American Institute of Mathematical Sciences

Advanced
February  2010, 27(1): 185-204. doi: 10.3934/dcds.2010.27.185

Long hitting time, slow decay of correlations and arithmetical properties

Stefano Galatolo 1, and  Pietro Peterlongo 2,

1. 

Dipartimento di Matematica Applicata, Via Buonarroti 1, Pisa, 56100, Italy
2. 

Scuola Normale Superiore, Piazza dei Cavalieri 7, Pisa, 56100, Italy

Received  March 2009 Revised  July 2009 Published  February 2010

Let $\tau _r(x,x_0)$ be the time needed for a point $x$ to enter for the first time in a ball $B_r(x_0)$ centered in $x_0$, with small radius $r$. We construct a class of translations on the two torus having particular arithmetic properties (Liouville components with intertwined denominators of convergents) not satisfying a logarithm law, i.e. such that for typical $x,x_0$

liminfr → 0 $ \frac{\log \tau _r(x,x_0)}{-\log r} = \infty.$


   By considering a suitable reparametrization of the flow generated by a suspension of this translation, using a previous construction by Fayad, we show the existence of a mixing system on three torus having the same properties. The speed of mixing of this example must be subpolynomial, because we also show that: in a system having polynomial decay of correlations, the limsupr → 0 of the above ratio of logarithms (which is also called the upper hitting time indicator) is bounded from above by a function of the local dimension and the speed of correlation decay.
   More generally, this shows that reparametrizations of torus translations having a Liouville component cannot be polynomially mixing.

Keywords: Arithmetical properties., Hitting time, Dimension, Decay of correlations.
Mathematics Subject Classification: Primary: 37A25, 37A45; Secondary: 37C4.
Citation: Stefano Galatolo, Pietro Peterlongo. Long hitting time, slow decay of correlations and arithmetical properties. Discrete & Continuous Dynamical Systems - A, 2010, 27 (1) : 185-204. doi: 10.3934/dcds.2010.27.185
[1]

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
[2]

Michiko Yuri. Polynomial decay of correlations for intermittent sofic systems. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 445-464. doi: 10.3934/dcds.2008.22.445
[3]

Vincent Lynch. Decay of correlations for non-Hölder observables. Discrete & Continuous Dynamical Systems - A, 2006, 16 (1) : 19-46. doi: 10.3934/dcds.2006.16.19
[4]

Ioannis Konstantoulas. Effective decay of multiple correlations in semidirect product actions. Journal of Modern Dynamics, 2016, 10: 81-111. doi: 10.3934/jmd.2016.10.81
[5]

Dong Han Kim, Luca Marchese, Stefano Marmi. Long hitting time for translation flows and L-shaped billiards. Journal of Modern Dynamics, 2019, 14: 291-353. doi: 10.3934/jmd.2019011
[6]

Karla Díaz-Ordaz. Decay of correlations for non-Hölder observables for one-dimensional expanding Lorenz-like maps. Discrete & Continuous Dynamical Systems - A, 2006, 15 (1) : 159-176. doi: 10.3934/dcds.2006.15.159
[7]

Lucio Boccardo, Maria Michaela Porzio. Some degenerate parabolic problems: Existence and decay properties. Discrete & Continuous Dynamical Systems - S, 2014, 7 (4) : 617-629. doi: 10.3934/dcdss.2014.7.617
[8]

Eskil Hansen, Alexander Ostermann. Dimension splitting for time dependent operators. Conference Publications, 2009, 2009 (Special) : 322-332. doi: 10.3934/proc.2009.2009.322
[9]

Ugo Boscain, Gregoire Charlot, Moussa Gaye, Paolo Mason. Local properties of almost-Riemannian structures in dimension 3. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4115-4147. doi: 10.3934/dcds.2015.35.4115
[10]

Moez Daoulatli. Rates of decay for the wave systems with time dependent damping. Discrete & Continuous Dynamical Systems - A, 2011, 31 (2) : 407-443. doi: 10.3934/dcds.2011.31.407
[11]

Qing Chen, Zhong Tan. Time decay of solutions to the compressible Euler equations with damping. Kinetic & Related Models, 2014, 7 (4) : 605-619. doi: 10.3934/krm.2014.7.605
[12]

Antonio Magaña, Alain Miranville, Ramón Quintanilla. On the time decay in phase–lag thermoelasticity with two temperatures. Electronic Research Archive, 2019, 27: 7-19. doi: 10.3934/era.2019007
[13]

Niclas Carlsson, Göran Högnäs. Asymptotic properties of a TCP model with time-outs. Discrete & Continuous Dynamical Systems - B, 2005, 5 (3) : 543-564. doi: 10.3934/dcdsb.2005.5.543
[14]

Gabriele Grillo, Matteo Muratori, Maria Michaela Porzio. Porous media equations with two weights: Smoothing and decay properties of energy solutions via Poincaré inequalities. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3599-3640. doi: 10.3934/dcds.2013.33.3599
[15]

Fred C. Pinto. Nonlinear stability and dynamical properties for a Kuramoto-Sivashinsky equation in space dimension two. Discrete & Continuous Dynamical Systems - A, 1999, 5 (1) : 117-136. doi: 10.3934/dcds.1999.5.117
[16]

Adam Kanigowski, Federico Rodriguez Hertz, Kurt Vinhage. On the non-equivalence of the Bernoulli and $ K$ properties in dimension four. Journal of Modern Dynamics, 2018, 13: 221-250. doi: 10.3934/jmd.2018019
[17]

Irene M. Gamba, Maria Pia Gualdani, Christof Sparber. A note on the time decay of solutions for the linearized Wigner-Poisson system. Kinetic & Related Models, 2009, 2 (1) : 181-189. doi: 10.3934/krm.2009.2.181
[18]

Zhuangyi Liu, Ramón Quintanilla. Time decay in dual-phase-lag thermoelasticity: Critical case. Communications on Pure & Applied Analysis, 2018, 17 (1) : 177-190. doi: 10.3934/cpaa.2018011
[19]

Huijiang Zhao. Large time decay estimates of solutions of nonlinear parabolic equations. Discrete & Continuous Dynamical Systems - A, 2002, 8 (1) : 69-114. doi: 10.3934/dcds.2002.8.69
[20]

Youcef Mammeri. On the decay in time of solutions of some generalized regularized long waves equations. Communications on Pure & Applied Analysis, 2008, 7 (3) : 513-532. doi: 10.3934/cpaa.2008.7.513

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (11)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences