39 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
Equidistribution of expanding translates of curves and Dirichlet's theorem on Diophantine approximation
We show that for almost all points on any analytic curve on R^{k} which is
not contained in a proper affine subspace, the Dirichlet's theorem on
simultaneous approximation, as well as its dual result for simultaneous
approximation of linear forms, cannot be improved. The result is obtained by
proving asymptotic equidistribution of evolution of a curve on a strongly
unstable leaf under certain partially hyperbolic flow on the space of
unimodular lattices in R^{k+1}. The proof involves ergodic properties of
unipotent flows on homogeneous spaces.Comment: 26 page
Decay of correlations for maps with uniformly contracting fibers and logarithm law for singular hyperbolic attractors
We consider two dimensional maps preserving a foliation which is uniformly
contracting and a one dimensional associated quotient map having exponential
convergence to equilibrium (iterates of Lebesgue measure converge exponentially
fast to physical measure). We prove that these maps have exponential decay of
correlations over a large class of observables. We use this result to deduce
exponential decay of correlations for the Poincare maps of a large class of
singular hyperbolic flows. From this we deduce logarithm laws for these flows.Comment: 39 pages; 03 figures; proof of Theorem 1 corrected; many typos
corrected; improvements on the statements and comments suggested by a
referee. Keywords: singular flows, singular-hyperbolic attractor, exponential
decay of correlations, exact dimensionality, logarithm la
Equidistribution for higher-rank Abelian actions on Heisenberg nilmanifolds
2010 Mathematics Subject Classification: Primary: 37C85, 37A17, 37A45; Secondary: 11K36, 11L07.We prove quantitative equidistribution results for actions of Abelian subgroups of the (2g + 1)-dimensional Heisenberg group acting on compact (2g + 1)-dimensional homogeneous nilmanifolds. The results are based on the study of the C∞-cohomology of the action of such groups, on tame estimates of the associated cohomological equations and on a renormalization method initially applied by Forni to surface flows and by Forni and the second author to other parabolic flows. As an application we obtain bounds for finite Theta sums defined by real quadratic forms in g variables, generalizing the classical results of Hardy and Littlewood [25, 26] and the optimal result of Fiedler, Jurkat, and Körner [17] to higher dimension.This work was partially done while L. Flaminio visited the Isaac Newton Institute in Cambridge, UK. He wishes to thank the Institute and the organizers of the program Interactions between Dynamics of Group Actions and Number Theory for their hospitality. L. Flaminio was supported in part by the Labex CEMPI (ANR-11-LABX-07). S. Cosentino was partially supported by CMAT - Centro de Matematica da Universidade do Minho, financed by the Strategic Project PEst-OE/MAT/UI0013/2014