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

Access to Document

10.3934/dcds.2004.10.581

Cite this

@article{e102fa10f7c04fffa13f07c9986b2abb,

title = "On the law of logarithm of the recurrence time",

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",

author = "Chihurn Kim and Kim, \{Dong Han\}",

year = "2004",

month = apr,

doi = "10.3934/dcds.2004.10.581",

language = "English",

volume = "10",

pages = "581--587",

journal = "Discrete and Continuous Dynamical Systems- Series A",

issn = "1078-0947",

publisher = "American Institute of Mathematical Sciences",

number = "3",

}

TY - JOUR

T1 - On the law of logarithm of the recurrence time

AU - Kim, Chihurn

AU - Kim, Dong Han

PY - 2004/4

Y1 - 2004/4

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

UR - https://www.scopus.com/pages/publications/1042267862

U2 - 10.3934/dcds.2004.10.581

DO - 10.3934/dcds.2004.10.581

M3 - Article

AN - SCOPUS:1042267862

SN - 1078-0947

VL - 10

SP - 581

EP - 587

JO - Discrete and Continuous Dynamical Systems- Series A

JF - Discrete and Continuous Dynamical Systems- Series A

IS - 3

ER -

On the law of logarithm of the recurrence time

Abstract

Keywords

Access to Document

Other files and links

Fingerprint

Cite this