Divergent trajectories and ℚ-rank

Weiss, Barak

doi:10.1007/bf02771984

Cited by 12 publications

(12 citation statements)

References 3 publications

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“…The study of singular systems can thus be viewed as a special case of the problem of understanding divergent trajectories of a one-parameter diagonal action where there are exactly one positive and one negative Lyapunov exponent. This viewpoint was initiated by S.G. Dani in [10], where Khintchine's results are generalized to this broader setup, and further developed by Barak Weiss ([29] and [30]) and by Kleinbock and Weiss [17].…”

Section: Introductionmentioning

confidence: 99%

Hausdorff dimension of singular vectors

Cheung¹,

Chevallier²

2016

Duke Math. J.

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We prove that the set of singular vectors in R d , d ≥ 2, has Hausdorff dimension d 2 d+1 and that the Hausdorff dimension of the set of ε-Dirichlet improvable vectors in R d is roughly d 2 d+1 plus a power of ε between d 2 and d. As a corollary, the set of divergent trajectories of the flow by diag(e t , . . . , e t , e −dt ) acting on SL d+1 (R)/ SL d+1 (Z) has Hausdorff codimension d d+1 . These results extend the work of the first author in [6].

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Section: Introductionmentioning

confidence: 99%

Hausdorff dimension of singular vectors

Cheung¹,

Chevallier²

2016

Duke Math. J.

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“…Since g t ph and g t h differ by the action of an element from a bounded set, nondivergence of g t h is equivalent to that of g t ph. Therefore, Further results about singular vectors and divergent trajectories can be found in the papers [11], [12], [22], and [23].…”

Section: Introductionmentioning

confidence: 99%

Hausdorff dimension of the set of singular pairs

Cheung

2011

Ann. Math.

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In this paper we show that the Hausdorff dimension of the set of singular pairs is . We also show that the action of diag(e t , e t , 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 generalization of the basic inequalities satisfied by convergents of continued fractions. As an illustration of the technique used to compute Hausdorff dimension, we reprove a result of I. J. Good asserting that the Hausdorff dimension of the set of real numbers with divergent partial quotients is 1 2.

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“…The case D = A was settled in [243]. Recently, Conjecture 25(1) was proved in [35] when rank Q G = 2 and in [257] in complete generality. We also mention that Conjecture 25(3) was checked in [256] for G = SL(4, R) for all diagonal subgroups D except…”

Section: Divergent Trajectoriesmentioning

confidence: 99%

Open problems in dynamics and related fields

Gorodnik¹

2007

Journal of Modern Dynamics

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Divergent trajectories and ℚ-rank

Cited by 12 publications

References 3 publications

Hausdorff dimension of singular vectors

Hausdorff dimension of singular vectors

Hausdorff dimension of the set of singular pairs

Open problems in dynamics and related fields

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