Abstract
Let
$g_0$
be a smooth pinched negatively curved Riemannian metric on a complete surface N, and let
$\Lambda _0$
be a basic hyperbolic set of the geodesic flow of
$g_0$
with Hausdorff dimension strictly smaller than two. Given a small smooth perturbation g of
$g_0$
and a smooth real-valued function f on the unit tangent bundle to N with respect to g, let
$L_{g,\Lambda ,f}$
(respectively
$M_{g,\Lambda ,f}$
) be the Lagrange (respectively Markov) spectrum of asymptotic highest (respectively highest) values of f along the geodesics in the hyperbolic continuation
$\Lambda $
of
$\Lambda _0$
. We prove that for generic choices of g and f, the Hausdorff dimensions of the sets
$L_{g,\Lambda , f}\cap (-\infty , t)$
vary continuously with
$t\in \mathbb {R}$
and, moreover,
$M_{g,\Lambda , f}\cap (-\infty , t)$
has the same Hausdorff dimension as
$L_{g,\Lambda , f}\cap (-\infty , t)$
for all
$t\in \mathbb {R}$
.
Exceptional directions for the Teichmüller geodesic flow and Hausdorff dimension
