# American Institute of Mathematical Sciences

• Previous Article
The construction of chaotic maps in the sense of Devaney on dendrites which commute to continuous maps on the unit interval
• DCDS Home
• This Issue
• Next Article
Polynomial growth of the derivative for diffeomorphisms on tori
February & March  2004, 11(2&3): 517-546. doi: 10.3934/dcds.2004.11.517

## Expanding interval maps with intermittent behaviour, physical measures and time scales

 1 Institut de Mathématiques de Luminy, Case 907. 163, avenue de Luminy, 13288 Marseille Cedex 9, France

Received  May 2001 Revised  February 2004 Published  June 2004

We exhibit a new family of piecewise monotonic expanding interval maps with interesting intermittent-like statistical behaviours. Among these maps, there are uniformly expanding ones for which a Lebesgue-typical orbit spends most of the time close to an "indifferent Cantor set" which plays the role of the usual neutral fixed point. There are also examples with an indifferent fixed point and an infinite absolutely continuous invariant measure. Like in the classical case, the Dirac mass at $0$ describes the statistical behaviour at usual time scale while the infinite one tells about the statistical behaviour at larger scales. But, here, there is another invariant measure describing the statistical behaviour of the ergodic sums at a third natural (intermediate) time scale.
To try to understand this last phenomenon, we propose a more general construction that yields an example for which we conjecture there is an infinite number of natural time scales.
Citation: Xavier Bressaud. Expanding interval maps with intermittent behaviour, physical measures and time scales. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 517-546. doi: 10.3934/dcds.2004.11.517
 [1] Romain Aimino, Huyi Hu, Matthew Nicol, Andrei Török, Sandro Vaienti. Polynomial loss of memory for maps of the interval with a neutral fixed point. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 793-806. doi: 10.3934/dcds.2015.35.793 [2] Michael Blank. Finite rank approximations of expanding maps with neutral singularities. Discrete & Continuous Dynamical Systems - A, 2008, 21 (3) : 749-762. doi: 10.3934/dcds.2008.21.749 [3] Roland Zweimüller. Asymptotic orbit complexity of infinite measure preserving transformations. Discrete & Continuous Dynamical Systems - A, 2006, 15 (1) : 353-366. doi: 10.3934/dcds.2006.15.353 [4] S. Eigen, A. B. Hajian, V. S. Prasad. Universal skyscraper templates for infinite measure preserving transformations. Discrete & Continuous Dynamical Systems - A, 2006, 16 (2) : 343-360. doi: 10.3934/dcds.2006.16.343 [5] Yakov Krasnov, Alexander Kononovich, Grigory Osharovich. On a structure of the fixed point set of homogeneous maps. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 1017-1027. doi: 10.3934/dcdss.2013.6.1017 [6] Luis Hernández-Corbato, Francisco R. Ruiz del Portal. Fixed point indices of planar continuous maps. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 2979-2995. doi: 10.3934/dcds.2015.35.2979 [7] Oliver Butterley. An alternative approach to generalised BV and the application to expanding interval maps. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3355-3363. doi: 10.3934/dcds.2013.33.3355 [8] Daniel Schnellmann. Typical points for one-parameter families of piecewise expanding maps of the interval. Discrete & Continuous Dynamical Systems - A, 2011, 31 (3) : 877-911. doi: 10.3934/dcds.2011.31.877 [9] Vaughn Climenhaga. A note on two approaches to the thermodynamic formalism. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 995-1005. doi: 10.3934/dcds.2010.27.995 [10] Nasab Yassine. Quantitative recurrence of some dynamical systems preserving an infinite measure in dimension one. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 343-361. doi: 10.3934/dcds.2018017 [11] Marco Lenci. Uniformly expanding Markov maps of the real line: Exactness and infinite mixing. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3867-3903. doi: 10.3934/dcds.2017163 [12] Jérôme Buzzi, Sylvie Ruette. Large entropy implies existence of a maximal entropy measure for interval maps. Discrete & Continuous Dynamical Systems - A, 2006, 14 (4) : 673-688. doi: 10.3934/dcds.2006.14.673 [13] Teck-Cheong Lim. On the largest common fixed point of a commuting family of isotone maps. Conference Publications, 2005, 2005 (Special) : 621-623. doi: 10.3934/proc.2005.2005.621 [14] Grzegorz Graff, Piotr Nowak-Przygodzki. Fixed point indices of iterations of $C^1$ maps in $R^3$. Discrete & Continuous Dynamical Systems - A, 2006, 16 (4) : 843-856. doi: 10.3934/dcds.2006.16.843 [15] Michael Jakobson, Lucia D. Simonelli. Countable Markov partitions suitable for thermodynamic formalism. Journal of Modern Dynamics, 2018, 13: 199-219. doi: 10.3934/jmd.2018018 [16] Yakov Pesin. On the work of Sarig on countable Markov chains and thermodynamic formalism. Journal of Modern Dynamics, 2014, 8 (1) : 1-14. doi: 10.3934/jmd.2014.8.1 [17] Manfred Denker, Yuri Kifer, Manuel Stadlbauer. Thermodynamic formalism for random countable Markov shifts. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 131-164. doi: 10.3934/dcds.2008.22.131 [18] Yongluo Cao, De-Jun Feng, Wen Huang. The thermodynamic formalism for sub-additive potentials. Discrete & Continuous Dynamical Systems - A, 2008, 20 (3) : 639-657. doi: 10.3934/dcds.2008.20.639 [19] Anna Mummert. The thermodynamic formalism for almost-additive sequences. Discrete & Continuous Dynamical Systems - A, 2006, 16 (2) : 435-454. doi: 10.3934/dcds.2006.16.435 [20] Luis Barreira. Nonadditive thermodynamic formalism: Equilibrium and Gibbs measures. Discrete & Continuous Dynamical Systems - A, 2006, 16 (2) : 279-305. doi: 10.3934/dcds.2006.16.279

2019 Impact Factor: 1.338