  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.
