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Logarithm laws and shrinking target properties.(English)Zbl 1184.37004

Author’s abstract: 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.

MSC:

37A05 Dynamical aspects of measure-preserving transformations
37A30 Ergodic theorems, spectral theory, Markov operators

Cite

References:

[1]Athreya J S, Ghosh A and Prasad A, Ultrametric Logarithm Laws I, Discrete and Continuous Dynamical Systems-S 2(2) (2009) 337–348 ·Zbl 1175.37009 ·doi:10.3934/dcdss.2009.2.337
[2]Athreya J S and Margulis G A, Logarithm laws for unipotent flows, I, preprint, arxiv.org/0811.2806 [math.DS]
[3]Athreya J S and Margulis G A, Logarithm laws for unipotent flows, II, in preparation ·Zbl 1184.37007
[4]Athreya J S and Minsky Y, in preparation
[5]Athreya J S and Ulcigrai C, in preparation
[6]Avila A and Forni G, Weak mixing for interval exchange transformations and translation flows, Ann. Math. 165 (2007) 637–664 ·Zbl 1136.37003 ·doi:10.4007/annals.2007.165.637
[7]Boshernitzan M and Chaika J, in preparation
[8]Chernov N and Kleinbock D Y, Dynamical Borel-Cantelli lemmas for Gibbs measures, Israel J. Math. 122 (2001) 1–27 ·Zbl 0997.37002 ·doi:10.1007/BF02809888
[9]Dolgopyat D, Limit theorems for partially hyperbolic systems, Trans. Amer. Math. Soc. 356(4) (2004) 1637–1689 (electronic) ·Zbl 1031.37031 ·doi:10.1090/S0002-9947-03-03335-X
[10]Fayad B, Mixing in the absence of the shrinking target property, Bull. London Math. Soc. 38(5) (2006) 829–838 ·Zbl 1194.37013 ·doi:10.1112/S0024609306018546
[11]Galotolo S and Kim D, The dynamical Borel-Cantelli lemma and the waiting time problems, preprint, arXiv:math/0610213v2 [math.DS]
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[14]Hill B and Velani S, The ergodic theory of shrinking targets, Invent. Math. 119 (1995) 175–198 ·Zbl 0834.28009 ·doi:10.1007/BF01245179
[15]Hersonsky S and Paulin F, Hausdorff dimension of diophantine geodesics in negatively curved manifolds, J. Reine Agnew. Math. 539 (2001) 29–43 ·Zbl 0994.53021 ·doi:10.1515/crll.2001.077
[16]Hersonsky S and Paulin F, A logarithm law for automorphism groups of trees, Arch. Math. (Basel) 88(2) (2007) 97–108 ·Zbl 1160.20020
[17]Hersonsky S and Paulin F, On the almost sure spiraling of geodesics in negatively curved manifolds, preprint, arXiv:0708.3389v2 [math.DG]
[18]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. ·Zbl 0878.58020
[19]Keynes HB and Newton D, A Minimal, Non-Uniquely Ergodic Interval Exchange Transformation, Math. Z. 148 (1976) 101–105 ·Zbl 0308.28014 ·doi:10.1007/BF01214699
[20]Kim D, The shrinking target property of irrational rotations, Nonlinearity 20 (2007) 1637–1643 ·Zbl 1116.37029 ·doi:10.1088/0951-7715/20/7/006
[21]Kleinbock D Y and Margulis G A, Logarithm laws for flows on homogeneous spaces, Invent. Math. 138(3) (1999) 451–494 ·Zbl 0934.22016 ·doi:10.1007/s002220050350
[22]Kurzweil J, On the metric theory of inhomogeneous Diophantine approximations, Studia Math. 15 (1955) 84–112 ·Zbl 0066.03702
[23]Masur H, Interval exchange transformations and measured foliations, Ann. Math. 115 (1982) 169–200 ·Zbl 0497.28012 ·doi:10.2307/1971341
[24]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)
[25]Maucourant F, Dynamical Borel-Cantelli lemma for hyperbolic spaces, Israel J. Math. 152 (2006) 143–155 ·Zbl 1129.53057 ·doi:10.1007/BF02771980
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[28]Sullivan D, Disjoint spheres, approximation by quadratic numbers and the logarithm law for geodesics, Acta Mathematica 149 (1982) 215–237 ·Zbl 0517.58028 ·doi:10.1007/BF02392354
[29]Tseng J, On circle rotations and the shrinking target properties, Discrete Contin. Dyn. Syst. 20(4) (2008) 1111–1122 ·Zbl 1151.37004 ·doi:10.3934/dcds.2008.20.1111
[30]Veech W, Gauss measures for transformations on the space of interval exchange maps, Ann. Math. 115 (1982) 201–242 ·Zbl 0486.28014 ·doi:10.2307/1971391
[31]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)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
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