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  2018, 12: 285-313. doi: 10.3934/jmd.2018011

Mixing properties for toral extensions of slowly mixing dynamical systems with finite and infinite measure

Ian Melbourne 1, and  Dalia Terhesiu 2,

1. 

Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK
2. 

Mathematics Department, University of Exeter, EX4 4QF, UK

Received  December 2015 Revised  April 29, 2018 Published  November 2018

We prove results on mixing and mixing rates for toral extensions of nonuniformly expanding maps with subexponential decay of correlations. Both the finite and infinite measure settings are considered. Under a Dolgo-pyat-type condition on nonexistence of approximate eigenfunctions, we prove that existing results for (possibly non-Markovian) nonuniformly expanding maps hold also for their toral extensions.

Keywords: Toral extensions, decay of correlations, infinite ergodic theory.
Mathematics Subject Classification: Primary: 37A25; Secondary: 37A40, 37D25, 60K05.
Citation: Ian Melbourne, Dalia Terhesiu. Mixing properties for toral extensions of slowly mixing dynamical systems with finite and infinite measure. Journal of Modern Dynamics, 2018, 12: 285-313. doi: 10.3934/jmd.2018011
References:
[1]

J. Aaronson, An Introduction to Infinite Ergodic Theory, Math. Surveys and Monographs, 50, Amer. Math. Soc., 1997. doi: 10.1090/surv/050.

[2]

J. Aaronson and M. Denker, Local limit theorems for partial sums of stationary sequences generated by Gibbs-Markov maps, Stoch. Dyn., 1 (2001), 193-237.  doi: 10.1142/S0219493701000114.

[3]

P. Bálint, O. Butterley and I. Melbourne, Polynomial decay of correlations for flows, including Lorentz gas examples, preprint, 2017.

[4]

H. Bruin, M. Holland and I. Melbourne, Subexponential decay of correlations for compact group extensions of nonuniformly expanding systems, Ergodic Theory Dynam. Systems, 25 (2005), 1719-1738.  doi: 10.1017/S014338570500026X.

[5]

H. Bruin and D. Terhesiu, Upper and lower bounds for the correlation function via inducing with general return times, Ergodic Theory Dynam. Systems, 38 (2018), 34-62.  doi: 10.1017/etds.2016.20.

[6]

N. I. Chernov and H.-K. Zhang, Billiards with polynomial mixing rates, Nonlinearity, 18 (2005), 1527-1553.  doi: 10.1088/0951-7715/18/4/006.

[7]

D. Dolgopyat, Prevalence of rapid mixing in hyperbolic flows, Ergodic Theory Dynam. Systems, 18 (1998), 1097-1114.  doi: 10.1017/S0143385798117431.

[8]

D. Dolgopyat, On mixing properties of compact group extensions of hyperbolic systems, Israel J. Math., 130 (2002), 157-205.  doi: 10.1007/BF02764076.

[9]

M. J. Field, I. Melbourne and A. Török, Stable ergodicity for smooth compact Lie group extensions of hyperbolic basic sets, Ergodic Theory Dynam. Systems, 25 (2005), 517-551.  doi: 10.1017/S0143385704000355.

[10]

M. J. Field, I. Melbourne and A. Török, Stability of mixing and rapid mixing for hyperbolic flows, Ann. of Math., 166 (2007), 269-291.  doi: 10.4007/annals.2007.166.269.

[11]

M. J. Field and W. Parry, Stable ergodicity of skew extensions by compact Lie groups, Topology, 38 (1999), 167-187.  doi: 10.1016/S0040-9383(98)00008-1.

[12]

S. Gouëzel, Sharp polynomial estimates for the decay of correlations, Israel J. Math., 139 (2004), 29-65.  doi: 10.1007/BF02787541.

[13]

S. Gouëzel, Vitesse de Décorrélation et Théorèmes Limites Pour les Applications non Uniformément Dilatantes, Ph. D. Thesis. Ecole Normale Supérieure, 2004.

[14]

S. Gouëzel, Berry-Esseen theorem and local limit theorem for nonuniformly expanding maps, Ann. Inst. H. Poincaré Probab. Statist., 41 (2005), 997-1024.  doi: 10.1016/j.anihpb.2004.09.002.

[15]

S. Gouëzel, Correlation asymptotics from large deviations in dynamical systems with infinite measure, Colloq. Math., 125 (2011), 193-212.  doi: 10.4064/cm125-2-5.

[16]

H. Hennion, Sur un théorème spectral et son application aux noyaux lipchitziens, Proc. Amer. Math. Soc., 118 (1993), 627-634.  doi: 10.2307/2160348.

[17]

H. Hu, Decay of correlations for piecewise smooth maps with indifferent fixed points, Ergodic Theory Dynam. Systems, 24 (2004), 495-524.  doi: 10.1017/S0143385703000671.

[18]

Y. Katznelson, An Introduction to Harmonic Analysis, Dover, New York, 1976.

[19]

C. Liverani, B. Saussol and S. Vaienti, A probabilistic approach to intermittency, Ergodic Theory Dynam. Systems, 19 (1999), 671-685.  doi: 10.1017/S0143385799133856.

[20]

R. Markarian, Billiards with polynomial decay of correlations, Ergodic Theory Dynam. Systems, 24 (2004), 177-197.  doi: 10.1017/S0143385703000270.

[21]

I. Melbourne, Rapid decay of correlations for nonuniformly hyperbolic flows, Trans. Amer. Math. Soc., 359 (2007), 2421-2441.  doi: 10.1090/S0002-9947-06-04267-X.

[22]

I. Melbourne, Superpolynomial and polynomial mixing for semiflows and flows, Nonlinearity, 31 (2018), R268-R316.  doi: 10.1088/1361-6544/aad309.

[23]

I. Melbourne and M. Nicol, Statistical properties of endomorphisms and compact group extensions, J. London Math. Soc., 70 (2004), 427-446.  doi: 10.1112/S0024610704005587.

[24]

I. Melbourne and D. Terhesiu, Operator renewal theory and mixing rates for dynamical systems with infinite measure, Invent. Math., 189 (2012), 61-110.  doi: 10.1007/s00222-011-0361-4.

[25]

I. Melbourne and D. Terhesiu, Decay of correlations for nonuniformly expanding systems with general return times, Ergodic Theory Dynam. Systems, 34 (2014), 893-918.  doi: 10.1017/etds.2012.158.

[26]

Y. Pomeau and P. Manneville, Intermittent transition to turbulence in dissipative dynamical systems, Comm. Math. Phys., 74 (1980), 189-197.  doi: 10.1007/BF01197757.

[27]

O. M. Sarig, Subexponential decay of correlations, Invent. Math., 150 (2002), 629-653.  doi: 10.1007/s00222-002-0248-5.

[28]

D. Terhesiu, Improved mixing rates for infinite measure preserving systems, Ergodic Theory Dynam. Systems, 35 (2015), 585-614.  doi: 10.1017/etds.2013.59.

[29]

M. Thaler, Estimates of the invariant densities of endomorphisms with indifferent fixed points, Israel J. Math., 37 (1980), 303-314.  doi: 10.1007/BF02788928.

[30]

L.-S. Young, Recurrence times and rates of mixing, Israel J. Math., 110 (1999), 153-188.  doi: 10.1007/BF02808180.

[31]

R. Zweimüller, Ergodic structure and invariant densities of non-Markovian interval maps with indifferent fixed points, Nonlinearity, 11 (1998), 1263-1276.  doi: 10.1088/0951-7715/11/5/005.

[32]

R. Zweimüller, Ergodic properties of infinite measure-preserving interval maps with indifferent fixed points, Ergodic Theory Dynam. Systems, 20 (2000), 1519-1549.  doi: 10.1017/S0143385700000821.

show all references

References:
[1]

J. Aaronson, An Introduction to Infinite Ergodic Theory, Math. Surveys and Monographs, 50, Amer. Math. Soc., 1997. doi: 10.1090/surv/050.

[2]

J. Aaronson and M. Denker, Local limit theorems for partial sums of stationary sequences generated by Gibbs-Markov maps, Stoch. Dyn., 1 (2001), 193-237.  doi: 10.1142/S0219493701000114.

[3]

P. Bálint, O. Butterley and I. Melbourne, Polynomial decay of correlations for flows, including Lorentz gas examples, preprint, 2017.

[4]

H. Bruin, M. Holland and I. Melbourne, Subexponential decay of correlations for compact group extensions of nonuniformly expanding systems, Ergodic Theory Dynam. Systems, 25 (2005), 1719-1738.  doi: 10.1017/S014338570500026X.

[5]

H. Bruin and D. Terhesiu, Upper and lower bounds for the correlation function via inducing with general return times, Ergodic Theory Dynam. Systems, 38 (2018), 34-62.  doi: 10.1017/etds.2016.20.

[6]

N. I. Chernov and H.-K. Zhang, Billiards with polynomial mixing rates, Nonlinearity, 18 (2005), 1527-1553.  doi: 10.1088/0951-7715/18/4/006.

[7]

D. Dolgopyat, Prevalence of rapid mixing in hyperbolic flows, Ergodic Theory Dynam. Systems, 18 (1998), 1097-1114.  doi: 10.1017/S0143385798117431.

[8]

D. Dolgopyat, On mixing properties of compact group extensions of hyperbolic systems, Israel J. Math., 130 (2002), 157-205.  doi: 10.1007/BF02764076.

[9]

M. J. Field, I. Melbourne and A. Török, Stable ergodicity for smooth compact Lie group extensions of hyperbolic basic sets, Ergodic Theory Dynam. Systems, 25 (2005), 517-551.  doi: 10.1017/S0143385704000355.

[10]

M. J. Field, I. Melbourne and A. Török, Stability of mixing and rapid mixing for hyperbolic flows, Ann. of Math., 166 (2007), 269-291.  doi: 10.4007/annals.2007.166.269.

[11]

M. J. Field and W. Parry, Stable ergodicity of skew extensions by compact Lie groups, Topology, 38 (1999), 167-187.  doi: 10.1016/S0040-9383(98)00008-1.

[12]

S. Gouëzel, Sharp polynomial estimates for the decay of correlations, Israel J. Math., 139 (2004), 29-65.  doi: 10.1007/BF02787541.

[13]

S. Gouëzel, Vitesse de Décorrélation et Théorèmes Limites Pour les Applications non Uniformément Dilatantes, Ph. D. Thesis. Ecole Normale Supérieure, 2004.

[14]

S. Gouëzel, Berry-Esseen theorem and local limit theorem for nonuniformly expanding maps, Ann. Inst. H. Poincaré Probab. Statist., 41 (2005), 997-1024.  doi: 10.1016/j.anihpb.2004.09.002.

[15]

S. Gouëzel, Correlation asymptotics from large deviations in dynamical systems with infinite measure, Colloq. Math., 125 (2011), 193-212.  doi: 10.4064/cm125-2-5.

[16]

H. Hennion, Sur un théorème spectral et son application aux noyaux lipchitziens, Proc. Amer. Math. Soc., 118 (1993), 627-634.  doi: 10.2307/2160348.

[17]

H. Hu, Decay of correlations for piecewise smooth maps with indifferent fixed points, Ergodic Theory Dynam. Systems, 24 (2004), 495-524.  doi: 10.1017/S0143385703000671.

[18]

Y. Katznelson, An Introduction to Harmonic Analysis, Dover, New York, 1976.

[19]

C. Liverani, B. Saussol and S. Vaienti, A probabilistic approach to intermittency, Ergodic Theory Dynam. Systems, 19 (1999), 671-685.  doi: 10.1017/S0143385799133856.

[20]

R. Markarian, Billiards with polynomial decay of correlations, Ergodic Theory Dynam. Systems, 24 (2004), 177-197.  doi: 10.1017/S0143385703000270.

[21]

I. Melbourne, Rapid decay of correlations for nonuniformly hyperbolic flows, Trans. Amer. Math. Soc., 359 (2007), 2421-2441.  doi: 10.1090/S0002-9947-06-04267-X.

[22]

I. Melbourne, Superpolynomial and polynomial mixing for semiflows and flows, Nonlinearity, 31 (2018), R268-R316.  doi: 10.1088/1361-6544/aad309.

[23]

I. Melbourne and M. Nicol, Statistical properties of endomorphisms and compact group extensions, J. London Math. Soc., 70 (2004), 427-446.  doi: 10.1112/S0024610704005587.

[24]

I. Melbourne and D. Terhesiu, Operator renewal theory and mixing rates for dynamical systems with infinite measure, Invent. Math., 189 (2012), 61-110.  doi: 10.1007/s00222-011-0361-4.

[25]

I. Melbourne and D. Terhesiu, Decay of correlations for nonuniformly expanding systems with general return times, Ergodic Theory Dynam. Systems, 34 (2014), 893-918.  doi: 10.1017/etds.2012.158.

[26]

Y. Pomeau and P. Manneville, Intermittent transition to turbulence in dissipative dynamical systems, Comm. Math. Phys., 74 (1980), 189-197.  doi: 10.1007/BF01197757.

[27]

O. M. Sarig, Subexponential decay of correlations, Invent. Math., 150 (2002), 629-653.  doi: 10.1007/s00222-002-0248-5.

[28]

D. Terhesiu, Improved mixing rates for infinite measure preserving systems, Ergodic Theory Dynam. Systems, 35 (2015), 585-614.  doi: 10.1017/etds.2013.59.

[29]

M. Thaler, Estimates of the invariant densities of endomorphisms with indifferent fixed points, Israel J. Math., 37 (1980), 303-314.  doi: 10.1007/BF02788928.

[30]

L.-S. Young, Recurrence times and rates of mixing, Israel J. Math., 110 (1999), 153-188.  doi: 10.1007/BF02808180.

[31]

R. Zweimüller, Ergodic structure and invariant densities of non-Markovian interval maps with indifferent fixed points, Nonlinearity, 11 (1998), 1263-1276.  doi: 10.1088/0951-7715/11/5/005.

[32]

R. Zweimüller, Ergodic properties of infinite measure-preserving interval maps with indifferent fixed points, Ergodic Theory Dynam. Systems, 20 (2000), 1519-1549.  doi: 10.1017/S0143385700000821.

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