# American Institute of Mathematical Sciences

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 and 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. [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. [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. [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. [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. [6] D. Szász, Multidimensional hyperbolic billiards, Contemp. Math., 698 (2017), 201-220.  doi: 10.1090/conm/698/14028. [7] M. Viana, Multidimensional non-hyperbolic attractors, Publ. Math. Inst. Hautes Études Sci., 85 (1997), 63-96. [8] L. S. Young, Statistical properties of dynamical systems with some hyperbolicity, Ann. of Math., 147 (1998), 585-650.  doi: 10.2307/120960. [9] L. S. Young, Recurrence times and rates of mixing, Israel J. Math., 110 (1999), 153-188.  doi: 10.1007/BF02808180.

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. [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. [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. [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. [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. [6] D. Szász, Multidimensional hyperbolic billiards, Contemp. Math., 698 (2017), 201-220.  doi: 10.1090/conm/698/14028. [7] M. Viana, Multidimensional non-hyperbolic attractors, Publ. Math. Inst. Hautes Études Sci., 85 (1997), 63-96. [8] L. S. Young, Statistical properties of dynamical systems with some hyperbolicity, Ann. of Math., 147 (1998), 585-650.  doi: 10.2307/120960. [9] L. S. Young, Recurrence times and rates of mixing, Israel J. Math., 110 (1999), 153-188.  doi: 10.1007/BF02808180.
 [1] Yakov Pesin, Vaughn Climenhaga. Open problems in the theory of non-uniform hyperbolicity. Discrete and Continuous Dynamical Systems, 2010, 27 (2) : 589-607. doi: 10.3934/dcds.2010.27.589 [2] Boris Kalinin, Victoria Sadovskaya. Normal forms for non-uniform contractions. Journal of Modern Dynamics, 2017, 11: 341-368. doi: 10.3934/jmd.2017014 [3] Nigel P. Byott, Mark Holland, Yiwei Zhang. On the mixing properties of piecewise expanding maps under composition with permutations. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3365-3390. doi: 10.3934/dcds.2013.33.3365 [4] Pablo G. Barrientos, Abbas Fakhari. Ergodicity of non-autonomous discrete systems with non-uniform expansion. Discrete and Continuous Dynamical Systems - B, 2020, 25 (4) : 1361-1382. doi: 10.3934/dcdsb.2019231 [5] Markus Bachmayr, Van Kien Nguyen. Identifiability of diffusion coefficients for source terms of non-uniform sign. Inverse Problems and Imaging, 2019, 13 (5) : 1007-1021. doi: 10.3934/ipi.2019045 [6] Boris Hasselblatt, Yakov Pesin, Jörg Schmeling. Pointwise hyperbolicity implies uniform hyperbolicity. Discrete and Continuous Dynamical Systems, 2014, 34 (7) : 2819-2827. doi: 10.3934/dcds.2014.34.2819 [7] Zhong-Jie Han, Gen-Qi Xu. Spectrum and dynamical behavior of a kind of planar network of non-uniform strings with non-collocated feedbacks. Networks and Heterogeneous Media, 2010, 5 (2) : 315-334. doi: 10.3934/nhm.2010.5.315 [8] Donald L. DeAngelis, Bo Zhang. Effects of dispersal in a non-uniform environment on population dynamics and competition: A patch model approach. Discrete and Continuous Dynamical Systems - B, 2014, 19 (10) : 3087-3104. doi: 10.3934/dcdsb.2014.19.3087 [9] Zhong-Jie Han, Gen-Qi Xu. Exponential decay in non-uniform porous-thermo-elasticity model of Lord-Shulman type. Discrete and Continuous Dynamical Systems - B, 2012, 17 (1) : 57-77. doi: 10.3934/dcdsb.2012.17.57 [10] Izumi Takagi, Conghui Zhang. Existence and stability of patterns in a reaction-diffusion-ODE system with hysteresis in non-uniform media. Discrete and Continuous Dynamical Systems, 2021, 41 (7) : 3109-3140. doi: 10.3934/dcds.2020400 [11] Hai Huyen Dam, Wing-Kuen Ling. Optimal design of finite precision and infinite precision non-uniform cosine modulated filter bank. Journal of Industrial and Management Optimization, 2019, 15 (1) : 97-112. doi: 10.3934/jimo.2018034 [12] Zhong-Jie Han, Gen-Qi Xu. Dynamical behavior of networks of non-uniform Timoshenko beams system with boundary time-delay inputs. Networks and Heterogeneous Media, 2011, 6 (2) : 297-327. doi: 10.3934/nhm.2011.6.297 [13] Grigor Nika, Bogdan Vernescu. Rate of convergence for a multi-scale model of dilute emulsions with non-uniform surface tension. Discrete and Continuous Dynamical Systems - S, 2016, 9 (5) : 1553-1564. doi: 10.3934/dcdss.2016062 [14] Ruilin Li, Xin Wang, Hongyuan Zha, Molei Tao. Improving sampling accuracy of stochastic gradient MCMC methods via non-uniform subsampling of gradients. Discrete and Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021157 [15] Mickaël Kourganoff. Uniform hyperbolicity in nonflat billiards. Discrete and Continuous Dynamical Systems, 2018, 38 (3) : 1145-1160. doi: 10.3934/dcds.2018048 [16] Alexander Zlotnik. The Numerov-Crank-Nicolson scheme on a non-uniform mesh for the time-dependent Schrödinger equation on the half-axis. Kinetic and Related Models, 2015, 8 (3) : 587-613. doi: 10.3934/krm.2015.8.587 [17] Victor Churchill, Rick Archibald, Anne Gelb. Edge-adaptive $\ell_2$ regularization image reconstruction from non-uniform Fourier data. Inverse Problems and Imaging, 2019, 13 (5) : 931-958. doi: 10.3934/ipi.2019042 [18] Fumihiko Nakamura. Asymptotic behavior of non-expanding piecewise linear maps in the presence of random noise. Discrete and Continuous Dynamical Systems - B, 2018, 23 (6) : 2457-2473. doi: 10.3934/dcdsb.2018055 [19] Ralf Spatzier, Lei Yang. Exponential mixing and smooth classification of commuting expanding maps. Journal of Modern Dynamics, 2017, 11: 263-312. doi: 10.3934/jmd.2017012 [20] Makoto Mori. Higher order mixing property of piecewise linear transformations. Discrete and Continuous Dynamical Systems, 2000, 6 (4) : 915-934. doi: 10.3934/dcds.2000.6.915

2020 Impact Factor: 1.392