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ITERATED LOGARITHM SPEED OF RETURN TIMES

Published online by Cambridge University Press:  04 October 2017

ŁUKASZ PAWELEC*
Affiliation:
Department of Mathematics and Mathematical Economics, Warsaw School of Economics, al. Niepodległości 162, 02–554 Warszawa, Poland email LPawel@sgh.waw.pl
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Abstract

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In a general setting of an ergodic dynamical system, we give a more accurate calculation of the speed of the recurrence of a point to itself (or to a fixed point). Precisely, we show that for a certain $\unicode[STIX]{x1D709}$ depending on the dimension of the space, $\liminf _{n\rightarrow +\infty }(n\log \log n)^{\unicode[STIX]{x1D709}}d(T^{n}x,x)=0$ almost everywhere and $\liminf _{n\rightarrow +\infty }(n\log \log n)^{\unicode[STIX]{x1D709}}d(T^{n}x,y)=0$ for almost all $x$ and $y$. This is done by assuming the exponential decay of correlations and making a weak assumption on the invariant measure.

Type
Research Article
Copyright
© 2017 Australian Mathematical Publishing Association Inc. 

References

Barreira, L. and Saussol, B., ‘Hausdorff dimension of measures via Poincaré recurrence’, Commun. Math. Phys. 219 (2001), 443463.Google Scholar
Boshernitzan, M. D., ‘Quantitative recurrence results’, Invent. Math. 113 (1993), 617631.Google Scholar
Boyarski, A. and Góra, P., Laws of Chaos: Invariant Measures and Dynamical Systems in One Dimension (Birkhäuser, Boston, 1997).Google Scholar
Mauldin, R. D. and Urbański, M., ‘Fractal measures for parabolic IFS’, Adv. Math. 136 (1998), 225253.Google Scholar
Mauldin, R. D. and Urbański, M., Graph Directed Markov Systems: Geometry and Dynamics of Limit Sets (Cambridge University Press, Cambridge, 2003).Google Scholar
Persson, T. and Rams, M., ‘On shrinking targets for piecewise expanding interval maps’, Ergod. Th. & Dynam. Syst. 37 (2017), 646663.CrossRefGoogle Scholar
Przytycki, F. and Urbański, M., Conformal Fractals: Ergodic Theory Methods (Cambridge University Press, Cambridge, 2010).Google Scholar
Sullivan, D., ‘Conformal dynamical systems’, in: Geometric Dynamics, Lecture Notes in Mathematics, 1007 (ed. Palis, J.) (Springer, Berlin, Heidelberg, 1983), 725752.Google Scholar
Urbański, M., ‘Recurrence rates for loosely Markov dynamical systems’, J. Aust. Math. Soc. 82 (2007), 3957.Google Scholar