On the law of logarithm of the recurrence time

Chihurn Kim; Dong Han Kim

doi:10.3934/dcds.2004.10.581

On the law of logarithm of the recurrence time

Chihurn Kim, Dong Han Kim

Department of Mathematics Education

Korea Advanced Institute of Science and Technology

Research output: Contribution to journal › Article › peer-review

11 Scopus citations

Abstract

Let T be a transformation from I = [0, 1) onto itself and let Q _n(x) be the subinterval [i/2ⁿ, (i+1)/2ⁿ), 0 ≤ i ≤ 2ⁿ containing x. Define K_n(x) = min{j ≥ 1: T^j(x) ∈ Q_n(x)} and K_n(x, y) = min{j ≥ 1: T^j-1(y) ∈ Q_n(x)}. For various transformations defined on I, we show that lim_n→∞ log K_n(x)/n = 1 and lim_n→∞log K_n(x, y)/n = 1 a.e.

Original language	English
Pages (from-to)	581-587
Number of pages	7
Journal	Discrete and Continuous Dynamical Systems- Series A
Volume	10
Issue number	3
DOIs	https://doi.org/10.3934/dcds.2004.10.581
State	Published - Apr 2004

Keywords

Recurrence time
The first return time
Waiting time

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abstract = "Let T be a transformation from I = [0, 1) onto itself and let Q n(x) be the subinterval [i/2n, (i+1)/2n), 0 ≤ i ≤ 2n containing x. Define Kn(x) = min{j ≥ 1: Tj(x) ∈ Qn(x)} and Kn(x, y) = min{j ≥ 1: Tj-1(y) ∈ Qn(x)}. For various transformations defined on I, we show that limn→∞ log Kn(x)/n = 1 and limn→∞log Kn(x, y)/n = 1 a.e.",

keywords = "Recurrence time, The first return time, Waiting time",

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AU - Kim, Chihurn

AU - Kim, Dong Han

PY - 2004/4

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N2 - Let T be a transformation from I = [0, 1) onto itself and let Q n(x) be the subinterval [i/2n, (i+1)/2n), 0 ≤ i ≤ 2n containing x. Define Kn(x) = min{j ≥ 1: Tj(x) ∈ Qn(x)} and Kn(x, y) = min{j ≥ 1: Tj-1(y) ∈ Qn(x)}. For various transformations defined on I, we show that limn→∞ log Kn(x)/n = 1 and limn→∞log Kn(x, y)/n = 1 a.e.

AB - Let T be a transformation from I = [0, 1) onto itself and let Q n(x) be the subinterval [i/2n, (i+1)/2n), 0 ≤ i ≤ 2n containing x. Define Kn(x) = min{j ≥ 1: Tj(x) ∈ Qn(x)} and Kn(x, y) = min{j ≥ 1: Tj-1(y) ∈ Qn(x)}. For various transformations defined on I, we show that limn→∞ log Kn(x)/n = 1 and limn→∞log Kn(x, y)/n = 1 a.e.

KW - Recurrence time

KW - The first return time

KW - Waiting time

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EP - 587

JO - Discrete and Continuous Dynamical Systems- Series A

JF - Discrete and Continuous Dynamical Systems- Series A

IS - 3

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On the law of logarithm of the recurrence time

Abstract

Keywords

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