2016, 36(3): 1465-1491. doi: 10.3934/dcds.2016.36.1465

Young towers for product systems

1. 

International Center for Theoretical Physics (ICTP), Strada Costiera 11, 34151 Trieste, Italy

2. 

International School for Advanced Studies (SISSA), Via Bonomea 265, 34136 Trieste, Italy

Received  February 2015 Revised  June 2015 Published  August 2015

We show that the direct product of maps with Young towers admits a Young tower whose return times decay at a rate which is bounded above by the slowest of the rates of decay of the return times of the component maps. An application of this result, together with other results in the literature, yields various statistical properties for the direct product of various classes of systems, including Lorenz-like maps, multimodal maps, piecewise $C^2$ interval maps with critical points and singularities, Hénon maps and partially hyperbolic systems.
Citation: Stefano Luzzatto, Marks Ruziboev. Young towers for product systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1465-1491. doi: 10.3934/dcds.2016.36.1465
References:
[1]

J. F. Alves, C. Dias and S. Luzzatto, Geometry of expanding absolutely continuous invariant measures and the liftability problem,, Ann. Inst. Henri Poincaré, 30 (2013), 101. doi: 10.1016/j.anihpc.2012.06.004.

[2]

J. F. Alves, J. M. Freitas, S. Luzzatto and S. Vaienti, From rates of mixing to recurrence times via large deviations,, Advances in Mathematics, 228 (2011), 1203. doi: 10.1016/j.aim.2011.06.014.

[3]

J. F. Alves and X. Li, Gibbs-Markov-Young structure with (stretched) exponential recurrence times for partially hyperbolic attractors,, Adv. Math., 279 (2015), 405. doi: 10.1016/j.aim.2015.02.017.

[4]

J. F. Alves, S. Luzzatto and V. Pinheiro, Markov structures and decay of correlations for non-uniformly expanding dynamical systems,, Ann. Inst. Henri Poincaré, 22 (2005), 817. doi: 10.1016/j.anihpc.2004.12.002.

[5]

J. F. Alves and V. Pinheiro, Slow rates of mixing for dynamical systems with hyperbolic structures,, J. Stat. Phys., 131 (2008), 505. doi: 10.1007/s10955-008-9482-6.

[6]

J. F. Alves and D. Schnellmann, Ergodic properties of Viana-like maps with singularities in the base dynamics,, Proceedings of the AMS, 141 (2013), 3943. doi: 10.1090/S0002-9939-2013-11680-1.

[7]

A. Avez, Propriétés ergodiques des endomorphisms dilatants des variétés compactes,, C.R. Acad. Sci. Paris Sér. A-B, 266 (1968), 610.

[8]

V. Baladi, Positive Transfer Operators and Decay of Correlations,, World Scientific, (2000). doi: 10.1142/9789812813633.

[9]

V. Baladi and S. Gouëzel, Stretched exponential bounds for the correlations of the Viana-Alves skew products,, Second Workshop on Dynamics and Randomness, (2002).

[10]

M. Benedicks and L. Carleson, The dynamics of the Hénon map,, Ann. Math., 122 (1985), 1. doi: 10.2307/1971367.

[11]

M. Benedicks and L.-S. Young, Sinai-Bowen-Ruelle measures for certain Hénon maps,, Invent. Math., 112 (1993), 541. doi: 10.1007/BF01232446.

[12]

V. I. Bogachev, Measure Theory, Vol. 1,, Springer, (2006).

[13]

R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms,, Springer Lecture Notes in Math., (1975).

[14]

H. Bruin, S. Luzzatto and S. van Strien, Decay of correlations in one-dimensional dynamics,, Annales de l'ENS, 36 (2003), 621. doi: 10.1016/S0012-9593(03)00025-9.

[15]

J. Buzzi and V. Maume-Deschamps, Decay of correlations on towers with non-Hölder Jacobian and non-exponential return time,, Discrete and Continuous Dynam. Systems, 12 (2005), 639. doi: 10.3934/dcds.2005.12.639.

[16]

J. Buzzi, O. Sester and M. Tsujii, Weakly expanding skew-products of quadratic maps,, Ergod. Th. Dynam. Syst., 23 (2003), 1401. doi: 10.1017/S0143385702001694.

[17]

N. Chernov, Statistical properties of piecewise smooth hyperbolic systems in high dimensions,, Discrete and Continuous Dynam. Systems, 5 (1999), 425. doi: 10.3934/dcds.1999.5.425.

[18]

N. Chernov, Decay of correlations and dispersing billiards,, J. Stat. Phys., 94 (1999), 513. doi: 10.1023/A:1004581304939.

[19]

N. Chernov and R. Markarian, Chaotic Billiards,, Mathematical Surveys and Monographs, (2006). doi: 10.1090/surv/127.

[20]

K. Diaz-Ordaz, Decay of correlations for non-Hölder observables for one-dimensional expanding Lorenz-Like maps,, Discrete and Continuous Dynam. Systems, 15 (2006), 159. doi: 10.3934/dcds.2006.15.159.

[21]

K. Diaz-Ordaz, M. P. Holland and S. Luzzatto, Statistical properties of one-dimensional maps with critical points and singularities,, Stochastics and Dynamics, 6 (2006), 423. doi: 10.1142/S0219493706001852.

[22]

S. Gouëzel, Decay of correlations for nonuniformly expanding systems,, Bull. Soc. Math. France, 134 (2006), 1.

[23]

F. Hofbauer and G. Keller, Ergodic properties of invariant measures for piecewise monotonic transformations,, Math. Z., 180 (1982), 119. doi: 10.1007/BF01215004.

[24]

M. Holland, Slowly mixing systems and intermittency maps,, Ergodic theory and Dynamical Systems, 25 (2004), 133. doi: 10.1017/S0143385704000343.

[25]

H. Hu, Decay of correlations for piecewise smooth maps with indifferent fixed points,, Ergodic theory and Dynamical Systems, 24 (2004), 495. doi: 10.1017/S0143385703000671.

[26]

G. Keller and T. Nowicki, Spectral theory, zeta functions and the distributions of points for Collet-Eckman maps,, Comm. Math. Phys., 149 (1992), 31. doi: 10.1007/BF02096623.

[27]

A. Lasota and J. A. Yorke, On the existence of invariant measures for piecewise monotonic transformations,, Transactions of The AMS, 186 (1973), 481. doi: 10.1090/S0002-9947-1973-0335758-1.

[28]

C. Liverani, Decay of correlations,, Annals Math., 142 (1995), 239. doi: 10.2307/2118636.

[29]

C. Liverani, Multidimensional expanding maps with singularities: A pedestrian approach,, Ergodic Theory and Dynamical Systems, 33 (2013), 168. doi: 10.1017/S0143385711000939.

[30]

A. Lopes, Entropy and large deviations,, Nonlinearity, 3 (1990), 527. doi: 10.1088/0951-7715/3/2/013.

[31]

S. Luzzatto, Stochastic-like Behaviour in Non-Uniformly Expanding Maps,, Handbook of Dynamical Systems Vol. 1B, (2006). doi: 10.1016/S1874-575X(06)80028-7.

[32]

S. Luzzatto and I. Melbourne, Statistical properties and decay of correlations for interval maps with critical points and singularities,, Commun. Math. Phys., 320 (2013), 21. doi: 10.1007/s00220-013-1709-y.

[33]

V. Lynch, Non-uniformly Expanding Dynamical Systems and Decay of Correlations for Non-Hölder Continuous Observables,, Ph.D thesis, (2003).

[34]

V. Lynch, Decay of correlations for non-Hölder observables,, Discrete and Continuous Dynam. Systems, 16 (2006), 19. doi: 10.3934/dcds.2006.16.19.

[35]

I. Melbourne and M. Nicol, Large deviations for nonuniformly hyperbolic systems,, Transactions of AMS, 360 (2008), 6661. doi: 10.1090/S0002-9947-08-04520-0.

[36]

I. Melbourne and M. Nicol, Almost sure invariance principle for nonuniformly hyperbolic systems,, Commun. Math. Phys., 260 (2005), 131. doi: 10.1007/s00220-005-1407-5.

[37]

P. Natalini and B. Palumbo, Inequalities for the Incomplete Gamma function,, Mathematical Inequalities & Applications, 3 (2000), 69. doi: 10.7153/mia-03-08.

[38]

T. Nowicki and S. van Strien, Absolutely continuous invariant measures for $C^2$ unimodal maps satisfying the Collet-Eckmann conditions,, Invent. Math., 93 (1988), 619. doi: 10.1007/BF01410202.

[39]

V. Pinheiro, Expanding Measures,, Ann. Inst. Henri Poincaré, 28 (2011), 889. doi: 10.1016/j.anihpc.2011.07.001.

[40]

M. Pollicott and M. Yuri, Statistical properties of maps with indifferent periodic points,, Commun. Math. Phys., 217 (2001), 503. doi: 10.1007/s002200100368.

[41]

D. Ruelle, A measure associated with Axiom A attractors,, Amer. J. Math., 98 (1976), 619. doi: 10.2307/2373810.

[42]

Y. Sinai, Gibbs measures in ergodic theory,, Russ. Math. Surveys, 27 (1972), 21.

[43]

Y. Sinai, Dynamical systems with elastic reflections, Ergodic properties of dispersing billiards,, Russ. Math. Surveys, 25 (1970), 141.

[44]

T. Tao and V. H. Vu, Additive Combinatorics,, Cambridge studies in advanced mathematics, (2006). doi: 10.1017/CBO9780511755149.

[45]

D. Thomine, A spectral gap for transfer operators of piecewise expanding maps,, Discrete and continuous time Dynam. Systems, 30 (2011), 917. doi: 10.3934/dcds.2011.30.917.

[46]

L.-S. Young, Decay of correlations for certain quadratic maps,, Comm. Math. Phys., 146 (1992), 123. doi: 10.1007/BF02099211.

[47]

L.-S. Young, Statistical properties of dynamical systems with some hyperbolicity,, Ann. Math., 147 (1998), 585. doi: 10.2307/120960.

[48]

L.-S. Young, Recurrence times and rates of mixing,, Israel J. Math., 110 (1999), 153. doi: 10.1007/BF02808180.

show all references

References:
[1]

J. F. Alves, C. Dias and S. Luzzatto, Geometry of expanding absolutely continuous invariant measures and the liftability problem,, Ann. Inst. Henri Poincaré, 30 (2013), 101. doi: 10.1016/j.anihpc.2012.06.004.

[2]

J. F. Alves, J. M. Freitas, S. Luzzatto and S. Vaienti, From rates of mixing to recurrence times via large deviations,, Advances in Mathematics, 228 (2011), 1203. doi: 10.1016/j.aim.2011.06.014.

[3]

J. F. Alves and X. Li, Gibbs-Markov-Young structure with (stretched) exponential recurrence times for partially hyperbolic attractors,, Adv. Math., 279 (2015), 405. doi: 10.1016/j.aim.2015.02.017.

[4]

J. F. Alves, S. Luzzatto and V. Pinheiro, Markov structures and decay of correlations for non-uniformly expanding dynamical systems,, Ann. Inst. Henri Poincaré, 22 (2005), 817. doi: 10.1016/j.anihpc.2004.12.002.

[5]

J. F. Alves and V. Pinheiro, Slow rates of mixing for dynamical systems with hyperbolic structures,, J. Stat. Phys., 131 (2008), 505. doi: 10.1007/s10955-008-9482-6.

[6]

J. F. Alves and D. Schnellmann, Ergodic properties of Viana-like maps with singularities in the base dynamics,, Proceedings of the AMS, 141 (2013), 3943. doi: 10.1090/S0002-9939-2013-11680-1.

[7]

A. Avez, Propriétés ergodiques des endomorphisms dilatants des variétés compactes,, C.R. Acad. Sci. Paris Sér. A-B, 266 (1968), 610.

[8]

V. Baladi, Positive Transfer Operators and Decay of Correlations,, World Scientific, (2000). doi: 10.1142/9789812813633.

[9]

V. Baladi and S. Gouëzel, Stretched exponential bounds for the correlations of the Viana-Alves skew products,, Second Workshop on Dynamics and Randomness, (2002).

[10]

M. Benedicks and L. Carleson, The dynamics of the Hénon map,, Ann. Math., 122 (1985), 1. doi: 10.2307/1971367.

[11]

M. Benedicks and L.-S. Young, Sinai-Bowen-Ruelle measures for certain Hénon maps,, Invent. Math., 112 (1993), 541. doi: 10.1007/BF01232446.

[12]

V. I. Bogachev, Measure Theory, Vol. 1,, Springer, (2006).

[13]

R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms,, Springer Lecture Notes in Math., (1975).

[14]

H. Bruin, S. Luzzatto and S. van Strien, Decay of correlations in one-dimensional dynamics,, Annales de l'ENS, 36 (2003), 621. doi: 10.1016/S0012-9593(03)00025-9.

[15]

J. Buzzi and V. Maume-Deschamps, Decay of correlations on towers with non-Hölder Jacobian and non-exponential return time,, Discrete and Continuous Dynam. Systems, 12 (2005), 639. doi: 10.3934/dcds.2005.12.639.

[16]

J. Buzzi, O. Sester and M. Tsujii, Weakly expanding skew-products of quadratic maps,, Ergod. Th. Dynam. Syst., 23 (2003), 1401. doi: 10.1017/S0143385702001694.

[17]

N. Chernov, Statistical properties of piecewise smooth hyperbolic systems in high dimensions,, Discrete and Continuous Dynam. Systems, 5 (1999), 425. doi: 10.3934/dcds.1999.5.425.

[18]

N. Chernov, Decay of correlations and dispersing billiards,, J. Stat. Phys., 94 (1999), 513. doi: 10.1023/A:1004581304939.

[19]

N. Chernov and R. Markarian, Chaotic Billiards,, Mathematical Surveys and Monographs, (2006). doi: 10.1090/surv/127.

[20]

K. Diaz-Ordaz, Decay of correlations for non-Hölder observables for one-dimensional expanding Lorenz-Like maps,, Discrete and Continuous Dynam. Systems, 15 (2006), 159. doi: 10.3934/dcds.2006.15.159.

[21]

K. Diaz-Ordaz, M. P. Holland and S. Luzzatto, Statistical properties of one-dimensional maps with critical points and singularities,, Stochastics and Dynamics, 6 (2006), 423. doi: 10.1142/S0219493706001852.

[22]

S. Gouëzel, Decay of correlations for nonuniformly expanding systems,, Bull. Soc. Math. France, 134 (2006), 1.

[23]

F. Hofbauer and G. Keller, Ergodic properties of invariant measures for piecewise monotonic transformations,, Math. Z., 180 (1982), 119. doi: 10.1007/BF01215004.

[24]

M. Holland, Slowly mixing systems and intermittency maps,, Ergodic theory and Dynamical Systems, 25 (2004), 133. doi: 10.1017/S0143385704000343.

[25]

H. Hu, Decay of correlations for piecewise smooth maps with indifferent fixed points,, Ergodic theory and Dynamical Systems, 24 (2004), 495. doi: 10.1017/S0143385703000671.

[26]

G. Keller and T. Nowicki, Spectral theory, zeta functions and the distributions of points for Collet-Eckman maps,, Comm. Math. Phys., 149 (1992), 31. doi: 10.1007/BF02096623.

[27]

A. Lasota and J. A. Yorke, On the existence of invariant measures for piecewise monotonic transformations,, Transactions of The AMS, 186 (1973), 481. doi: 10.1090/S0002-9947-1973-0335758-1.

[28]

C. Liverani, Decay of correlations,, Annals Math., 142 (1995), 239. doi: 10.2307/2118636.

[29]

C. Liverani, Multidimensional expanding maps with singularities: A pedestrian approach,, Ergodic Theory and Dynamical Systems, 33 (2013), 168. doi: 10.1017/S0143385711000939.

[30]

A. Lopes, Entropy and large deviations,, Nonlinearity, 3 (1990), 527. doi: 10.1088/0951-7715/3/2/013.

[31]

S. Luzzatto, Stochastic-like Behaviour in Non-Uniformly Expanding Maps,, Handbook of Dynamical Systems Vol. 1B, (2006). doi: 10.1016/S1874-575X(06)80028-7.

[32]

S. Luzzatto and I. Melbourne, Statistical properties and decay of correlations for interval maps with critical points and singularities,, Commun. Math. Phys., 320 (2013), 21. doi: 10.1007/s00220-013-1709-y.

[33]

V. Lynch, Non-uniformly Expanding Dynamical Systems and Decay of Correlations for Non-Hölder Continuous Observables,, Ph.D thesis, (2003).

[34]

V. Lynch, Decay of correlations for non-Hölder observables,, Discrete and Continuous Dynam. Systems, 16 (2006), 19. doi: 10.3934/dcds.2006.16.19.

[35]

I. Melbourne and M. Nicol, Large deviations for nonuniformly hyperbolic systems,, Transactions of AMS, 360 (2008), 6661. doi: 10.1090/S0002-9947-08-04520-0.

[36]

I. Melbourne and M. Nicol, Almost sure invariance principle for nonuniformly hyperbolic systems,, Commun. Math. Phys., 260 (2005), 131. doi: 10.1007/s00220-005-1407-5.

[37]

P. Natalini and B. Palumbo, Inequalities for the Incomplete Gamma function,, Mathematical Inequalities & Applications, 3 (2000), 69. doi: 10.7153/mia-03-08.

[38]

T. Nowicki and S. van Strien, Absolutely continuous invariant measures for $C^2$ unimodal maps satisfying the Collet-Eckmann conditions,, Invent. Math., 93 (1988), 619. doi: 10.1007/BF01410202.

[39]

V. Pinheiro, Expanding Measures,, Ann. Inst. Henri Poincaré, 28 (2011), 889. doi: 10.1016/j.anihpc.2011.07.001.

[40]

M. Pollicott and M. Yuri, Statistical properties of maps with indifferent periodic points,, Commun. Math. Phys., 217 (2001), 503. doi: 10.1007/s002200100368.

[41]

D. Ruelle, A measure associated with Axiom A attractors,, Amer. J. Math., 98 (1976), 619. doi: 10.2307/2373810.

[42]

Y. Sinai, Gibbs measures in ergodic theory,, Russ. Math. Surveys, 27 (1972), 21.

[43]

Y. Sinai, Dynamical systems with elastic reflections, Ergodic properties of dispersing billiards,, Russ. Math. Surveys, 25 (1970), 141.

[44]

T. Tao and V. H. Vu, Additive Combinatorics,, Cambridge studies in advanced mathematics, (2006). doi: 10.1017/CBO9780511755149.

[45]

D. Thomine, A spectral gap for transfer operators of piecewise expanding maps,, Discrete and continuous time Dynam. Systems, 30 (2011), 917. doi: 10.3934/dcds.2011.30.917.

[46]

L.-S. Young, Decay of correlations for certain quadratic maps,, Comm. Math. Phys., 146 (1992), 123. doi: 10.1007/BF02099211.

[47]

L.-S. Young, Statistical properties of dynamical systems with some hyperbolicity,, Ann. Math., 147 (1998), 585. doi: 10.2307/120960.

[48]

L.-S. Young, Recurrence times and rates of mixing,, Israel J. Math., 110 (1999), 153. doi: 10.1007/BF02808180.

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