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Yitwah Cheung

Publications and source records attributed to Yitwah Cheung.

14 recordsLinked to original sources
arXiv2024

BCZ map is weakly mixing

The BCZ map was introduced in 2001 by Boca, Cobeli and Zaharescu as a tool to study the statistical properties of Farey sequences, whose relation to Riemann Hypothesis dates back to Franel and Landau. Later, J. Athreya and the first author observed that the BCZ map arises as a Poincare section of horocycle flow, establishing both ergodicity as well as zero measure-theoretic entropy. In this article, we prove that the BCZ map is weakly mixing, answering the last remaining question about the BCZ map raised in a 2006 survey by Boca and Zaharescu. The proof uses a self-similarity property of the BCZ map that derives from a well-known fact that horocycle flow is renormalized by the geodesic flow, a property already observed in arXiv:1206.6597. We note that the questions of mixing and rigidity remain open.

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arXiv2022

Levy-Khintchin Theorem for best simultaneous Diophantine approximations

We extend two results about the ordinary continued fraction expansion to best simultaneous Diophantine approximations of vectors or matrices. The first is Levy-Khintchin Theorem about the almost sure growth rate of the denominators of the convergents. The second is a Theorem of Bosma, Hendrik and Wiedijk about the almost sure limit distribution of the sequence of products $q_n d(q_nθ, Z)$ where the $q_n$'s are the denominators of the convergents associated with the real number $θ$ by the ordinary continued fraction algorithm. Beside these two main results, we show that when $d\ge2$, for almost all vectors $θ\in R^d$, $\liminf_{n\to\infty} q_{n+k}d(q_nθ, Z^d)=0$ for all positive integers $k$, where $(q_n)_{n\in N}$ is the sequence of best approximation denominators of $θ$.

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arXiv2021

About the value of the two dimensional Levy's constant

We give a numerical approximation of the Lévy constant on the growth of the denominators of the best Diophantine approximations in dimension 2 with respect to the euclidean norm. This constant is expressed as an integral on a surface of dimension 7. We reduce the computation of this integral to a triple integral, whose numerical evaluation was carried out in \cite{Xieu}.

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arXiv2019

Siegel-Veech transforms are in $L^2$

Let $\mathcal{H}$ denote a connected component of a stratum of translation surfaces. We show that the Siegel-Veech transform of a bounded compactly supported function on $\mathbb{R}^2$ is in $L^2(\mathcal{H}, μ)$, where $μ$ is Lebesgue measure on $\mathcal{H}$, and give applications to bounding error terms for counting problems for saddle connections. We also propose a new invariant associated to $SL(2, \mathbb{R})$-invariant measures on strata satisfying certain integrability conditions.

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arXiv2016

Hausdorff dimension and uniform exponents in dimension two

In this paper we prove the Hausdorff dimension of the set of (nondegenerate) singular two-dimensional vectors with uniform exponent $μ$ $\in$ (1/2, 1) is 2(1 -- $μ$) when $μ$ $\ge$ $\sqrt$ 2/2, whereas for $μ$ \textless{} $\sqrt$ 2/2 it is greater than 2(1 -- $μ$) and at most (3 -- 2$μ$)(1 -- $μ$)/(1 + $μ$ + $μ$ 2). We also establish that this dimension tends to 4/3 (which is the dimension of the set of singular two-dimensional vectors) when $μ$ tends to 1/2. These results improve upon previous estimates of R. Baker, joint work of the first author with M. Laurent, and unpublished work of M. Laurent. We also prove a lower bound on the packing dimension that is strictly greater than the Hausdorff dimension for $μ$ $\ge$ 0.565. .. .

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arXiv2012

A Poincaré section for horocycle flow on the space of lattices

We construct a Poincaré section for the horocycle flow on the modular surface $SL(2, \R)/SL(2, \Z)$, and study the associated first return map, which coincides with a transformation (the {\it BCZ map}) defined by Boca-Cobeli-Zaharescu. We classify ergodic invariant measures for this map and prove equidistribution of periodic orbits. As corollaries, we obtain results on the average depth of cusp excursions and on the distribution of gaps for Farey sequences and slopes of lattice vectors.

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arXiv2011

Divergent directions in some periodic wind-tree models

The periodic wind-tree model is a family T(a,b) of billiards in the plane in which identical rectangular scatterers of size axb are disposed at each integer point. It was proven by P. Hubert, S. Lelièvre and S. Troubetzkoy (arXiv:0912.2891v1) that for a residual set of parameters (a,b) the billiard flow in T(a,b) is recurrent in almost every direction. We prove that for many parameters (a,b) there exists a set S of angles of positive Hausdorff dimension such that every billiard trajectory in T(a,b) with initial angle in S is self-avoiding. In particular, the flow in a direction of S is divergent.

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arXiv2010

Dichotomy for the Hausdorff dimension of the set of nonergodic directions

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 of P'erez Marco, i.e. the sum of (loglog q_{k+1})/q_k is finite, where q_k is the the denominator of the kth convergent of the length of the barrier.

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arXiv2008

Hausdorff dimension of the set of singular pairs

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(e^t,e^t,e^{-2t}) on SL(3,R)/SL(3,Z) 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.

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arXiv2007

Slow Divergence and Unique Ergodicity

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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arXiv2007

Topological Dichotomy and Strict Ergodicity for Translation Surfaces

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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arXiv2006

Unique Ergodicity of Translation Flows

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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