontinuous entropy spectrum

We study the regularity of the entropy spectrum of the Lyapunov exponents for hyperbolic maps on surfaces. It is well-known that the entropy spectrum is a concave upper semi-continuous function which is analytic on the interior of the set Lyapunov exponents. In this paper we construct a family of horseshoes with a discontinuous entropy spectrum at the boundary of the set of Lyapunov exponents.