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Aperiodic sequences and aperiodic geodesics


We introduce a quantitative condition on orbits of dynamical systems, which measures their aperiodicity. We show the existence of sequences in the Bernoulli shift and geodesics on closed hyperbolic manifolds which are as aperiodic as possible with respect to this condition.

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[1]Bernik, V. I. and Dodson, M. M.. Metric Diophantine Approximation on Manifolds. Vol. 137. Cambridge University Press, Cambridge, 1999.
[2]Boshernitzan, M. D.. Quantitative recurrence results. Invent. Math. 112 (1993), 617631.
[3]Bridson, M. R. and Haeflinger, A.. Metric Spaces of Non-positive Curvature. Springer, Berlin, 1999.
[4]Eberlein, P.. Geometry of Nonpositively Curved Manifolds. University of Chicago Press, Chicago, 1994.
[5]Einsiedler, M. and Ward, T.. Ergodic Theory: with a view towards Number Theory. Springer, London, 2011.
[6]Furstenberg, H.. Recurrence in Ergodic Theory and Combinatorial Number Theory. Vol. 2. Princeton University Press, Princeton, NJ, 1981.
[7]Haas, A.. Geodesic cusp excursions and metric diophantine approximation. Math. Res. Lett. 16 (2009), 6785.
[8]Heintze, E. and Im Hof, H. C.. Geometry of horospheres. J. Differential Geom. 12 (4) (1977), 481491.
[9]Hersonsky, S. and Paulin, F.. Diophantine approximation for negatively curved manifolds. Math. Z. 241 (2002), 181226.
[10]Hersonsky, S. and Paulin, F.. On the almost sure spiraling of geodesics in negatively curved manifolds. J. Differential Geom. (2) 85 (2010), 271314.
[11]Morse, M. and Hedlund, G.. Unending chess, symbolic dynamics and a problem in semigroups. Duke Math. J. 11 (1944), 17.
[12]Ornstein, D. and Weiss, B.. Entropy and recurrence rates for stationary random fields. IEEE Trans. Inform. Theory 48 (6) (2002), 16941697.
[13]Parkkonen, J. and Paulin, F.. Prescribing the behaviour of geodesics in negative curvature. Geom. Topol. 14 (2010), 277392.
[14]Parkkonen, J. and Paulin, F.. Spiraling spectra of geodesic lines in negatively curved manifolds. Math. Z. 268 (2011), 101142.
[15]Patterson, S. J.. Diophantine approximation in Fuchsian groups. Philos. Trans. R. Soc. Lond. Ser. A 282 (1976), 527563.
[16]Robinson, J. C.. Dimensions, Embeddings and Attractors. Vol. 186. Cambridge University Press, Cambridge, 2011.
[17]Sullivan, D.. Disjoint spheres, approximation by imaginary quadratic numbers, and the logarithm law for geodesics. Acta Math. 149 (1982), 215237.
[18]Velani, S. J.. Diophantine approximation and Hausdorff-dimension in Fuchsian groups. Math. Proc. Cambridge Philos. Soc. 113 (1993), 343354.
[19]Ballmann, W., Gromov, M. and Schroeder, V.. Manifolds of Nonpositive Curvature. Vol. 61. Birkhäuser, Boston, 1985.
[20]Walters, P.. Introduction to Ergodic Theory. Vol. 79. Springer, New York, 1981.
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Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
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