# American Institute of Mathematical Sciences

March  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, 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, Analyse Non Linéaire, 30 (2013), 101-120. doi: 10.1016/j.anihpc.2012.06.004.  Google Scholar [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), 1203-1236. doi: 10.1016/j.aim.2011.06.014.  Google Scholar [3] J. F. Alves and X. Li, Gibbs-Markov-Young structure with (stretched) exponential recurrence times for partially hyperbolic attractors, Adv. Math., 279 (2015), 405-437. doi: 10.1016/j.aim.2015.02.017.  Google Scholar [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, Analyse Non Linéaire, 22 (2005), 817-839. doi: 10.1016/j.anihpc.2004.12.002.  Google Scholar [5] J. F. Alves and V. Pinheiro, Slow rates of mixing for dynamical systems with hyperbolic structures, J. Stat. Phys., 131 (2008), 505-534. doi: 10.1007/s10955-008-9482-6.  Google Scholar [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), 3943-3955. doi: 10.1090/S0002-9939-2013-11680-1.  Google Scholar [7] A. Avez, Propriétés ergodiques des endomorphisms dilatants des variétés compactes, C.R. Acad. Sci. Paris Sér. A-B, 266 (1968), 610-612.  Google Scholar [8] V. Baladi, Positive Transfer Operators and Decay of Correlations, World Scientific, 2000. doi: 10.1142/9789812813633.  Google Scholar [9] V. Baladi and S. Gouëzel, Stretched exponential bounds for the correlations of the Viana-Alves skew products, Second Workshop on Dynamics and Randomness, Universidad de Chile, 2002. Google Scholar [10] M. Benedicks and L. Carleson, The dynamics of the Hénon map, Ann. Math., 122 (1985), 1-25. doi: 10.2307/1971367.  Google Scholar [11] M. Benedicks and L.-S. Young, Sinai-Bowen-Ruelle measures for certain Hénon maps, Invent. Math., 112 (1993), 541-576. doi: 10.1007/BF01232446.  Google Scholar [12] V. I. Bogachev, Measure Theory, Vol. 1, Springer, 2006. Google Scholar [13] R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Springer Lecture Notes in Math., 1975.  Google Scholar [14] H. Bruin, S. Luzzatto and S. van Strien, Decay of correlations in one-dimensional dynamics, Annales de l'ENS, 36 (2003), 621-646. doi: 10.1016/S0012-9593(03)00025-9.  Google Scholar [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), 639-656. doi: 10.3934/dcds.2005.12.639.  Google Scholar [16] J. Buzzi, O. Sester and M. Tsujii, Weakly expanding skew-products of quadratic maps, Ergod. Th. Dynam. Syst., 23 (2003), 1401-1414. doi: 10.1017/S0143385702001694.  Google Scholar [17] N. Chernov, Statistical properties of piecewise smooth hyperbolic systems in high dimensions, Discrete and Continuous Dynam. Systems, 5 (1999), 425-448. doi: 10.3934/dcds.1999.5.425.  Google Scholar [18] N. Chernov, Decay of correlations and dispersing billiards, J. Stat. Phys., 94 (1999), 513-556. doi: 10.1023/A:1004581304939.  Google Scholar [19] N. Chernov and R. Markarian, Chaotic Billiards, Mathematical Surveys and Monographs, Vol. 127, Amer. Math. Soc., Providence, RI, 2006. doi: 10.1090/surv/127.  Google Scholar [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), 159-176. doi: 10.3934/dcds.2006.15.159.  Google Scholar [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), 423-458. doi: 10.1142/S0219493706001852.  Google Scholar [22] S. Gouëzel, Decay of correlations for nonuniformly expanding systems, Bull. Soc. Math. France, 134 (2006), 1-31.  Google Scholar [23] F. Hofbauer and G. Keller, Ergodic properties of invariant measures for piecewise monotonic transformations, Math. Z., 180 (1982), 119-140. doi: 10.1007/BF01215004.  Google Scholar [24] M. Holland, Slowly mixing systems and intermittency maps, Ergodic theory and Dynamical Systems, 25 (2004), 133-159. doi: 10.1017/S0143385704000343.  Google Scholar [25] H. Hu, Decay of correlations for piecewise smooth maps with indifferent fixed points, Ergodic theory and Dynamical Systems, 24 (2004), 495-524. doi: 10.1017/S0143385703000671.  Google Scholar [26] G. Keller and T. Nowicki, Spectral theory, zeta functions and the distributions of points for Collet-Eckman maps, Comm. Math. Phys., 149 (1992), 31-69. doi: 10.1007/BF02096623.  Google Scholar [27] A. Lasota and J. A. Yorke, On the existence of invariant measures for piecewise monotonic transformations, Transactions of The AMS, 186 (1973), 481-488. doi: 10.1090/S0002-9947-1973-0335758-1.  Google Scholar [28] C. Liverani, Decay of correlations, Annals Math., 142 (1995), 239-301. doi: 10.2307/2118636.  Google Scholar [29] C. Liverani, Multidimensional expanding maps with singularities: A pedestrian approach, Ergodic Theory and Dynamical Systems, 33 (2013), 168-182. doi: 10.1017/S0143385711000939.  Google Scholar [30] A. Lopes, Entropy and large deviations, Nonlinearity, 3 (1990), 527-546. doi: 10.1088/0951-7715/3/2/013.  Google Scholar [31] S. Luzzatto, Stochastic-like Behaviour in Non-Uniformly Expanding Maps, Handbook of Dynamical Systems Vol. 1B, Elsevier, 2006. doi: 10.1016/S1874-575X(06)80028-7.  Google Scholar [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), 21-35. doi: 10.1007/s00220-013-1709-y.  Google Scholar [33] V. Lynch, Non-uniformly Expanding Dynamical Systems and Decay of Correlations for Non-Hölder Continuous Observables, Ph.D thesis, University of Warwick, 2003. Google Scholar [34] V. Lynch, Decay of correlations for non-Hölder observables, Discrete and Continuous Dynam. Systems, 16 (2006), 19-46. doi: 10.3934/dcds.2006.16.19.  Google Scholar [35] I. Melbourne and M. Nicol, Large deviations for nonuniformly hyperbolic systems, Transactions of AMS, 360 (2008), 6661-6676. doi: 10.1090/S0002-9947-08-04520-0.  Google Scholar [36] I. Melbourne and M. Nicol, Almost sure invariance principle for nonuniformly hyperbolic systems, Commun. Math. Phys., 260 (2005), 131-146. doi: 10.1007/s00220-005-1407-5.  Google Scholar [37] P. Natalini and B. Palumbo, Inequalities for the Incomplete Gamma function, Mathematical Inequalities & Applications, 3 (2000), 69-77. doi: 10.7153/mia-03-08.  Google Scholar [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), 619-635. doi: 10.1007/BF01410202.  Google Scholar [39] V. Pinheiro, Expanding Measures， Ann. Inst. Henri Poincaré, Analyse Non Linéaire, 28 (2011), 889-939. doi: 10.1016/j.anihpc.2011.07.001.  Google Scholar [40] M. Pollicott and M. Yuri, Statistical properties of maps with indifferent periodic points, Commun. Math. Phys., 217 (2001), 503-520. doi: 10.1007/s002200100368.  Google Scholar [41] D. Ruelle, A measure associated with Axiom A attractors, Amer. J. Math., 98 (1976), 619-654. doi: 10.2307/2373810.  Google Scholar [42] Y. Sinai, Gibbs measures in ergodic theory, Russ. Math. Surveys, 27 (1972), 21-64.  Google Scholar [43] Y. Sinai, Dynamical systems with elastic reflections, Ergodic properties of dispersing billiards, Russ. Math. Surveys, 25 (1970), 141-192.  Google Scholar [44] T. Tao and V. H. Vu, Additive Combinatorics, Cambridge studies in advanced mathematics, 105, Cambridge University Press, Cambridge, 2006. doi: 10.1017/CBO9780511755149.  Google Scholar [45] D. Thomine, A spectral gap for transfer operators of piecewise expanding maps, Discrete and continuous time Dynam. Systems, 30 (2011), 917-944. doi: 10.3934/dcds.2011.30.917.  Google Scholar [46] L.-S. Young, Decay of correlations for certain quadratic maps, Comm. Math. Phys., 146 (1992), 123-138. doi: 10.1007/BF02099211.  Google Scholar [47] L.-S. Young, Statistical properties of dynamical systems with some hyperbolicity, Ann. Math., 147 (1998), 585-650. doi: 10.2307/120960.  Google Scholar [48] L.-S. Young, Recurrence times and rates of mixing, Israel J. Math., 110 (1999), 153-188. doi: 10.1007/BF02808180.  Google Scholar

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, Analyse Non Linéaire, 30 (2013), 101-120. doi: 10.1016/j.anihpc.2012.06.004.  Google Scholar [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), 1203-1236. doi: 10.1016/j.aim.2011.06.014.  Google Scholar [3] J. F. Alves and X. Li, Gibbs-Markov-Young structure with (stretched) exponential recurrence times for partially hyperbolic attractors, Adv. Math., 279 (2015), 405-437. doi: 10.1016/j.aim.2015.02.017.  Google Scholar [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, Analyse Non Linéaire, 22 (2005), 817-839. doi: 10.1016/j.anihpc.2004.12.002.  Google Scholar [5] J. F. Alves and V. Pinheiro, Slow rates of mixing for dynamical systems with hyperbolic structures, J. Stat. Phys., 131 (2008), 505-534. doi: 10.1007/s10955-008-9482-6.  Google Scholar [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), 3943-3955. doi: 10.1090/S0002-9939-2013-11680-1.  Google Scholar [7] A. Avez, Propriétés ergodiques des endomorphisms dilatants des variétés compactes, C.R. Acad. Sci. Paris Sér. A-B, 266 (1968), 610-612.  Google Scholar [8] V. Baladi, Positive Transfer Operators and Decay of Correlations, World Scientific, 2000. doi: 10.1142/9789812813633.  Google Scholar [9] V. Baladi and S. Gouëzel, Stretched exponential bounds for the correlations of the Viana-Alves skew products, Second Workshop on Dynamics and Randomness, Universidad de Chile, 2002. Google Scholar [10] M. Benedicks and L. Carleson, The dynamics of the Hénon map, Ann. Math., 122 (1985), 1-25. doi: 10.2307/1971367.  Google Scholar [11] M. Benedicks and L.-S. Young, Sinai-Bowen-Ruelle measures for certain Hénon maps, Invent. Math., 112 (1993), 541-576. doi: 10.1007/BF01232446.  Google Scholar [12] V. I. Bogachev, Measure Theory, Vol. 1, Springer, 2006. Google Scholar [13] R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Springer Lecture Notes in Math., 1975.  Google Scholar [14] H. Bruin, S. Luzzatto and S. van Strien, Decay of correlations in one-dimensional dynamics, Annales de l'ENS, 36 (2003), 621-646. doi: 10.1016/S0012-9593(03)00025-9.  Google Scholar [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), 639-656. doi: 10.3934/dcds.2005.12.639.  Google Scholar [16] J. Buzzi, O. Sester and M. Tsujii, Weakly expanding skew-products of quadratic maps, Ergod. Th. Dynam. Syst., 23 (2003), 1401-1414. doi: 10.1017/S0143385702001694.  Google Scholar [17] N. Chernov, Statistical properties of piecewise smooth hyperbolic systems in high dimensions, Discrete and Continuous Dynam. Systems, 5 (1999), 425-448. doi: 10.3934/dcds.1999.5.425.  Google Scholar [18] N. Chernov, Decay of correlations and dispersing billiards, J. Stat. Phys., 94 (1999), 513-556. doi: 10.1023/A:1004581304939.  Google Scholar [19] N. Chernov and R. Markarian, Chaotic Billiards, Mathematical Surveys and Monographs, Vol. 127, Amer. Math. Soc., Providence, RI, 2006. doi: 10.1090/surv/127.  Google Scholar [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), 159-176. doi: 10.3934/dcds.2006.15.159.  Google Scholar [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), 423-458. doi: 10.1142/S0219493706001852.  Google Scholar [22] S. Gouëzel, Decay of correlations for nonuniformly expanding systems, Bull. Soc. Math. France, 134 (2006), 1-31.  Google Scholar [23] F. Hofbauer and G. Keller, Ergodic properties of invariant measures for piecewise monotonic transformations, Math. Z., 180 (1982), 119-140. doi: 10.1007/BF01215004.  Google Scholar [24] M. Holland, Slowly mixing systems and intermittency maps, Ergodic theory and Dynamical Systems, 25 (2004), 133-159. doi: 10.1017/S0143385704000343.  Google Scholar [25] H. Hu, Decay of correlations for piecewise smooth maps with indifferent fixed points, Ergodic theory and Dynamical Systems, 24 (2004), 495-524. doi: 10.1017/S0143385703000671.  Google Scholar [26] G. Keller and T. Nowicki, Spectral theory, zeta functions and the distributions of points for Collet-Eckman maps, Comm. Math. Phys., 149 (1992), 31-69. doi: 10.1007/BF02096623.  Google Scholar [27] A. Lasota and J. A. Yorke, On the existence of invariant measures for piecewise monotonic transformations, Transactions of The AMS, 186 (1973), 481-488. doi: 10.1090/S0002-9947-1973-0335758-1.  Google Scholar [28] C. Liverani, Decay of correlations, Annals Math., 142 (1995), 239-301. doi: 10.2307/2118636.  Google Scholar [29] C. Liverani, Multidimensional expanding maps with singularities: A pedestrian approach, Ergodic Theory and Dynamical Systems, 33 (2013), 168-182. doi: 10.1017/S0143385711000939.  Google Scholar [30] A. Lopes, Entropy and large deviations, Nonlinearity, 3 (1990), 527-546. doi: 10.1088/0951-7715/3/2/013.  Google Scholar [31] S. Luzzatto, Stochastic-like Behaviour in Non-Uniformly Expanding Maps, Handbook of Dynamical Systems Vol. 1B, Elsevier, 2006. doi: 10.1016/S1874-575X(06)80028-7.  Google Scholar [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), 21-35. doi: 10.1007/s00220-013-1709-y.  Google Scholar [33] V. Lynch, Non-uniformly Expanding Dynamical Systems and Decay of Correlations for Non-Hölder Continuous Observables, Ph.D thesis, University of Warwick, 2003. Google Scholar [34] V. Lynch, Decay of correlations for non-Hölder observables, Discrete and Continuous Dynam. Systems, 16 (2006), 19-46. doi: 10.3934/dcds.2006.16.19.  Google Scholar [35] I. Melbourne and M. Nicol, Large deviations for nonuniformly hyperbolic systems, Transactions of AMS, 360 (2008), 6661-6676. doi: 10.1090/S0002-9947-08-04520-0.  Google Scholar [36] I. Melbourne and M. Nicol, Almost sure invariance principle for nonuniformly hyperbolic systems, Commun. Math. Phys., 260 (2005), 131-146. doi: 10.1007/s00220-005-1407-5.  Google Scholar [37] P. Natalini and B. Palumbo, Inequalities for the Incomplete Gamma function, Mathematical Inequalities & Applications, 3 (2000), 69-77. doi: 10.7153/mia-03-08.  Google Scholar [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), 619-635. doi: 10.1007/BF01410202.  Google Scholar [39] V. Pinheiro, Expanding Measures， Ann. Inst. Henri Poincaré, Analyse Non Linéaire, 28 (2011), 889-939. doi: 10.1016/j.anihpc.2011.07.001.  Google Scholar [40] M. Pollicott and M. Yuri, Statistical properties of maps with indifferent periodic points, Commun. Math. Phys., 217 (2001), 503-520. doi: 10.1007/s002200100368.  Google Scholar [41] D. Ruelle, A measure associated with Axiom A attractors, Amer. J. Math., 98 (1976), 619-654. doi: 10.2307/2373810.  Google Scholar [42] Y. Sinai, Gibbs measures in ergodic theory, Russ. Math. Surveys, 27 (1972), 21-64.  Google Scholar [43] Y. Sinai, Dynamical systems with elastic reflections, Ergodic properties of dispersing billiards, Russ. Math. Surveys, 25 (1970), 141-192.  Google Scholar [44] T. Tao and V. H. Vu, Additive Combinatorics, Cambridge studies in advanced mathematics, 105, Cambridge University Press, Cambridge, 2006. doi: 10.1017/CBO9780511755149.  Google Scholar [45] D. Thomine, A spectral gap for transfer operators of piecewise expanding maps, Discrete and continuous time Dynam. Systems, 30 (2011), 917-944. doi: 10.3934/dcds.2011.30.917.  Google Scholar [46] L.-S. Young, Decay of correlations for certain quadratic maps, Comm. Math. Phys., 146 (1992), 123-138. doi: 10.1007/BF02099211.  Google Scholar [47] L.-S. Young, Statistical properties of dynamical systems with some hyperbolicity, Ann. Math., 147 (1998), 585-650. doi: 10.2307/120960.  Google Scholar [48] L.-S. Young, Recurrence times and rates of mixing, Israel J. Math., 110 (1999), 153-188. doi: 10.1007/BF02808180.  Google Scholar
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