Quantitative recurrence results

Boshernitzan, Michael

doi:10.1007/bf01244320

Cited by 140 publications

(128 citation statements)

References 9 publications

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“…Mariusz Urbanski [4] and S is small enough. Suppose that the Perron-Frobenius operator £ M preserves C b (X) and H ?…”

Section: Loosely Markov Mapsmentioning

confidence: 99%

Recurrence rates for loosely Markov dynamical systems

Urbański¹

2007

J. Aust. Math. Soc.

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The concept of loosely Markov dynamical systems is introduced. We show that for these systems the recurrence rates and pointwise dimensions coincide. The systems generated by hyperbolic exponential maps, arbitrary rational functions of the Riemann sphere, and measurable dynamical systems generated by infinite conformal iterated function systems are all checked to be loosely Markov.2000 Mathematics subject classification: primary 37C45; secondary 37C40, 37F35, 37A25.

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“…Mariusz Urbanski [4] and S is small enough. Suppose that the Perron-Frobenius operator £ M preserves C b (X) and H ?…”

Section: Loosely Markov Mapsmentioning

confidence: 99%

Recurrence rates for loosely Markov dynamical systems

Urbański¹

2007

J. Aust. Math. Soc.

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“…A natural approach is to use the metric of the space X and consider a decreasing sequence of balls ( [7], [4], [3], [25], [26], [18], [16]) in the above question. More precisely, let us define τ r (x) to be the first return time of x into the ball B(x, r) centered in x and with radius r. It turns out that in many chaotic systems the asymptotic behaviour of τ r (x), when r is small is related to the local dimension of the invariant measure at x:…”

Section: Introductionmentioning

confidence: 99%

The recurrence time for ergodic systems with infinite invariant measures

Galatolo¹,

Kim²,

Park³

2006

Nonlinearity

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Abstract. We investigate quantitative recurrence in systems having an infinite measure. We extend the Ornstein-Weiss theorem for a general class of infinite systems estimating return time in decreasing sequences of cylinders. Then we restrict to a class of one dimensional maps with indifferent fixed points and calculate quantitative recurrence in sequences of balls, obtaining that this is related to the behavior of the map near the fixed points.

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“…Results of Boshernitzan imply that dim H (µ) ≥ 1 2 for any T -invariant Borel probability measure for almost every α, β and for any α, β which are algebraic [Bo1]. The Hausdorff dimension of Ω has been computed before in [BK] for one particular example.…”

Section: Statement Of Resultsmentioning

confidence: 99%