January  2022, 42(1): 353-368. doi: 10.3934/dcds.2021120

Inducing schemes for multi-dimensional piecewise expanding maps

Dipartimento di Matematica, II Università di Roma (Tor Vergata), Via della Ricerca Scientifica, 00133 Roma, Italy

Received  November 2020 Revised  June 2021 Published  January 2022 Early access  September 2021

Fund Project: I thank the anonymous referee for valuable suggestions. The author was supported by the European Advanced Grant StochExtHomog (ERC AdG 320977) and by the PRIN Grant Regular and stochastic behaviour in dynamical systems (PRIN 2017S35EHN)

We construct inducing schemes for general multi-dimensional piecewise expanding maps where the base transformation is Gibbs-Markov and the return times have exponential tails. Such structures are a crucial tool in proving statistical properties of dynamical systems with some hyperbolicity. As an application we check the conditions for the first return map of a class of multi-dimensional non-Markov, non-conformal intermittent maps.

Citation: Peyman Eslami. Inducing schemes for multi-dimensional piecewise expanding maps. Discrete & Continuous Dynamical Systems, 2022, 42 (1) : 353-368. doi: 10.3934/dcds.2021120
References:
[1]

J. Alves, SRB measures for non-hyperbolic systems with multidimensional expansion, Ann. Sci. École Norm. Sup., 33 (2000), 1-32.  doi: 10.1016/S0012-9593(00)00101-4.  Google Scholar

[2]

P. Bálint and I. P. Tóth, Exponential decay of correlations in multi-dimensional dispersing billiards, Ann. Henri Poincaré, 9 (2008), 1309-1369.  doi: 10.1007/s00023-008-0389-1.  Google Scholar

[3]

N. Chernov, Statistical properties of piecewise smooth hyperbolic systems in high dimensions, Discrete Contin. Dynam. Systems, 5 (1999), 425-448.  doi: 10.3934/dcds.1999.5.425.  Google Scholar

[4]

P. Eslami, S. Vaienti and I. Melbourne, Sharp statistical properties for a family of multidimensional non-Markovian non-conformal intermittent maps, Adv. Math., 388 (2021). doi: 10.1016/j.aim.2021.107853.  Google Scholar

[5]

H. Hu and S. Vaienti, Absolutely continuous invariant measures for non-uniformly expanding maps, Ergodic Theory Dynam. Systems, 29 (2009), 1185-1215.  doi: 10.1017/S0143385708000576.  Google Scholar

[6]

D. Szász, Multidimensional hyperbolic billiards, Contemp. Math., 698 (2017), 201-220.  doi: 10.1090/conm/698/14028.  Google Scholar

[7]

M. Viana, Multidimensional non-hyperbolic attractors, Publ. Math. Inst. Hautes Études Sci., 85 (1997), 63-96.   Google Scholar

[8]

L. S. Young, Statistical properties of dynamical systems with some hyperbolicity, Ann. of Math., 147 (1998), 585-650.  doi: 10.2307/120960.  Google Scholar

[9]

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. Alves, SRB measures for non-hyperbolic systems with multidimensional expansion, Ann. Sci. École Norm. Sup., 33 (2000), 1-32.  doi: 10.1016/S0012-9593(00)00101-4.  Google Scholar

[2]

P. Bálint and I. P. Tóth, Exponential decay of correlations in multi-dimensional dispersing billiards, Ann. Henri Poincaré, 9 (2008), 1309-1369.  doi: 10.1007/s00023-008-0389-1.  Google Scholar

[3]

N. Chernov, Statistical properties of piecewise smooth hyperbolic systems in high dimensions, Discrete Contin. Dynam. Systems, 5 (1999), 425-448.  doi: 10.3934/dcds.1999.5.425.  Google Scholar

[4]

P. Eslami, S. Vaienti and I. Melbourne, Sharp statistical properties for a family of multidimensional non-Markovian non-conformal intermittent maps, Adv. Math., 388 (2021). doi: 10.1016/j.aim.2021.107853.  Google Scholar

[5]

H. Hu and S. Vaienti, Absolutely continuous invariant measures for non-uniformly expanding maps, Ergodic Theory Dynam. Systems, 29 (2009), 1185-1215.  doi: 10.1017/S0143385708000576.  Google Scholar

[6]

D. Szász, Multidimensional hyperbolic billiards, Contemp. Math., 698 (2017), 201-220.  doi: 10.1090/conm/698/14028.  Google Scholar

[7]

M. Viana, Multidimensional non-hyperbolic attractors, Publ. Math. Inst. Hautes Études Sci., 85 (1997), 63-96.   Google Scholar

[8]

L. S. Young, Statistical properties of dynamical systems with some hyperbolicity, Ann. of Math., 147 (1998), 585-650.  doi: 10.2307/120960.  Google Scholar

[9]

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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