Logarithm laws and shrinking target properties
Tóm tắt
We survey some of the recent developments in the study of logarithm laws and shrinking target properties for various families of dynamical systems. We discuss connections to geometry, diophantine approximation and probability theory.
Tài liệu tham khảo
Athreya J S, Ghosh A and Prasad A, Ultrametric Logarithm Laws I, Discrete and Continuous Dynamical Systems-S 2(2) (2009) 337–348
Athreya J S and Margulis G A, Logarithm laws for unipotent flows, I, preprint, arxiv.org/0811.2806 [math.DS]
Athreya J S and Margulis G A, Logarithm laws for unipotent flows, II, in preparation
Athreya J S and Minsky Y, in preparation
Athreya J S and Ulcigrai C, in preparation
Avila A and Forni G, Weak mixing for interval exchange transformations and translation flows, Ann. Math. 165 (2007) 637–664
Boshernitzan M and Chaika J, in preparation
Chernov N and Kleinbock D Y, Dynamical Borel-Cantelli lemmas for Gibbs measures, Israel J. Math. 122 (2001) 1–27
Dolgopyat D, Limit theorems for partially hyperbolic systems, Trans. Amer. Math. Soc. 356(4) (2004) 1637–1689 (electronic)
Fayad B, Mixing in the absence of the shrinking target property, Bull. London Math. Soc. 38(5) (2006) 829–838
Galotolo S and Kim D, The dynamical Borel-Cantelli lemma and the waiting time problems, preprint, arXiv:math/0610213v2 [math.DS]
Gorodnik A and Shah N, Khinchin theorem for integral points on quadratic varieties, preprint, arXiv:math/0804.3530v1 [math.NT]
Hedlund G, Fuchsian groups and transitive horocycles, Duke Math. J. 2(3) (1936) 530–542
Hill B and Velani S, The ergodic theory of shrinking targets, Invent. Math. 119 (1995) 175–198
Hersonsky S and Paulin F, Hausdorff dimension of diophantine geodesics in negatively curved manifolds, J. Reine Agnew. Math. 539 (2001) 29–43
Hersonsky S and Paulin F, A logarithm law for automorphism groups of trees, Arch. Math. (Basel) 88(2) (2007) 97–108
Hersonsky S and Paulin F, On the almost sure spiraling of geodesics in negatively curved manifolds, preprint, arXiv:0708.3389v2 [math.DG]
Katok A and Hasselblatt B, Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its Applications 54 (Cambridge University Press) (1995) 822 pp.
Keynes HB and Newton D, A Minimal, Non-Uniquely Ergodic Interval Exchange Transformation, Math. Z. 148 (1976) 101–105
Kim D, The shrinking target property of irrational rotations, Nonlinearity 20 (2007) 1637–1643
Kleinbock D Y and Margulis G A, Logarithm laws for flows on homogeneous spaces, Invent. Math. 138(3) (1999) 451–494
Kurzweil J, On the metric theory of inhomogeneous Diophantine approximations, Studia Math. 15 (1955) 84–112
Masur H, Interval exchange transformations and measured foliations, Ann. Math. 115 (1982) 169–200
Masur H, Logarithmic lawfor geodesics in moduli space, Mapping class groups and moduli spaces of Riemann surfaces (Gttingen, 1991/Seattle, WA, 1991), 229–245, Contemp. Math., 150, Amer. Math. Soc. (RI: Providence) (1993)
Maucourant F, Dynamical Borel-Cantelli lemma for hyperbolic spaces, Israel J. Math. 152 (2006) 143–155
Philipp W, Some metrical theorems in number theory, Pacific J. Math. 20 (1967) 109–127
Stratmann B and Velani S, The Patterson measure for geometrically finite groups with parabolic elements, new and old, Proc. London Math. Soc. (1995) 197–220
Sullivan D, Disjoint spheres, approximation by quadratic numbers and the logarithm law for geodesics, Acta Mathematica 149 (1982) 215–237
Tseng J, On circle rotations and the shrinking target properties, Discrete Contin. Dyn. Syst. 20(4) (2008) 1111–1122
Veech W, Gauss measures for transformations on the space of interval exchange maps, Ann. Math. 115 (1982) 201–242
Yoccoz J-C, Continued Fraction Algorithms for Interval Exchange Maps: an Introduction, in “Frontiers in Number Theory, Geometry and Physics, Proceedings of the Spring School at Les Houches, France (March 2003)