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March  2021, 41(3): 1051-1069. doi: 10.3934/dcds.2020309

## Pomeau-Manneville maps are global-local mixing

 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.

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

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##### 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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