Statistical stability for multidimensional piecewise expanding maps

Alves, José; Pumariño, Antonio; Vigil, Enrique

doi:10.1090/proc/13518

Statistical stability for multidimensional piecewise expanding maps
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by José F. Alves, Antonio Pumariño and Enrique Vigil

Proc. Amer. Math. Soc. 145 (2017), 3057-3068

DOI: https://doi.org/10.1090/proc/13518

Published electronically: February 22, 2017

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Abstract:

We present sufficient conditions for the (strong) statistical stability of some classes of multidimensional piecewise expanding maps. As a consequence we get that a certain natural two-dimensional extension of the classical one-dimensional family of tent maps is statistically stable.

References

Kourosh Adl-Zarabi, Absolutely continuous invariant measures for piecewise expanding $C^2$ transformations in $\textbf {R}^n$ on domains with cusps on the boundaries, Ergodic Theory Dynam. Systems 16 (1996), no. 1, 1–18. MR 1375124, DOI 10.1017/S0143385700008683
José Ferreira Alves, SRB measures for non-hyperbolic systems with multidimensional expansion, Ann. Sci. École Norm. Sup. (4) 33 (2000), no. 1, 1–32 (English, with English and French summaries). MR 1743717, DOI 10.1016/S0012-9593(00)00101-4
José F. Alves, Strong statistical stability of non-uniformly expanding maps, Nonlinearity 17 (2004), no. 4, 1193–1215. MR 2069701, DOI 10.1088/0951-7715/17/4/004
José F. Alves and Marcelo Viana, Statistical stability for robust classes of maps with non-uniform expansion, Ergodic Theory Dynam. Systems 22 (2002), no. 1, 1–32. MR 1889563, DOI 10.1017/S0143385702000019
J. Buzzi, Absolutely continuous invariant measures for generic multi-dimensional piecewise affine expanding maps, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 9 (1999), no. 9, 1743–1750. Discrete dynamical systems. MR 1728733, DOI 10.1142/S021812749900122X
Jérôme Buzzi, Absolutely continuous invariant probability measures for arbitrary expanding piecewise $\mathbf R$-analytic mappings of the plane, Ergodic Theory Dynam. Systems 20 (2000), no. 3, 697–708. MR 1764923, DOI 10.1017/S0143385700000377
Lawrence C. Evans and Ronald F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. MR 1158660
Enrico Giusti, Minimal surfaces and functions of bounded variation, Monographs in Mathematics, vol. 80, Birkhäuser Verlag, Basel, 1984. MR 775682, DOI 10.1007/978-1-4684-9486-0
P. Góra and A. Boyarsky, Absolutely continuous invariant measures for piecewise expanding $C^2$ transformation in $\textbf {R}^N$, Israel J. Math. 67 (1989), no. 3, 272–286. MR 1029902, DOI 10.1007/BF02764946
PawełGóra and Abraham Boyarsky, On functions of bounded variation in higher dimensions, Amer. Math. Monthly 99 (1992), no. 2, 159–160. MR 1144357, DOI 10.2307/2324185
Gerhard Keller, Ergodicité et mesures invariantes pour les transformations dilatantes par morceaux d’une région bornée du plan, C. R. Acad. Sci. Paris Sér. A-B 289 (1979), no. 12, A625–A627 (French, with English summary). MR 556443
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). MR 335758, DOI 10.1090/S0002-9947-1973-0335758-1
Antonio Pumariño, José Ángel Rodríguez, Joan Carles Tatjer, and Enrique Vigil, Expanding Baker maps as models for the dynamics emerging from 3D-homoclinic bifurcations, Discrete Contin. Dyn. Syst. Ser. B 19 (2014), no. 2, 523–541. MR 3170197, DOI 10.3934/dcdsb.2014.19.523
Antonio Pumariño, José Ángel Rodríguez, Joan Carles Tatjer, and Enrique Vigil, Chaotic dynamics for two-dimensional tent maps, Nonlinearity 28 (2015), no. 2, 407–434. MR 3303174, DOI 10.1088/0951-7715/28/2/407
Antonio Pumariño and Joan Carles Tatjer, Dynamics near homoclinic bifurcations of three-dimensional dissipative diffeomorphisms, Nonlinearity 19 (2006), no. 12, 2833–2852. MR 2273761, DOI 10.1088/0951-7715/19/12/006
Antonio Pumariño and Joan Carles Tatjer, Attractors for return maps near homoclinic tangencies of three-dimensional dissipative diffeomorphisms, Discrete Contin. Dyn. Syst. Ser. B 8 (2007), no. 4, 971–1005. MR 2342131, DOI 10.3934/dcdsb.2007.8.971
Benoît Saussol, Absolutely continuous invariant measures for multidimensional expanding maps, Israel J. Math. 116 (2000), 223–248. MR 1759406, DOI 10.1007/BF02773219
Joan Carles Tatjer, Three-dimensional dissipative diffeomorphisms with homoclinic tangencies, Ergodic Theory Dynam. Systems 21 (2001), no. 1, 249–302. MR 1826668, DOI 10.1017/S0143385701001146
Masato Tsujii, Absolutely continuous invariant measures for piecewise real-analytic expanding maps on the plane, Comm. Math. Phys. 208 (2000), no. 3, 605–622. MR 1736328, DOI 10.1007/s002200050003
Masato Tsujii, Absolutely continuous invariant measures for expanding piecewise linear maps, Invent. Math. 143 (2001), no. 2, 349–373. MR 1835391, DOI 10.1007/PL00005797