• 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

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. BonannoP. 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. BonannoP. 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. LiveraniB. 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. BonannoP. 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. BonannoP. 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. LiveraniB. 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]

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

[2]

Seung-Yeal Ha, Myeongju Kang, Bora Moon. Collective behaviors of a Winfree ensemble on an infinite cylinder. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2749-2779. doi: 10.3934/dcdsb.2020204

[3]

A. Kochergin. Well-approximable angles and mixing for flows on T^2 with nonsingular fixed points. Electronic Research Announcements, 2004, 10: 113-121.

[4]

Marcelo Messias. Periodic perturbation of quadratic systems with two infinite heteroclinic cycles. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1881-1899. doi: 10.3934/dcds.2012.32.1881

[5]

Pengfei Wang, Mengyi Zhang, Huan Su. Input-to-state stability of infinite-dimensional stochastic nonlinear systems. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021066

[6]

V. Vijayakumar, R. Udhayakumar, K. Kavitha. On the approximate controllability of neutral integro-differential inclusions of Sobolev-type with infinite delay. Evolution Equations & Control Theory, 2021, 10 (2) : 271-296. doi: 10.3934/eect.2020066

[7]

Vakhtang Putkaradze, Stuart Rogers. Numerical simulations of a rolling ball robot actuated by internal point masses. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 143-207. doi: 10.3934/naco.2020021

[8]

M. R. S. Kulenović, J. Marcotte, O. Merino. Properties of basins of attraction for planar discrete cooperative maps. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2721-2737. doi: 10.3934/dcdsb.2020202

[9]

Ronald E. Mickens. Positivity preserving discrete model for the coupled ODE's modeling glycolysis. Conference Publications, 2003, 2003 (Special) : 623-629. doi: 10.3934/proc.2003.2003.623

[10]

Alina Chertock, Alexander Kurganov, Mária Lukáčová-Medvi${\rm{\check{d}}}$ová, Șeyma Nur Özcan. An asymptotic preserving scheme for kinetic chemotaxis models in two space dimensions. Kinetic & Related Models, 2019, 12 (1) : 195-216. doi: 10.3934/krm.2019009

[11]

Carlos Gutierrez, Nguyen Van Chau. A remark on an eigenvalue condition for the global injectivity of differentiable maps of $R^2$. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 397-402. doi: 10.3934/dcds.2007.17.397

[12]

Fumihiko Nakamura. Asymptotic behavior of non-expanding piecewise linear maps in the presence of random noise. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2457-2473. doi: 10.3934/dcdsb.2018055

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (102)
  • HTML views (242)
  • Cited by (0)

Other articles
by authors

[Back to Top]