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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 - A, 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

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