American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Existence uniqueness and regularity theory for elliptic equations with complex-valued potentials
  • DCDS Home
  • This Issue
  • Next Article
    Properties of the minimizers for a constrained minimization problem arising in Kirchhoff equation
March  2021, 41(3): 1051-1069. doi: 10.3934/dcds.2020309

Pomeau-Manneville maps are global-local mixing

Claudio Bonanno 1,, and  Marco Lenci 2,

1. 

Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy
2. 

Dipartimento di Matematica, Università di Bologna, Piazza di Porta San Donato 5, 40126 Bologna, Italy, and, Istituto Nazionale di Fisica Nucleare, Sezione di Bologna, Via Irnerio 46, 40126 Bologna, Italy

* Corresponding author: Claudio Bonanno

Received  March 2020 Revised  June 2020 Published  August 2020

We prove that a large class of expanding maps of the unit interval with a $ C^2 $-regular indifferent fixed point in 0 and full increasing branches are global-local mixing. This class includes the standard Pomeau-Manneville maps $ T(x) = x + x^{p+1} $ mod 1 ($ p \ge 1 $), the Liverani-Saussol-Vaienti maps (with index $ p \ge 1 $) and many generalizations thereof.

Keywords: Infinite measure-preserving maps, infinite-volume mixing, indifferent fixed point.
Mathematics Subject Classification: Primary:37A40, 37A25, 37E05;Secondary:37D25, 37C25.
Citation: Claudio Bonanno, Marco Lenci. Pomeau-Manneville maps are global-local mixing. Discrete & Continuous Dynamical Systems, 2021, 41 (3) : 1051-1069. doi: 10.3934/dcds.2020309
References:
[1]

C. Bonanno, P. Giulietti and M. Lenci, Global-local mixing for the Boole map, Chaos Solitons Fractals, 111 (2018), 55-61.  doi: 10.1016/j.chaos.2018.03.020.  Google Scholar

[2]

C. Bonanno, P. Giulietti and M. Lenci, Infinite mixing for one-dimensional maps with an indifferent fixed point, Nonlinearity, 31 (2018), 5180-5213.  doi: 10.1088/1361-6544/aadc04.  Google Scholar

[3]

D. Dolgopyat and P. Nándori, Infinite measure mixing for some mechanical systems, preprint, arXiv: 1812.01174. Google Scholar

[4]

P. Giulietti, A. Hammerlindl and D. Ravotti, Quantitative global-local mixing mixing for accessible skew products, preprint, arXiv: 2006.06539. Google Scholar

[5]

M. Lenci, On infinite-volume mixing, Comm. Math. Phys., 298 (2010), 485-514.  doi: 10.1007/s00220-010-1043-6.  Google Scholar

[6]

M. Lenci, Exactness, K-property and infinite mixing, Publ. Mat. Urug., 14 (2013), 159-170.   Google Scholar

[7]

M. Lenci, Uniformly expanding Markov maps of the real line: Exactness and infinite mixing, Discrete Contin. Dyn. Syst., 37 (2017), 3867-3903.  doi: 10.3934/dcds.2017163.  Google Scholar

[8]

M. Lin, Mixing for Markov operators, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 19 (1971), 231-242.  doi: 10.1007/BF00534111.  Google Scholar

[9]

C. Liverani, B. Saussol and S. Vaienti, A probabilistic approach to intermittency, Ergodic Theory Dynam. Systems, 19 (1999), 671-685.  doi: 10.1017/S0143385799133856.  Google Scholar

[10]

P. Manneville, Intermittency, self-similarity and $1/f$ spectrum in dissipative dynamical systems, J. Physique, 41 (1980), 1235-1243.  doi: 10.1051/jphys:0198000410110123500.  Google Scholar

[11]

Y. Pomeau and P. Manneville, Intermittent transition to turbulence in dissipative dynamical systems, Comm. Math. Phys., 74 (1980), 189-197.  doi: 10.1007/BF01197757.  Google Scholar

[12]

M. Thaler, Estimates of the invariant densities of endomorphisms with indifferent fixed points, Israel J. Math., 37 (1980), 303-314.  doi: 10.1007/BF02788928.  Google Scholar

[13]

M. Thaler, Transformations on $[0,\,1]$ with infinite invariant measures, Israel J. Math., 46 (1983), 67-96.  doi: 10.1007/BF02760623.  Google Scholar

[14]

M. Thaler, Infinite ergodic theory. Examples: One-dimensional maps with indifferent fixed points, Course Notes, Marseille, 2001. Available at http://www.uni-salzburg.at/fileadmin/oracle\_file\_imports/1543201.PDF. Google Scholar

show all references

References:
[1]

C. Bonanno, P. Giulietti and M. Lenci, Global-local mixing for the Boole map, Chaos Solitons Fractals, 111 (2018), 55-61.  doi: 10.1016/j.chaos.2018.03.020.  Google Scholar

[2]

C. Bonanno, P. Giulietti and M. Lenci, Infinite mixing for one-dimensional maps with an indifferent fixed point, Nonlinearity, 31 (2018), 5180-5213.  doi: 10.1088/1361-6544/aadc04.  Google Scholar

[3]

D. Dolgopyat and P. Nándori, Infinite measure mixing for some mechanical systems, preprint, arXiv: 1812.01174. Google Scholar

[4]

P. Giulietti, A. Hammerlindl and D. Ravotti, Quantitative global-local mixing mixing for accessible skew products, preprint, arXiv: 2006.06539. Google Scholar

[5]

M. Lenci, On infinite-volume mixing, Comm. Math. Phys., 298 (2010), 485-514.  doi: 10.1007/s00220-010-1043-6.  Google Scholar

[6]

M. Lenci, Exactness, K-property and infinite mixing, Publ. Mat. Urug., 14 (2013), 159-170.   Google Scholar

[7]

M. Lenci, Uniformly expanding Markov maps of the real line: Exactness and infinite mixing, Discrete Contin. Dyn. Syst., 37 (2017), 3867-3903.  doi: 10.3934/dcds.2017163.  Google Scholar

[8]

M. Lin, Mixing for Markov operators, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 19 (1971), 231-242.  doi: 10.1007/BF00534111.  Google Scholar

[9]

C. Liverani, B. Saussol and S. Vaienti, A probabilistic approach to intermittency, Ergodic Theory Dynam. Systems, 19 (1999), 671-685.  doi: 10.1017/S0143385799133856.  Google Scholar

[10]

P. Manneville, Intermittency, self-similarity and $1/f$ spectrum in dissipative dynamical systems, J. Physique, 41 (1980), 1235-1243.  doi: 10.1051/jphys:0198000410110123500.  Google Scholar

[11]

Y. Pomeau and P. Manneville, Intermittent transition to turbulence in dissipative dynamical systems, Comm. Math. Phys., 74 (1980), 189-197.  doi: 10.1007/BF01197757.  Google Scholar

[12]

M. Thaler, Estimates of the invariant densities of endomorphisms with indifferent fixed points, Israel J. Math., 37 (1980), 303-314.  doi: 10.1007/BF02788928.  Google Scholar

[13]

M. Thaler, Transformations on $[0,\,1]$ with infinite invariant measures, Israel J. Math., 46 (1983), 67-96.  doi: 10.1007/BF02760623.  Google Scholar

[14]

M. Thaler, Infinite ergodic theory. Examples: One-dimensional maps with indifferent fixed points, Course Notes, Marseille, 2001. Available at http://www.uni-salzburg.at/fileadmin/oracle\_file\_imports/1543201.PDF. Google Scholar

[1]

Roland Zweimüller. Asymptotic orbit complexity of infinite measure preserving transformations. Discrete & Continuous Dynamical Systems, 2006, 15 (1) : 353-366. doi: 10.3934/dcds.2006.15.353
[2]

S. Eigen, A. B. Hajian, V. S. Prasad. Universal skyscraper templates for infinite measure preserving transformations. Discrete & Continuous Dynamical Systems, 2006, 16 (2) : 343-360. doi: 10.3934/dcds.2006.16.343
[3]

Marco Lenci. Uniformly expanding Markov maps of the real line: Exactness and infinite mixing. Discrete & Continuous Dynamical Systems, 2017, 37 (7) : 3867-3903. doi: 10.3934/dcds.2017163
[4]

Nasab Yassine. Quantitative recurrence of some dynamical systems preserving an infinite measure in dimension one. Discrete & Continuous Dynamical Systems, 2018, 38 (1) : 343-361. doi: 10.3934/dcds.2018017
[5]

Ian Melbourne, Dalia Terhesiu. Mixing properties for toral extensions of slowly mixing dynamical systems with finite and infinite measure. Journal of Modern Dynamics, 2018, 12: 285-313. doi: 10.3934/jmd.2018011
[6]

Rui Kuang, Xiangdong Ye. The return times set and mixing for measure preserving transformations. Discrete & Continuous Dynamical Systems, 2007, 18 (4) : 817-827. doi: 10.3934/dcds.2007.18.817
[7]

H. E. Lomelí, J. D. Meiss. Generating forms for exact volume-preserving maps. Discrete & Continuous Dynamical Systems - S, 2009, 2 (2) : 361-377. doi: 10.3934/dcdss.2009.2.361
[8]

Roland Gunesch, Anatole Katok. Construction of weakly mixing diffeomorphisms preserving measurable Riemannian metric and smooth measure. Discrete & Continuous Dynamical Systems, 2000, 6 (1) : 61-88. doi: 10.3934/dcds.2000.6.61
[9]

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
[10]

Luis Hernández-Corbato, Francisco R. Ruiz del Portal. Fixed point indices of planar continuous maps. Discrete & Continuous Dynamical Systems, 2015, 35 (7) : 2979-2995. doi: 10.3934/dcds.2015.35.2979
[11]

Mark F. Demers. Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps. Discrete & Continuous Dynamical Systems, 2021, 41 (1) : 217-256. doi: 10.3934/dcds.2020217
[12]

Charlene Kalle, Niels Langeveld, Marta Maggioni, Sara Munday. Matching for a family of infinite measure continued fraction transformations. Discrete & Continuous Dynamical Systems, 2020, 40 (11) : 6309-6330. doi: 10.3934/dcds.2020281
[13]

Rafael de la Llave, Jason D. Mireles James. Parameterization of invariant manifolds by reducibility for volume preserving and symplectic maps. Discrete & Continuous Dynamical Systems, 2012, 32 (12) : 4321-4360. doi: 10.3934/dcds.2012.32.4321
[14]

Robert Carlson. Dirichlet to Neumann maps for infinite quantum graphs. Networks & Heterogeneous Media, 2012, 7 (3) : 483-501. doi: 10.3934/nhm.2012.7.483
[15]

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
[16]

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, 2015, 35 (3) : 793-806. doi: 10.3934/dcds.2015.35.793
[17]

Grzegorz Graff, Piotr Nowak-Przygodzki. Fixed point indices of iterations of $C^1$ maps in $R^3$. Discrete & Continuous Dynamical Systems, 2006, 16 (4) : 843-856. doi: 10.3934/dcds.2006.16.843
[18]

Simeon Reich, Alexander J. Zaslavski. Convergence of generic infinite products of homogeneous order-preserving mappings. Discrete & Continuous Dynamical Systems, 1999, 5 (4) : 929-945. doi: 10.3934/dcds.1999.5.929
[19]

Gang Li, Xianwen Zhang. A Vlasov-Poisson plasma of infinite mass with a point charge. Kinetic & Related Models, 2018, 11 (2) : 303-336. doi: 10.3934/krm.2018015
[20]

Huiyan Xue, Antonella Zanna. Generating functions and volume preserving mappings. Discrete & Continuous Dynamical Systems, 2014, 34 (3) : 1229-1249. doi: 10.3934/dcds.2014.34.1229

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (121)
  • HTML views (247)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences