American Institute of Mathematical Sciences

Advanced
July  2004, 10(3): 581-587. doi: 10.3934/dcds.2004.10.581

On the law of logarithm of the recurrence time

Chihurn Kim 1, and  Dong Han Kim 2,

1. 

Department of Mathematics, Korea Advanced Institute of Science and Technology, Daejeon, 305-701, South Korea
2. 

School of Mathematics, Korea Institute for Advanced Study, Seoul, 130-722, South Korea

Received  November 2002 Revised  May 2003 Published  January 2004

Let $T$ be a transformation from $I=[0,1)$ onto itself and let $Q_n(x)$ be the subinterval $[i/2^n,(i+1)/2^n)$, $0 \leq i < 2^n$ containing $x$. Define $K_n (x) =$min{$j\geq 1:T^j (x)\in Q_n(x)$} and $K_n(x,y) =$min{$j\geq 1:T^{j-1} (y) \in Q_n(x)$}. For various transformations defined on $I$, we show that

$ \lim_{n\to\infty}\frac{\log K_n(x)}{n}=1 \quad$and$\quad \lim_{n\to\infty}\frac{\log K_n(x,y)}{n}=1 \quad $a.e.

Keywords: waiting time., the first return time, Recurrence time.
Mathematics Subject Classification: Primary: 28D05, 37E0.
Citation: Chihurn Kim, Dong Han Kim. On the law of logarithm of the recurrence time. Discrete & Continuous Dynamical Systems - A, 2004, 10 (3) : 581-587. doi: 10.3934/dcds.2004.10.581
[1]

Jean-René Chazottes, Renaud Leplaideur. Fluctuations of the nth return time for Axiom A diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2005, 13 (2) : 399-411. doi: 10.3934/dcds.2005.13.399
[2]

Piotr Oprocha. Chain recurrence in multidimensional time discrete dynamical systems. Discrete & Continuous Dynamical Systems - A, 2008, 20 (4) : 1039-1056. doi: 10.3934/dcds.2008.20.1039
[3]

Hideaki Takagi. Unified and refined analysis of the response time and waiting time in the M/M/m FCFS preemptive-resume priority queue. Journal of Industrial & Management Optimization, 2017, 13 (4) : 1945-1973. doi: 10.3934/jimo.2017026
[4]

Yutaka Sakuma, Atsushi Inoie, Ken’ichi Kawanishi, Masakiyo Miyazawa. Tail asymptotics for waiting time distribution of an M/M/s queue with general impatient time. Journal of Industrial & Management Optimization, 2011, 7 (3) : 593-606. doi: 10.3934/jimo.2011.7.593
[5]

Daniel Glasscock, Andreas Koutsogiannis, Florian Karl Richter. Multiplicative combinatorial properties of return time sets in minimal dynamical systems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5891-5921. doi: 10.3934/dcds.2019258
[6]

Andrey Shishkov. Waiting time of propagation and the backward motion of interfaces in thin-film flow theory. Conference Publications, 2007, 2007 (Special) : 938-945. doi: 10.3934/proc.2007.2007.938
[7]

Zsolt Saffer, Wuyi Yue. A dual tandem queueing system with GI service time at the first queue. Journal of Industrial & Management Optimization, 2014, 10 (1) : 167-192. doi: 10.3934/jimo.2014.10.167
[8]

Jérôme Buzzi, Véronique Maume-Deschamps. Decay of correlations on towers with non-Hölder Jacobian and non-exponential return time. Discrete & Continuous Dynamical Systems - A, 2005, 12 (4) : 639-656. doi: 10.3934/dcds.2005.12.639
[9]

Y. Charles Li, Hong Yang. On the arrow of time. Discrete & Continuous Dynamical Systems - S, 2014, 7 (6) : 1287-1303. doi: 10.3934/dcdss.2014.7.1287
[10]

Simão P. S. Santos, Natália Martins, Delfim F. M. Torres. Variational problems of Herglotz type with time delay: DuBois--Reymond condition and Noether's first theorem. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4593-4610. doi: 10.3934/dcds.2015.35.4593
[11]

Atsushi Kawamoto. Hölder stability estimate in an inverse source problem for a first and half order time fractional diffusion equation. Inverse Problems & Imaging, 2018, 12 (2) : 315-330. doi: 10.3934/ipi.2018014
[12]

Y. Gong, X. Xiang. A class of optimal control problems of systems governed by the first order linear dynamic equations on time scales. Journal of Industrial & Management Optimization, 2009, 5 (1) : 1-10. doi: 10.3934/jimo.2009.5.1
[13]

Qiuying Li, Lifang Huang, Jianshe Yu. Modulation of first-passage time for bursty gene expression via random signals. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1261-1277. doi: 10.3934/mbe.2017065
[14]

Lars Olsen. First return times: multifractal spectra and divergence points. Discrete & Continuous Dynamical Systems - A, 2004, 10 (3) : 635-656. doi: 10.3934/dcds.2004.10.635
[15]

Massimiliano Tamborrino. Approximation of the first passage time density of a Wiener process to an exponentially decaying boundary by two-piecewise linear threshold. Application to neuronal spiking activity. Mathematical Biosciences & Engineering, 2016, 13 (3) : 613-629. doi: 10.3934/mbe.2016011
[16]

Jiao Song, Jiang-Lun Wu, Fangzhou Huang. First jump time in simulation of sampling trajectories of affine jump-diffusions driven by $ \alpha $-stable white noise. Communications on Pure & Applied Analysis, 2020, 19 (8) : 4127-4142. doi: 10.3934/cpaa.2020184
[17]

Arno Berger. On finite-time hyperbolicity. Communications on Pure & Applied Analysis, 2011, 10 (3) : 963-981. doi: 10.3934/cpaa.2011.10.963
[18]

Sören Bartels, Jakob Keck. Adaptive time stepping in elastoplasticity. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020323
[19]

Anna Maria Cherubini, Giorgio Metafune, Francesco Paparella. On the stopping time of a bouncing ball. Discrete & Continuous Dynamical Systems - B, 2008, 10 (1) : 43-72. doi: 10.3934/dcdsb.2008.10.43
[20]

S. Mohamad, K. Gopalsamy. Neuronal dynamics in time varying enviroments: Continuous and discrete time models. Discrete & Continuous Dynamical Systems - A, 2000, 6 (4) : 841-860. doi: 10.3934/dcds.2000.6.841

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (36)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences