December  2020, 40(12): 6575-6609. doi: 10.3934/dcds.2020294

Local limit theorems for suspended semiflows

1. 

School of Math. Sciences, Tel Aviv University, 69978 Tel Aviv, Israel, Webpage: http://www.math.tau.ac.il/~aro

2. 

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

 

Dedicated to the memory of Nat Friedman

Received  June 2019 Revised  July 2020 Published  December 2020 Early access  August 2020

Fund Project: Aaronson's research was partially supported by ISF grant No. 1289/17

We prove local limit theorems for a cocycle over a semiflow by establishing topological, mixing properties of the associated skew product semiflow. We also establish conditional rational weak mixing of certain skew product semiflows and various mixing properties including order 2 rational weak mixing of hyperbolic geodesic flows on the tangent spaces of cyclic covers.

Citation: Jon Aaronson, Dalia Terhesiu. Local limit theorems for suspended semiflows. Discrete and Continuous Dynamical Systems, 2020, 40 (12) : 6575-6609. doi: 10.3934/dcds.2020294
References:
[1]

Jon Aaronson and Manfred Denker, The Poincaré series of $\Bbb C\setminus\Bbb Z$, Ergodic Theory Dynam. Systems, 19 (1999), 1-20.  doi: 10.1017/S0143385799126592.

[2]

Jon Aaronson and Manfred Denker, Local limit theorems for partial sums of stationary sequences generated by {G}ibbs-{M}arkov maps, Stoch. Dyn., 1 (2001), 193-237.  doi: 10.1142/S0219493701000114.

[3]

Jon. AaronsonManfred DenkerOmri Sarig and Rol Zweimüller, Aperiodicity of cocycles and conditional local limit theorems, Stoch. Dyn., 4 (2004), 31-62.  doi: 10.1142/S0219493704000936.

[4]

Jon Aaronson and Hitoshi Nakada, On the mixing coefficients of piecewise monotonic maps, Israel J. Math., 148 (2005), 1-10.  doi: 10.1007/BF02775429.

[5]

Jon Aaronson and Hitoshi Nakada, On multiple recurrence and other properties of 'nice' infinite measure-preserving transformations, Ergodic Theory Dynam. Systems, 37 (2017), 1345-1368.  doi: 10.1017/etds.2015.108.

[6]

Rufus Bowen, Symbolic dynamics for hyperbolic flows, Amer. J. Math., 95 (1973), 429-460.  doi: 10.2307/2373793.

[7]

Leo Breiman, Probability, Addison-Wesley Publishing Company, Reading, Mass.-London-Don Mills, Ont., 1968.

[8]

Rufus Bowen and Peter Walters, Expansive one-parameter flows, J. Differential Equations, 12 (1972), 180-193.  doi: 10.1016/0022-0396(72)90013-7.

[9]

D. Dolgopyat and P. Nándori, On mixing and the local central limit theorem for hyperbolic flows, Ergodic Theory Dynam. Systems, 40 (2020), 142–174. doi: 10.1017/etds.2018.29.

[10]

R. A. Doney, A bivariate local limit theorem, J. Multivariate Anal., 36 (1991), 95-102.  doi: 10.1016/0047-259X(91)90093-H.

[11]

Nathaniel A. Friedman, Mixing transformations in an infinite measure space, In Studies in probability and ergodic theory, Adv. in Math. Suppl. Stud., 2, 167–184. Academic Press, New York-London, 1978.

[12]

Y. Guivarc'h and J. Hardy, Théorèmes limites pour une classe de chaînes de Markov et applications aux difféomorphismes d'Anosov, Ann. Inst. H. Poincaré Probab. Statist., 24 (1988), 73-98. 

[13]

Hubert Hennion and Loïc Hervé, Limit Theorems for Markov Chains and Stochastic Properties of Dynamical Systems by Quasi-Compactness, volume 1766 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2001. doi: 10.1007/b87874.

[14]

E. Hopf, Ergodentheorie, Ergebnisse der Mathematik und ihrer Grenzgebiete, 5. Bd. Julius Springer, 1937.

[15]

Eberhard Hopf, Ergodic theory and the geodesic flow on surfaces of constant negative curvature, Bull. Amer. Math. Soc., 77 (1971), 863-877.  doi: 10.1090/S0002-9904-1971-12799-4.

[16]

Yukiko Iwata, A generalized local limit theorem for mixing semi-flows, Hokkaido Math. J., 37 (2008), 215-240.  doi: 10.14492/hokmj/1253539585.

[17]

Zbigniew J. Jurek and J. David Mason, Operator-Limit Distributions in Probability Theory, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1993.

[18]

K. Krickeberg, Recent results on mixing in topological measure spaces, In Probability and Information Theory (Proc. Internat. Sympos., McMaster Univ., Hamilton, Ont., 1968), 1969, pages 178–185. Springer, Berlin.

[19]

Gregory F. Lawler and Vlada Limic,. Random Walk: A Modern Introduction, volume 123 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2010. doi: 10.1017/CBO9780511750854.

[20]

A. Lasota and James A. Yorke, On the existence of invariant measures for piecewise monotonic transformations, Trans. Amer. Math. Soc., 186 (1973), 481–488, (1974). doi: 10.1090/S0002-9947-1973-0335758-1.

[21]

Elliott W. Montroll and George H. Weiss, Random walks on lattices. II, J. Mathematical Phys., 6 (1965), 167-181.  doi: 10.1063/1.1704269.

[22]

William Parry and Mark Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque, 187-188 (1990), 268 pp.

[23]

Françoise Pène, Planar Lorentz process in a random scenery, Ann. Inst. Henri Poincaré Probab. Stat., 45 (2009), 818-839.  doi: 10.1214/08-AIHP191.

[24]

V. V. Petrov, Sums of Independent Random Variables, Translated from the Russian by A. A. Brown. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 82. Springer-Verlag, New York-Heidelberg, 1975.

[25]

Mary Rees, Checking ergodicity of some geodesic flows with infinite Gibbs measure, Ergodic Theory Dynamical Systems, 1 (1981), 107-133.  doi: 10.1017/S0143385700001206.

[26]

Marek Rychlik, Bounded variation and invariant measures, Studia Math., 76 (1983), 69-80.  doi: 10.4064/sm-76-1-69-80.

[27]

Jorge D. Samur, Convergence of sums of mixing triangular arrays of random vectors with stationary rows, Ann. Probab., 12 (1984), 390-426.  doi: 10.1214/aop/1176993297.

[28]

Omri Sarig, Lecture notes on thermodynamic formalism for topological markov shifts, 2009.

[29]

Ken-iti Sato, Lévy Processes and Infinitely Divisible Distributions, volume 68 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1999. Translated from the 1990 Japanese original, Revised by the author.

[30]

Richard Sharp, Closed orbits in homology classes for Anosov flows, Ergodic Theory Dynam. Systems, 13 (1993), 387-408.  doi: 10.1017/S0143385700007434.

[31]

Ja. G. Sinai, Gibbs measures in ergodic theory, (Russian) Uspehi Mat. Nauk, 27 (1972), 21–64.

[32]

Rita Solomyak, A short proof of ergodicity of Babillot-Ledrappier measures, Proc. Amer. Math. Soc., 129 (2001), 3589-3591.  doi: 10.1090/S0002-9939-01-06181-0.

[33]

Charles Stone, On local and ratio limit theorems, In Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66), Vol. II: Contributions to Probability Theory, Part 2, pages 217–224. Univ. California Press, Berkeley, Calif., 1967.

[34]

Domokos Szász and Tamás Varjú, Local limit theorem for the Lorentz process and its recurrence in the plane, Ergodic Theory Dynam. Systems, 24 (2004), 257-278.  doi: 10.1017/S0143385703000439.

[35]

Domokos Szász and Tamás Varjú, Limit laws and recurrence for the planar Lorentz process with infinite horizon, J. Stat. Phys., 129 (2007), 59-80.  doi: 10.1007/s10955-007-9367-0.

[36]

Damien Thomine, Local time and first return time for periodic semi-flows, Israel J. Math., 215 (2016), 53-98.  doi: 10.1007/s11856-016-1326-5.

[37]

Simon Waddington, Large deviation asymptotics for Anosov flows, Ann. Inst. H. Poincaré Anal. Non Linéaire, 13 (1996), 445-484.  doi: 10.1016/S0294-1449(16)30110-X.

[38]

Roland 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.

show all references

References:
[1]

Jon Aaronson and Manfred Denker, The Poincaré series of $\Bbb C\setminus\Bbb Z$, Ergodic Theory Dynam. Systems, 19 (1999), 1-20.  doi: 10.1017/S0143385799126592.

[2]

Jon Aaronson and Manfred Denker, Local limit theorems for partial sums of stationary sequences generated by {G}ibbs-{M}arkov maps, Stoch. Dyn., 1 (2001), 193-237.  doi: 10.1142/S0219493701000114.

[3]

Jon. AaronsonManfred DenkerOmri Sarig and Rol Zweimüller, Aperiodicity of cocycles and conditional local limit theorems, Stoch. Dyn., 4 (2004), 31-62.  doi: 10.1142/S0219493704000936.

[4]

Jon Aaronson and Hitoshi Nakada, On the mixing coefficients of piecewise monotonic maps, Israel J. Math., 148 (2005), 1-10.  doi: 10.1007/BF02775429.

[5]

Jon Aaronson and Hitoshi Nakada, On multiple recurrence and other properties of 'nice' infinite measure-preserving transformations, Ergodic Theory Dynam. Systems, 37 (2017), 1345-1368.  doi: 10.1017/etds.2015.108.

[6]

Rufus Bowen, Symbolic dynamics for hyperbolic flows, Amer. J. Math., 95 (1973), 429-460.  doi: 10.2307/2373793.

[7]

Leo Breiman, Probability, Addison-Wesley Publishing Company, Reading, Mass.-London-Don Mills, Ont., 1968.

[8]

Rufus Bowen and Peter Walters, Expansive one-parameter flows, J. Differential Equations, 12 (1972), 180-193.  doi: 10.1016/0022-0396(72)90013-7.

[9]

D. Dolgopyat and P. Nándori, On mixing and the local central limit theorem for hyperbolic flows, Ergodic Theory Dynam. Systems, 40 (2020), 142–174. doi: 10.1017/etds.2018.29.

[10]

R. A. Doney, A bivariate local limit theorem, J. Multivariate Anal., 36 (1991), 95-102.  doi: 10.1016/0047-259X(91)90093-H.

[11]

Nathaniel A. Friedman, Mixing transformations in an infinite measure space, In Studies in probability and ergodic theory, Adv. in Math. Suppl. Stud., 2, 167–184. Academic Press, New York-London, 1978.

[12]

Y. Guivarc'h and J. Hardy, Théorèmes limites pour une classe de chaînes de Markov et applications aux difféomorphismes d'Anosov, Ann. Inst. H. Poincaré Probab. Statist., 24 (1988), 73-98. 

[13]

Hubert Hennion and Loïc Hervé, Limit Theorems for Markov Chains and Stochastic Properties of Dynamical Systems by Quasi-Compactness, volume 1766 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2001. doi: 10.1007/b87874.

[14]

E. Hopf, Ergodentheorie, Ergebnisse der Mathematik und ihrer Grenzgebiete, 5. Bd. Julius Springer, 1937.

[15]

Eberhard Hopf, Ergodic theory and the geodesic flow on surfaces of constant negative curvature, Bull. Amer. Math. Soc., 77 (1971), 863-877.  doi: 10.1090/S0002-9904-1971-12799-4.

[16]

Yukiko Iwata, A generalized local limit theorem for mixing semi-flows, Hokkaido Math. J., 37 (2008), 215-240.  doi: 10.14492/hokmj/1253539585.

[17]

Zbigniew J. Jurek and J. David Mason, Operator-Limit Distributions in Probability Theory, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1993.

[18]

K. Krickeberg, Recent results on mixing in topological measure spaces, In Probability and Information Theory (Proc. Internat. Sympos., McMaster Univ., Hamilton, Ont., 1968), 1969, pages 178–185. Springer, Berlin.

[19]

Gregory F. Lawler and Vlada Limic,. Random Walk: A Modern Introduction, volume 123 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2010. doi: 10.1017/CBO9780511750854.

[20]

A. Lasota and James A. Yorke, On the existence of invariant measures for piecewise monotonic transformations, Trans. Amer. Math. Soc., 186 (1973), 481–488, (1974). doi: 10.1090/S0002-9947-1973-0335758-1.

[21]

Elliott W. Montroll and George H. Weiss, Random walks on lattices. II, J. Mathematical Phys., 6 (1965), 167-181.  doi: 10.1063/1.1704269.

[22]

William Parry and Mark Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque, 187-188 (1990), 268 pp.

[23]

Françoise Pène, Planar Lorentz process in a random scenery, Ann. Inst. Henri Poincaré Probab. Stat., 45 (2009), 818-839.  doi: 10.1214/08-AIHP191.

[24]

V. V. Petrov, Sums of Independent Random Variables, Translated from the Russian by A. A. Brown. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 82. Springer-Verlag, New York-Heidelberg, 1975.

[25]

Mary Rees, Checking ergodicity of some geodesic flows with infinite Gibbs measure, Ergodic Theory Dynamical Systems, 1 (1981), 107-133.  doi: 10.1017/S0143385700001206.

[26]

Marek Rychlik, Bounded variation and invariant measures, Studia Math., 76 (1983), 69-80.  doi: 10.4064/sm-76-1-69-80.

[27]

Jorge D. Samur, Convergence of sums of mixing triangular arrays of random vectors with stationary rows, Ann. Probab., 12 (1984), 390-426.  doi: 10.1214/aop/1176993297.

[28]

Omri Sarig, Lecture notes on thermodynamic formalism for topological markov shifts, 2009.

[29]

Ken-iti Sato, Lévy Processes and Infinitely Divisible Distributions, volume 68 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1999. Translated from the 1990 Japanese original, Revised by the author.

[30]

Richard Sharp, Closed orbits in homology classes for Anosov flows, Ergodic Theory Dynam. Systems, 13 (1993), 387-408.  doi: 10.1017/S0143385700007434.

[31]

Ja. G. Sinai, Gibbs measures in ergodic theory, (Russian) Uspehi Mat. Nauk, 27 (1972), 21–64.

[32]

Rita Solomyak, A short proof of ergodicity of Babillot-Ledrappier measures, Proc. Amer. Math. Soc., 129 (2001), 3589-3591.  doi: 10.1090/S0002-9939-01-06181-0.

[33]

Charles Stone, On local and ratio limit theorems, In Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66), Vol. II: Contributions to Probability Theory, Part 2, pages 217–224. Univ. California Press, Berkeley, Calif., 1967.

[34]

Domokos Szász and Tamás Varjú, Local limit theorem for the Lorentz process and its recurrence in the plane, Ergodic Theory Dynam. Systems, 24 (2004), 257-278.  doi: 10.1017/S0143385703000439.

[35]

Domokos Szász and Tamás Varjú, Limit laws and recurrence for the planar Lorentz process with infinite horizon, J. Stat. Phys., 129 (2007), 59-80.  doi: 10.1007/s10955-007-9367-0.

[36]

Damien Thomine, Local time and first return time for periodic semi-flows, Israel J. Math., 215 (2016), 53-98.  doi: 10.1007/s11856-016-1326-5.

[37]

Simon Waddington, Large deviation asymptotics for Anosov flows, Ann. Inst. H. Poincaré Anal. Non Linéaire, 13 (1996), 445-484.  doi: 10.1016/S0294-1449(16)30110-X.

[38]

Roland 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.

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