Exceptional directions for the Teichmüller geodesic flow and Hausdorff dimension

AbstractLet (M,gλ) be a 𝒞2-family of complete convex-cocompact metrics with pinched negative sectional curvatures on a fixed manifold. We show that the topological entropy htop(gλ) of the geodesic flow is a 𝒞1 function of λ and we give an explicit formula for its derivative. We apply this to show that if ρλ(Γ)⊂PSL2(ℂ) is an analytic family of convex-cocompact faithful representations of a Kleinian group Γ, then the Hausdorff dimension of the limit set Λρλ(Γ) is a 𝒞1 function of λ. Finally, we give a variation formula for Λρλ (Γ).

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A large deviations bound for the Teichmüller flow on the moduli space of abelian differentials

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385710000349 ◽

2010 ◽

Vol 31 (4) ◽

pp. 1043-1071 ◽

Cited By ~ 5

Author(s):

VÍTOR ARAÚJO ◽

ALEXANDER I. BUFETOV

Keyword(s):

Dynamical Systems ◽

Large Deviations ◽

Moduli Space ◽

Geodesic Flow ◽

Large Deviation ◽

Suspension Flows ◽

Abelian Differentials ◽

Quantitative Recurrence ◽

Teichmuller Geodesic Flow ◽

Teichmüller Flow

AbstractLarge deviation rates are obtained for suspension flows over symbolic dynamical systems with a countable alphabet. We use a method employed previously by the first author [Large deviations bound for semiflows over a non-uniformly expanding base. Bull. Braz. Math. Soc. (N.S.)38(3) (2007), 335–376], which follows that of Young [Some large deviation results for dynamical systems. Trans. Amer. Math. Soc.318(2) (1990), 525–543]. As a corollary of the main results, we obtain a large deviation bound for the Teichmüller flow on the moduli space of abelian differentials, extending earlier work of Athreya [Quantitative recurrence and large deviations for Teichmuller geodesic flow. Geom. Dedicata119 (2006), 121–140].

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Continuity of Hausdorff dimension across generic dynamical Lagrange and Markov spectra II

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2021.18 ◽

2021 ◽

pp. 1-10

Author(s):

ALINE CERQUEIRA ◽

CARLOS G. MOREIRA ◽

SERGIO ROMAÑA

Keyword(s):

Hausdorff Dimension ◽

Tangent Bundle ◽

Geodesic Flow ◽

Riemannian Metric ◽

Hausdorff Dimensions ◽

Hyperbolic Set ◽

Negatively Curved ◽

Smooth Perturbation ◽

Unit Tangent Bundle ◽

Markov Spectrum

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 $t\in \mathbb {R}$ .

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On the Hausdorff Dimension of CAT(κ) Surfaces

Analysis and Geometry in Metric Spaces ◽

10.1515/agms-2016-0010 ◽

2016 ◽

Vol 4 (1) ◽

Author(s):

David Constantine ◽

Jean-François Lafont

Keyword(s):

Hausdorff Dimension ◽

Geodesic Flow ◽

Short Proof ◽

Closed Surface ◽

Upper And Lower Bounds ◽

Two Dimensional ◽

Dimensional Hausdorff Measure ◽

Dimension 2 ◽

Uniformity Condition ◽

Metric Balls

AbstractWe prove that a closed surface with a CAT(κ) metric has Hausdorff dimension = 2, and that there are uniform upper and lower bounds on the two-dimensional Hausdorff measure of small metric balls. We also discuss a connection between this uniformity condition and some results on the dynamics of the geodesic flow for such surfaces. Finally,we give a short proof of topological entropy rigidity for geodesic flow on certain CAT(−1) manifolds.

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