July  2017, 37(7): 3867-3903. doi: 10.3934/dcds.2017163

Uniformly expanding Markov maps of the real line: Exactness and infinite mixing

1. 

Dipartimento di Matematica, Università di Bologna, Piazza di Porta S. Donato 5,40126 Bologna, Italy

2. 

Istituto Nazionale di Fisica Nucleare, Sezione di Bologna, Via Irnerio 46,40126 Bologna, Italy

Received  January 2015 Revised  February 2017 Published  April 2017

Fund Project: The author was partially supported by PRIN Grant 2012AZS52J 001 (MIUR, Italy).

We give a fairly complete characterization of the exact components of a large class of uniformly expanding Markov maps of $ {\mathbb{R}}$. Using this result, for a class of $ $ $\mathbb{Z}$-invariant maps and finite modifications thereof, we prove certain properties of infinite mixing recently introduced by the author.

Citation: Marco Lenci. Uniformly expanding Markov maps of the real line: Exactness and infinite mixing. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3867-3903. doi: 10.3934/dcds.2017163
References:
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J. Aaronson, An Introduction to Infinite Ergodic Theory Mathematical Surveys and Monographs, 50. American Mathematical Society, Providence, RI, 1997. doi: 10.1090/surv/050.  Google Scholar

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R. Artuso and G. Cristadoro, Weak chaos and anomalous transport: A deterministic approach, Commun. Nonlinear Sci. Numer. Simul., 8 (2003), 137-148.  doi: 10.1016/S1007-5704(03)00025-X.  Google Scholar

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R. Artuso and G. Cristadoro, Anomalous transport: A deterministic approach, Phys. Rev. Lett. , 90 (2003), 244101. Google Scholar

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G. Atkinson, Recurrence for co-cycles and random walks, J. London. Math. Soc.(2), 13 (1976), 486-488.  doi: 10.1112/jlms/s2-13.3.486.  Google Scholar

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A. BianchiG. CristadoroM. Lenci and M. Ligabò, Random walks in a one-dimensional Lévy random environment, J. Stat. Phys., 163 (2016), 22-40.  doi: 10.1007/s10955-016-1469-0.  Google Scholar

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A. Boyarsky and P. Góra, Laws of Chaos. Invariant Measures and Dynamical Systems in One Dimension Probability and its Applications. Birkhäuser, Boston, MA, 1997. doi: 10.1007/978-1-4612-2024-4.  Google Scholar

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P. Bugiel, Approximation for the measures of ergodic transformations of the real line, Z. Wahrsch. Verw. Gebiete, 59 (1982), 27-38.  doi: 10.1007/BF00575523.  Google Scholar

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P. Bugiel, On the exactness of a class of endomorphisms of the real line, Univ. Iagel. Acta Math., 25 (1985), 53-65.   Google Scholar

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D. DolgopyatD. Szász and T. Varjú, Limit theorems for locally perturbed planar Lorentz processes, Duke Math. J., 148 (2009), 459-499.  doi: 10.1215/00127094-2009-031.  Google Scholar

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R. G. Gallager, Stochastic Processes. Theory for Applications Cambridge University Press, Cambridge, 2013.  Google Scholar

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A.B. Hajian and S. Kakutani, Weakly wandering sets and invariant measures, Trans. Amer. Math. Soc., 110 (1964), 136-151.  doi: 10.1090/S0002-9947-1964-0154961-1.  Google Scholar

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A. Iksanov and A. Yu. Pilipenko, A functional limit theorem for locally perturbed random walks, preprint, arXiv: 1504.06930. Google Scholar

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G. KellerP. Howard and R. Klages, Continuity properties of transport coefficients in simple maps, Nonlinearity, 21 (2008), 1719-1743.  doi: 10.1088/0951-7715/21/8/003.  Google Scholar

[16]

R. Klages, Suppression and enhancement of diffusion in disordered dynamical systems Phys. Rev. E, 65 (2002), 055203(R). doi: 10.1103/PhysRevE.65.055203.  Google Scholar

[17]

M. Lenci, Aperiodic Lorentz gas: recurrence and ergodicity, Ergodic Theory Dynam. Systems, 23 (2003), 869-883.  doi: 10.1017/S0143385702001529.  Google Scholar

[18]

M. Lenci, Typicality of recurrence for Lorentz gases, Ergodic Theory Dynam. Systems, 26 (2006), 799-820.  doi: 10.1017/S0143385706000022.  Google Scholar

[19]

M. Lenci, Central Limit Theorem and recurrence for random walks in bistochastic random environments J. Math. Phys. , 49 (2008), 125213, 9pp. doi: 10.1063/1.3005226.  Google Scholar

[20]

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

[21]

M. Lenci, Infinite-volume mixing for dynamical systems preserving an infinite measure, Procedia IUTAM, 5 (2012), 204-219.  doi: 10.1016/j.piutam.2012.06.028.  Google Scholar

[22]

M. Lenci, Random walks in random environments without ellipticity, Stochastic Process. Appl., 123 (2013), 1750-1764.  doi: 10.1016/j.spa.2013.01.007.  Google Scholar

[23]

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

[24]

M. Lenci, A simple proof of the exactness of expanding maps of the interval with an indifferent fixed point, Chaos Solitons Fractals, 82 (2016), 148-154.  doi: 10.1016/j.chaos.2015.11.024.  Google Scholar

[25]

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

[26]

T. Miernowski and A. Nogueira, Exactness of the Euclidean algorithm and of the Rauzy induction on the space of interval exchange transformations, Ergodic Theory Dynam. Systems, 33 (2013), 221-246.  doi: 10.1017/S014338571100085X.  Google Scholar

[27]

P. Nándori, Recurrence properties of a special type of heavy-tailed random walk, J. Stat. Phys., 142 (2011), 342-355.  doi: 10.1007/s10955-010-0116-4.  Google Scholar

[28]

D. Paulin and D. Szász, Locally perturbed random walks with unbounded jumps, J. Stat. Phys., 141 (2010), 1116-1130.  doi: 10.1007/s10955-010-0078-6.  Google Scholar

[29]

H. G. Schuster and W. Just, Deterministic Chaos: An Introduction 4th edition. Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim, 2005. doi: 10.1002/3527604804.  Google Scholar

[30]

A. N. Shiryayev, Probability Graduate Texts in Mathematics, 95. Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4899-0018-0.  Google Scholar

[31]

P. Walters, An Introduction to Ergodic Theory Graduate Texts in Mathematics, 79. Springer-Verlag, New York-Berlin, 1982.  Google Scholar

[32]

A.Yu. Pilipenko and Yu.E. Prikhod'ko, On the limit behavior of a sequence of Markov processes perturbed in a neighborhood of the singular point, Ukrainian Math. J., 67 (2015), 564-583.  doi: 10.1007/s11253-015-1101-5.  Google Scholar

show all references

References:
[1]

J. Aaronson, An Introduction to Infinite Ergodic Theory Mathematical Surveys and Monographs, 50. American Mathematical Society, Providence, RI, 1997. doi: 10.1090/surv/050.  Google Scholar

[3]

R. Adler and B. Weiss, The ergodic infinite measure preserving transformation of Boole, Israel J. Math., 16 (1973), 263-278.  doi: 10.1007/BF02756706.  Google Scholar

[4]

R. Artuso and G. Cristadoro, Weak chaos and anomalous transport: A deterministic approach, Commun. Nonlinear Sci. Numer. Simul., 8 (2003), 137-148.  doi: 10.1016/S1007-5704(03)00025-X.  Google Scholar

[5]

R. Artuso and G. Cristadoro, Anomalous transport: A deterministic approach, Phys. Rev. Lett. , 90 (2003), 244101. Google Scholar

[6]

G. Atkinson, Recurrence for co-cycles and random walks, J. London. Math. Soc.(2), 13 (1976), 486-488.  doi: 10.1112/jlms/s2-13.3.486.  Google Scholar

[7]

A. BianchiG. CristadoroM. Lenci and M. Ligabò, Random walks in a one-dimensional Lévy random environment, J. Stat. Phys., 163 (2016), 22-40.  doi: 10.1007/s10955-016-1469-0.  Google Scholar

[8]

A. Boyarsky and P. Góra, Laws of Chaos. Invariant Measures and Dynamical Systems in One Dimension Probability and its Applications. Birkhäuser, Boston, MA, 1997. doi: 10.1007/978-1-4612-2024-4.  Google Scholar

[9]

P. Bugiel, Approximation for the measures of ergodic transformations of the real line, Z. Wahrsch. Verw. Gebiete, 59 (1982), 27-38.  doi: 10.1007/BF00575523.  Google Scholar

[10]

P. Bugiel, On the exactness of a class of endomorphisms of the real line, Univ. Iagel. Acta Math., 25 (1985), 53-65.   Google Scholar

[11]

D. DolgopyatD. Szász and T. Varjú, Limit theorems for locally perturbed planar Lorentz processes, Duke Math. J., 148 (2009), 459-499.  doi: 10.1215/00127094-2009-031.  Google Scholar

[12]

R. G. Gallager, Stochastic Processes. Theory for Applications Cambridge University Press, Cambridge, 2013.  Google Scholar

[13]

A.B. Hajian and S. Kakutani, Weakly wandering sets and invariant measures, Trans. Amer. Math. Soc., 110 (1964), 136-151.  doi: 10.1090/S0002-9947-1964-0154961-1.  Google Scholar

[14]

A. Iksanov and A. Yu. Pilipenko, A functional limit theorem for locally perturbed random walks, preprint, arXiv: 1504.06930. Google Scholar

[15]

G. KellerP. Howard and R. Klages, Continuity properties of transport coefficients in simple maps, Nonlinearity, 21 (2008), 1719-1743.  doi: 10.1088/0951-7715/21/8/003.  Google Scholar

[16]

R. Klages, Suppression and enhancement of diffusion in disordered dynamical systems Phys. Rev. E, 65 (2002), 055203(R). doi: 10.1103/PhysRevE.65.055203.  Google Scholar

[17]

M. Lenci, Aperiodic Lorentz gas: recurrence and ergodicity, Ergodic Theory Dynam. Systems, 23 (2003), 869-883.  doi: 10.1017/S0143385702001529.  Google Scholar

[18]

M. Lenci, Typicality of recurrence for Lorentz gases, Ergodic Theory Dynam. Systems, 26 (2006), 799-820.  doi: 10.1017/S0143385706000022.  Google Scholar

[19]

M. Lenci, Central Limit Theorem and recurrence for random walks in bistochastic random environments J. Math. Phys. , 49 (2008), 125213, 9pp. doi: 10.1063/1.3005226.  Google Scholar

[20]

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

[21]

M. Lenci, Infinite-volume mixing for dynamical systems preserving an infinite measure, Procedia IUTAM, 5 (2012), 204-219.  doi: 10.1016/j.piutam.2012.06.028.  Google Scholar

[22]

M. Lenci, Random walks in random environments without ellipticity, Stochastic Process. Appl., 123 (2013), 1750-1764.  doi: 10.1016/j.spa.2013.01.007.  Google Scholar

[23]

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

[24]

M. Lenci, A simple proof of the exactness of expanding maps of the interval with an indifferent fixed point, Chaos Solitons Fractals, 82 (2016), 148-154.  doi: 10.1016/j.chaos.2015.11.024.  Google Scholar

[25]

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

[26]

T. Miernowski and A. Nogueira, Exactness of the Euclidean algorithm and of the Rauzy induction on the space of interval exchange transformations, Ergodic Theory Dynam. Systems, 33 (2013), 221-246.  doi: 10.1017/S014338571100085X.  Google Scholar

[27]

P. Nándori, Recurrence properties of a special type of heavy-tailed random walk, J. Stat. Phys., 142 (2011), 342-355.  doi: 10.1007/s10955-010-0116-4.  Google Scholar

[28]

D. Paulin and D. Szász, Locally perturbed random walks with unbounded jumps, J. Stat. Phys., 141 (2010), 1116-1130.  doi: 10.1007/s10955-010-0078-6.  Google Scholar

[29]

H. G. Schuster and W. Just, Deterministic Chaos: An Introduction 4th edition. Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim, 2005. doi: 10.1002/3527604804.  Google Scholar

[30]

A. N. Shiryayev, Probability Graduate Texts in Mathematics, 95. Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4899-0018-0.  Google Scholar

[31]

P. Walters, An Introduction to Ergodic Theory Graduate Texts in Mathematics, 79. Springer-Verlag, New York-Berlin, 1982.  Google Scholar

[32]

A.Yu. Pilipenko and Yu.E. Prikhod'ko, On the limit behavior of a sequence of Markov processes perturbed in a neighborhood of the singular point, Ukrainian Math. J., 67 (2015), 564-583.  doi: 10.1007/s11253-015-1101-5.  Google Scholar

Figure 1.  A uniformly expanding Markov map ${\mathbb{R}} \to {\mathbb{R}}$.
Figure 2.  An example of a quasi-lift of an expanding circle map.
Figure 3.  A finite modification of a quasi-lift of a circle map, constructed with the procedure given in Section 3.2, for the case $\mu_o = m$.
Figure 4.  A map $T$ associated a random walk. The marks on the abscissa indicate the Markov intervals $I_{jk}$, while those on the ordinate represent the intervals $[k, k+1]$.
Figure 5.  A rough sketch of the map $T$ of Counterexample 2. The bold segments on the abscissa indicate the set $X$. The bold parts of the graph of $T$ represent $T|_X$, which is invertible.
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