80 research outputs found
Logarithm laws for flows on homogeneous spaces
We prove that almost all geodesics on a noncompact locally symmetric space of
finite volume grow with a logarithmic speed -- the higher rank generalization
of a theorem of D. Sullivan (1982). More generally, under certain conditions on
a sequence of subsets of a homogeneous space ( a semisimple
Lie group, a non-uniform lattice) and a sequence of elements of
we prove that for almost all points of the space, one has for infinitely many .
The main tool is exponential decay of correlation coefficients of smooth
functions on . Besides the aforementioned application to geodesic
flows, as a corollary we obtain a new proof of the classical Khinchin-Groshev
theorem in simultaneous Diophantine approximation, and settle a related
conjecture recently made by M. Skriganov
Khintchine-type theorems on manifolds: the convergence case for standard and multiplicative versions
An analogue of the convergence part of the Khintchine-Groshev theorem, as
well as its multiplicative version, is proved for nondegenerate smooth
submanifolds in . The proof combines methods from metric number
theory with a new approach involving the geometry of lattices in Euclidean
spaces.Comment: 27 page
Singular systems of linear forms and non-escape of mass in the space of lattices
Singular systems of linear forms were introduced by Khintchine
in the 1920s, and it was shown by Dani in the 1980s that they
are in one-to-one correspondence with certain divergent orbits of oneparameter
diagonal groups on the space of lattices. We give a (conjecturally
sharp) upper bound on the Hausdor dimension of the set of
singular systems of linear forms (equivalently the set of lattices with divergent
trajectories) as well as the dimension of the set of lattices with
trajectories `escaping on average' (a notion weaker than divergence).
This extends work by Cheung, as well as by Chevallier and Cheung.
Our method di ers considerably from that of Cheung and Chevallier,
and is based on the technique of integral inequalities developed by Eskin,
Margulis and Mozes
Singular systems of linear forms and non-escape of mass in the space of lattices
Singular systems of linear forms were introduced by Khintchine
in the 1920s, and it was shown by Dani in the 1980s that they
are in one-to-one correspondence with certain divergent orbits of oneparameter
diagonal groups on the space of lattices. We give a (conjecturally
sharp) upper bound on the Hausdor dimension of the set of
singular systems of linear forms (equivalently the set of lattices with divergent
trajectories) as well as the dimension of the set of lattices with
trajectories `escaping on average' (a notion weaker than divergence).
This extends work by Cheung, as well as by Chevallier and Cheung.
Our method di ers considerably from that of Cheung and Chevallier,
and is based on the technique of integral inequalities developed by Eskin,
Margulis and Mozes
Schmidt games and Markov partitions
Let T be a C^2-expanding self-map of a compact, connected, smooth, Riemannian
manifold M. We correct a minor gap in the proof of a theorem from the
literature: the set of points whose forward orbits are nondense has full
Hausdorff dimension. Our correction allows us to strengthen the theorem.
Combining the correction with Schmidt games, we generalize the theorem in
dimension one: given a point x in M, the set of points whose forward orbit
closures miss x is a winning set.Comment: 32 page
Extremality and dynamically defined measures, part I: Diophantine properties of quasi-decaying measures
We present a new method of proving the Diophantine extremality of various dynamically defined measures, vastly expanding the class of measures known to be extremal. This generalizes and improves the celebrated theorem of Kleinbock and Margulis (’98) resolving Sprindžuk’s conjecture, as well as its extension by Kleinbock, Lindenstrauss, and Weiss (’04), hereafter abbreviated KLW. As applications we prove the extremality of all hyperbolic measures of smooth dynamical systems with sufficiently large Hausdorff dimension, of the Patterson–Sullivan measures of all nonplanar geometrically finite groups, and of the Gibbs measures (including conformal measures) of infinite iterated function systems. The key technical idea, which has led to a plethora of new applications, is a significant weakening of KLW’s sufficient conditions for extremality. In Part I, we introduce and develop a systematic account of two classes of measures, which we call quasi-decaying and weakly quasi-decaying. We prove that weak quasi-decay implies strong extremality in the matrix approximation framework, thus proving a conjecture of KLW. We also prove the “inherited exponent of irrationality” version of this theorem, describing the relationship between the Diophantine properties of certain subspaces of the space of matrices and measures supported on these subspaces. In subsequent papers, we exhibit numerous examples of quasi-decaying measures, in support of the thesis that “almost any measure from dynamics and/or fractal geometry is quasi-decaying”. We also discuss examples of non-extremal measures coming from dynamics, illustrating where the theory must halt
Non-planarity and metric Diophantine approximation for systems of linear forms
In this paper we develop a general theory of metric Diophantine approximation for systems of linear forms. A new notion of `weak non-planarity' of manifolds and more generally measures on the space of mxn matrices over R is introduced and studied. This notion generalises the one of non-planarity in R^n and is used to establish strong (Diophantine) extremality of manifolds and measures. The notion of weak non-planarity is shown to be `near optimal' in a certain sense. Beyond the above main theme of the paper, we also develop a corresponding theory of inhomogeneous and weighted Diophantine approximation. In particular, we extend the recent inhomogeneous transference results due to Beresnevich and Velani and use them to bring the inhomogeneous theory in balance with its homogeneous counterpart
Khinchin theorem for integral points on quadratic varieties
We prove an analogue the Khinchin theorem for the Diophantine approximation
by integer vectors lying on a quadratic variety. The proof is based on the
study of a dynamical system on a homogeneous space of the orthogonal group. We
show that in this system, generic trajectories visit a family of shrinking
subsets infinitely often.Comment: 19 page
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