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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 |
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.
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. |
| [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. |
| [3] |
D. Dolgopyat and P. Nándori, Infinite measure mixing for some mechanical systems, preprint, arXiv: 1812.01174. |
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P. Giulietti, A. Hammerlindl and D. Ravotti, Quantitative global-local mixing mixing for accessible skew products, preprint, arXiv: 2006.06539. |
| [5] |
M. Lenci,
On infinite-volume mixing, Comm. Math. Phys., 298 (2010), 485-514.
doi: 10.1007/s00220-010-1043-6. |
| [6] |
M. Lenci,
Exactness, K-property and infinite mixing, Publ. Mat. Urug., 14 (2013), 159-170.
|
| [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. |
| [8] |
M. Lin,
Mixing for Markov operators, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 19 (1971), 231-242.
doi: 10.1007/BF00534111. |
| [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. |
| [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. |
| [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. |
| [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. |
| [13] |
M. Thaler,
Transformations on $[0,\,1]$ with infinite invariant measures, Israel J. Math., 46 (1983), 67-96.
doi: 10.1007/BF02760623. |
| [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. |
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. |
| [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. |
| [3] |
D. Dolgopyat and P. Nándori, Infinite measure mixing for some mechanical systems, preprint, arXiv: 1812.01174. |
| [4] |
P. Giulietti, A. Hammerlindl and D. Ravotti, Quantitative global-local mixing mixing for accessible skew products, preprint, arXiv: 2006.06539. |
| [5] |
M. Lenci,
On infinite-volume mixing, Comm. Math. Phys., 298 (2010), 485-514.
doi: 10.1007/s00220-010-1043-6. |
| [6] |
M. Lenci,
Exactness, K-property and infinite mixing, Publ. Mat. Urug., 14 (2013), 159-170.
|
| [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. |
| [8] |
M. Lin,
Mixing for Markov operators, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 19 (1971), 231-242.
doi: 10.1007/BF00534111. |
| [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. |
| [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. |
| [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. |
| [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. |
| [13] |
M. Thaler,
Transformations on $[0,\,1]$ with infinite invariant measures, Israel J. Math., 46 (1983), 67-96.
doi: 10.1007/BF02760623. |
| [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. |
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