Yitwah Cheung's Homepage

Contact Information
  Office: Thorton Hall, Room 950
  Phone: (415) 338-1805
  Email: ycheung - AT - sfsu - DOT - edu

  • I am teaching MATH 226 Calculus I and MATH 228 Calculus III this semester.

    Curriculum Vitae
  • Here is a short cv.

  • (with N. Chevallier) Hausdorff dimension of singular vectors, [pdf], submitted
      Abstract: We prove the Hausdorff dimension of the set of singular vectors in Rd is d2/d+1 for d>1.
  • (with B. Weiss) TBD, in preparation
      Abstract: We give examples of divergent trajectories satisfying a conjecture of Wolfgang Schmidt on successive minima of a lattice, with arbitrarily slow rates of divergence.
  • (with A. Eskin) Slow Divergence and Unique Ergodicity [ pdf ], arXiv:0711.0240v1., preprint
      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.
  • (with J. Chaika and H. Masur) Winning games for bounded geodesics in Teichmuller discs [ pdf ], Journal Modern Dynamics, to appear
      Abstract: We prove that for the flat surface defined by a holomorphic quadratic differential the set of directions such that the corresponding Teichmueller geodesic lies in a compact set in the corresponding stratum is a winning set in Schmidt game. This generalizes a classical result in the case of the torus due to Schmidt and strengthens a result of Kleinbock and Weiss.
  • (with J. Athreya) A Poincare section for the horocycle flow on the space of lattices [ pdf], Int. Math. Res. Not., to appear
      Abstract: In this paper, we show that the "BCZ map" introduced by Boca-Cobeli-Zaharescu is a Poincare section of the horocycle flow on the modular surface. Using this discovery, we develop the ergodic properties of the BCZ map, proving in particular that it is ergodic and has zero entropy with respect to Lebesgue measure. We also show that many earlier results on statistical properties of Farey fractions follow from the equidistribution principle for closed horocycle orbits. As one further application of the BCZ map, we compute the average depth of a horocycle orbit relative to the number of visits into the cusp.
  • (with A. Goetz and A. Quas) Piecewise Isometries, Uniform Distribution and 3 log 2 - π2/8 [ pdf ], Ergod. Th. Dynam. Sys., 32, (2012), 1862-1888.
      Abstract: We use analytic tools to study a simple family of piecewise isometries of the plane parameterized by an angle parameter. In previous work we showed the existence of large numbers of periodic points, each surrounded by a 'periodic island'. We also proved conservativity of the systems as infinite measure-preserving transformations. In experiments it is observed that the periodic islands fill up a large part of the phase space and it has been asked whether the periodic islands form a set of full measure. In this paper we study the periodic islands around an important family of periodic orbits and and demonstrate that for all angle parameters that are irrational multiples of π the islands have asymptotic density in the plane of 3 log 2 - π2/8 ≈ 0.846.

  • (with P. Hubert and H. Masur) Dichotomy for the Hausdorff dimension of the set of nonergodic directions [ pdf ], Inventiones, 183, (2011), 337-383.
      Abstract: We consider billiards in a (1/2)-by-1 rectangle with a barrier midway along a vertical side. Let NE be the set of directions theta such that the flow in direction theta is not ergodic. We show that the Hausdorff dimension of the set NE is either 0 or 1/2, with the latter occurring if and only if the length of the barrier satisfies the condition that the sum of (loglog qk+1)/qk is finite, where qk is the denominator of the kth convergent of the length of the barrier.
  • Hausdorff dimension of the set of Singular Pairs [ pdf ], Annals of Math., 173, (2011), 127-167.
      Abstract: In this paper we show that the Hausdorff dimension of the set of singular pairs is 4/3. We also show that the action of diag(et,et,e-2t) on SL3R/SL3Z admits divergent trajectories that exit to infinity at arbitrarily slow prescribed rates, answering a question of A.N. Starkov. As a by-product of the analysis, we obtain a higher dimensional generalisation of the basic inequalities satisfied by convergents of continued fractions. As an illustration of the techniques used to compute Hausdorff dimension, we show that the set of real numbers with divergent partial quotients has Hausdorff dimension 1/2.
  • (with P.Hubert and H.Masur) Topological Dichotomy and Strict Ergodicity for Translation Surfaces [ pdf ] Ergod. Th. Dynam. Sys., 28 (2008), 1729--1748.
      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.
  • (with A. Eskin) Unique Ergodicity of Translation Flows [ pdf | ps ] Fields Institute Communications 51 (2007) 213-222.
      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.
  • 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.
      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.
  • (with H. Masur) Minimal nonergodic directions on genus 2 translation surfaces [ pdf | ps ] Ergod. Th. Dynam. Sys. 26 (2006), 341--351.
      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.
  • (with H. Masur) A divergent Teichmuller geodesic with uniquely ergodic vertical foliation [ pdf | ps ] (Israel J. Math. 152 (2006), 1--15.
      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.
  • Slowly divergent geodesics in moduli space [ pdf | ps ] (Conform. Geom. Dyn. 8 (2004), 167--189.)
      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.
  • Hausdorff dimension of the set of nonergodic directions [ pdf | ps ] (Ann. of Math., 158 (2003), 661--678.)
      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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      Last updated: August 28, 2012