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Teichmüller geodesics with $ d$-dimensional limit sets
Mixing properties for toral extensions of slowly mixing dynamical systems with finite and infinite measure
1. | Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK |
2. | Mathematics Department, University of Exeter, EX4 4QF, UK |
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.
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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