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| Preprints |
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(with A. Eskin) Slow Divergence and Unique Ergodicity [ pdf ], to appear |
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Abstract: Masur showed that a Teichmuller geodesic that is recurrent in the moduli space of closed Riemann surfaces is necessarily determined by a quadratic differential with a uniquely ergodic vertical foliation. In this paper, we show that a divergent Teichmuller geodesic satisfying a certain slow rate of divergence is also necessarily determined by a quadratic differential with unique ergodic vertical foliation. As an application, we sketch a proof of a complete characterization of the set of nonergodic directions in any double cover of the flat torus branched over two points. |
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Hausdorff dimension of the set of Singular Pairs [ pdf | ps ], submitted |
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Abstract: In this paper the Hausdorff dimension of the set of singular pairs is determined to be 4/3. The proof involves a detailed analysis of the sequence of best approximations associated to any point in Rd, generalising the theory of continued fractions from the perspective of the approximation properties of convergents. In terms of the homogeneous flow on the space SL3R/SL3Z induced by the one-parameter subgroup diag(et,et,e-2t) acting by left multiplication, the main result says that the set of points lying on divergent trajectories of the flow has Hausdorff dimension 71/3. As a further application, we show that there exist divergent trajectories of the flow that exit to infinity at arbitrarily slow prescribed rates, affirmatively answering a question of A.N. Starkov. |
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| Papers |
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(with P.Hubert and H.Masur) Topological Dichotomy and Strict Ergodicity for Translation Surfaces [ pdf ] Ergod. Th. Dynam. Sys., to appear |
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Abstract: In this paper the authors find examples of translation surfaces that have infinitely generated Veech groups, satisfy the topological dichotomy property that for every direction either the flow in that direction is completely periodic or minimal, and yet have minimal but non uniquely ergodic directions. |
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Hausdorff dimension of the set of points on divergent trajectories of a homogeneous flow on a product space [pdf | ps] Ergod. Th. Dynam. Sys., 27 (2007), 65--85. |
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Abstract: In this paper we compute the Hausdorff dimension of the set D_n of points on divergent trajectories of the homogeneous flow induced by a certain one-parameter subgroup of G=SL(2,R) acting by left multiplication on the product space G^n/Gamma^n, where Gamma=SL(2,Z). We prove that the Hausdorff dimension of D_n equals 3n-(1/2) for any n greater than one.
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(with A. Eskin) Unique Ergodicity of Translation Flows [ pdf | ps ] Fields Institute Communications 51 (2007) 203-222. |
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Abstract: This preliminary report contains a sketch of the proof of the following result: a slowly divergent Teichmuller geodesic satisfying a certain logarithmic law is determined by a uniquely ergodic measured foliation. |
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(with H. Masur) Minimal nonergodic directions on genus 2 translation surfaces [ pdf | ps ] Ergod. Th. Dynam. Sys. 26 (2006), 341--351. |
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Abstract: It is well-known that on any Veech surface, the dynamics in any minimal direction is uniquely ergodic. In this paper, it is shown that for any genus 2 translation surface which is not a Veech surface there are uncountably many minimal but not uniquely ergodic directions. Slides [ pdf | ps ] for talk at Midwest Dynamics Seminar, October 2005. |
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(with H. Masur) A divergent Teichmuller geodesic with uniquely ergodic vertical foliation [ pdf | ps ] (Israel J. Math. 152 (2006), 1--15. |
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Abstract: We construct an example of a quadratic differential whose vertical foliation is uniquely ergodic and such that the Teichmuller geodesic determined by the quadratic differential diverges in the moduli space of Riemann surfaces. |
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Slowly divergent geodesics in moduli space [ pdf | ps ] (Conform. Geom. Dyn. 8 (2004), 167--189.) |
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Abstract: Slowly divergent geodesics in the moduli space of Riemann surfaces of genus at least 2 are constructed via cyclic branched covers of the torus. Nonergodic examples (i.e. geodesics whose defining quadratic differential has nonergodic vertical foliation) diverging to infinity at sublinear rates are constructed using a Diophantine condition. Examples with an arbitrarily slow prescribed growth rate are also exhibited. |
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Hausdorff dimension of the set of nonergodic directions [ pdf | ps ] (Ann. of Math., 158 (2003), 661--678.) |
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Abstract: It is known that nonergodic directions in a rational billard form a subset of the unit circle with Hausdorff dimension at most 1/2. Explicit examples realizing the dimension 1/2 are constructed using Diophantine numbers and continued fractions. A lower estimate on the number of primitive lattice points in certain subsets of the plane is used in the construction. |
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