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May  2016, 36(5): 2873-2886. doi: 10.3934/dcds.2016.36.2873

## Sinai-Ruelle-Bowen measures for piecewise hyperbolic maps with two directions of instability in three-dimensional spaces

 1 Department of Mathematics, Michigan State University, East Lansing, MI 48824, United States

Received  July 2015 Revised  August 2015 Published  October 2015

A class of piecewise twice-differentiable Lozi-like maps in three-dimensional Euclidean spaces is introduced, and the existence of Sinai-Ruelle-Bowen measures is studied, where the dimension of the instability is equal to two. Further, an example with computer simulations is provided to illustrate the theoretical results.
Citation: Xu Zhang. Sinai-Ruelle-Bowen measures for piecewise hyperbolic maps with two directions of instability in three-dimensional spaces. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2873-2886. doi: 10.3934/dcds.2016.36.2873
##### References:
 [1] V. Baladi and S. Gouëzel, Banach spaces for piecewise cone-hyperbolic maps,, J. Mod. Dyn., 4 (2010), 91. doi: 10.3934/jmd.2010.4.91. Google Scholar [2] L. Barreira and Ya. Pesin, Lectures on Lyapunov exponents and smooth ergodic theory,, in Smooth Ergodic Theory and its Applications (Seattle, (1999), 3. doi: 10.1090/pspum/069/1858534. Google Scholar [3] L. Barreira and Ya. Pesin, Lyapunov Exponents and Smooth Ergodic Theory,, Univ. Lect. Series, (2002). Google Scholar [4] M. Benedicks and L. Carleson, The dynamics of the Hénon map,, Ann. of Math., 133 (1991), 73. doi: 10.2307/2944326. Google Scholar [5] M. Benedicks and L. S. Young, Sinai-Bowen-Ruelle measure for certain Hénon maps,, Invent. Math., 112 (1993), 541. doi: 10.1007/BF01232446. Google Scholar [6] M. Di Bernardo, M. I. Feigin, S. J. Hogan and M. E. Homer, Local analysis of C-bifurcation in $n$-dimensional piecewise-smooth dynamical systems,, Chaos Solitons Fractals, 10 (1999), 1881. doi: 10.1016/S0960-0779(98)00317-8. Google Scholar [7] R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms,, Lect. Notes Math., (1975). Google Scholar [8] A. Boyarsky and P. Góra, Laws of Chaos: Invariant Measures and Dynamical Systems in One Dimension. Probability and its Applications,, Birkhäuser, (1997). doi: 10.1007/978-1-4612-2024-4. Google Scholar [9] N. Chernov and R. Markarian, Chaotic Billiards,, Mathematical Surveys and Monographs, (2006). doi: 10.1090/surv/127. Google Scholar [10] P. Collet and Y. Levy, Ergodic properties of the Lozi mappings,, Commun. Math. Phys., 93 (1984), 461. doi: 10.1007/BF01212290. Google Scholar [11] M. Demers and C. Liverani, Stability of statistical properties in two-dimensional piecewise hyperbolic maps,, Trans. Amer. Math. Soc., 360 (2008), 4777. doi: 10.1090/S0002-9947-08-04464-4. Google Scholar [12] Z. Elhadj, Lozi Mappings. Theory and Applications,, CRC Press, (2014). Google Scholar [13] L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions,, Studies in Advanced Mathematics, (1992). Google Scholar [14] M. Jakobson and S. Newhouse, A two-dimensional version of the folklore theorem,, in Sinai's Moscow Seminar on Dynamical Systems, (1996), 89. Google Scholar [15] M. Jessa, Data encryption algorithms using one dimensional chaotic maps,, IEEE Int. Symp. on Circuits and Systems, (2000), 28. doi: 10.1109/ISCAS.2000.857194. Google Scholar [16] A. Katok, J. M. Strelcyn, A. Ledrappier and F. Przytycki, Invariant Manifolds, Entropy and Billiards; Smooth Maps with Singularities,, Lect. Notes. Math., (1222). Google Scholar [17] T. Kohda, Y. Ookubo and K. Ishii, A color image communication using YIQ signals by spread spectrum techniques,, Proc. IEEE Int. Symp. Spread Spectrum Techn. Appl., 3 (1998), 743. doi: 10.1109/ISSSTA.1998.722476. Google Scholar [18] A. Lasota and J. A. Yorke, On the existence of invariant measures for piecewise monotonic transformations,, Trans. Amer. Math. Soc., 186 (1973), 481. doi: 10.1090/S0002-9947-1973-0335758-1. Google Scholar [19] C. Liverani, Multidimensional expanding maps with singularities: A pedestrian approach,, Ergodic Theory Dynam. Systems, 33 (2013), 168. doi: 10.1017/S0143385711000939. Google Scholar [20] R. May, Simple mathematical models with very complicated dynamics,, Chapter: The Theory of Chaotic Attractors, (2004), 85. doi: 10.1007/978-0-387-21830-4_7. Google Scholar [21] Ya. B. Pesin, Dynamical systems with generalized hyperbolic attractors: Hyperbolic, ergodic and topological properties,, Ergodic Theory Dynam. Systems, 12 (1992), 123. doi: 10.1017/S0143385700006635. Google Scholar [22] Ya. B. Pesin and Ya. G. Sinai, Gibbs measures for partially hyperbolic attractors,, Ergodic Theory Dynam. Systems, 2 (1982), 417. doi: 10.1017/S014338570000170X. Google Scholar [23] C. Robinson, Dynamical Systems: Stability, Symbolic Dynamics and Chaos,, CRC Press, (1999). Google Scholar [24] F. Sánchez-Salas, Sinai-Ruelle-Bowen measures for piecewise hyperbolic transformations,, Divulg. Mat., 9 (2001), 35. Google Scholar [25] O. M. Sarig, Subexponential decay of corrlations,, Invent. Math., 150 (2002), 629. doi: 10.1007/s00222-002-0248-5. Google Scholar [26] Ya. G. Sinai, Gibbs measures in ergodic theory (Russian),, Uspehi Mat. Nauk, 27 (1972), 21. Google Scholar [27] L. S. Young, Bowen-Ruelle measures for certain piecewise hyperbolic maps,, Trans. Amer. Math. Soc., 287 (1985), 41. doi: 10.1090/S0002-9947-1985-0766205-1. Google Scholar [28] L. S. Young, Ergodic theory of differentiable dynamical systems,, in Real and Complex Dynamical Systems (eds. B. Branner and P. Hjorth), (1995), 293. Google Scholar [29] L. S. Young, Statistical properties of dynamical systems with some hyperbolicity,, Ann. of Math., 147 (1998), 585. doi: 10.2307/120960. Google Scholar [30] L. S. Young, Recurrence times and rates of mixing,, Isr. J. Math., 110 (1999), 153. doi: 10.1007/BF02808180. Google Scholar [31] L. S. Young, What are SRB measures, and which dynamical systems have them?, J. Statist. Phys., 108 (2002), 733. doi: 10.1023/A:1019762724717. Google Scholar

show all references

##### References:
 [1] V. Baladi and S. Gouëzel, Banach spaces for piecewise cone-hyperbolic maps,, J. Mod. Dyn., 4 (2010), 91. doi: 10.3934/jmd.2010.4.91. Google Scholar [2] L. Barreira and Ya. Pesin, Lectures on Lyapunov exponents and smooth ergodic theory,, in Smooth Ergodic Theory and its Applications (Seattle, (1999), 3. doi: 10.1090/pspum/069/1858534. Google Scholar [3] L. Barreira and Ya. Pesin, Lyapunov Exponents and Smooth Ergodic Theory,, Univ. Lect. Series, (2002). Google Scholar [4] M. Benedicks and L. Carleson, The dynamics of the Hénon map,, Ann. of Math., 133 (1991), 73. doi: 10.2307/2944326. Google Scholar [5] M. Benedicks and L. S. Young, Sinai-Bowen-Ruelle measure for certain Hénon maps,, Invent. Math., 112 (1993), 541. doi: 10.1007/BF01232446. Google Scholar [6] M. Di Bernardo, M. I. Feigin, S. J. Hogan and M. E. Homer, Local analysis of C-bifurcation in $n$-dimensional piecewise-smooth dynamical systems,, Chaos Solitons Fractals, 10 (1999), 1881. doi: 10.1016/S0960-0779(98)00317-8. Google Scholar [7] R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms,, Lect. Notes Math., (1975). Google Scholar [8] A. Boyarsky and P. Góra, Laws of Chaos: Invariant Measures and Dynamical Systems in One Dimension. Probability and its Applications,, Birkhäuser, (1997). doi: 10.1007/978-1-4612-2024-4. Google Scholar [9] N. Chernov and R. Markarian, Chaotic Billiards,, Mathematical Surveys and Monographs, (2006). doi: 10.1090/surv/127. Google Scholar [10] P. Collet and Y. Levy, Ergodic properties of the Lozi mappings,, Commun. Math. Phys., 93 (1984), 461. doi: 10.1007/BF01212290. Google Scholar [11] M. Demers and C. Liverani, Stability of statistical properties in two-dimensional piecewise hyperbolic maps,, Trans. Amer. Math. Soc., 360 (2008), 4777. doi: 10.1090/S0002-9947-08-04464-4. Google Scholar [12] Z. Elhadj, Lozi Mappings. Theory and Applications,, CRC Press, (2014). Google Scholar [13] L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions,, Studies in Advanced Mathematics, (1992). Google Scholar [14] M. Jakobson and S. Newhouse, A two-dimensional version of the folklore theorem,, in Sinai's Moscow Seminar on Dynamical Systems, (1996), 89. Google Scholar [15] M. Jessa, Data encryption algorithms using one dimensional chaotic maps,, IEEE Int. Symp. on Circuits and Systems, (2000), 28. doi: 10.1109/ISCAS.2000.857194. Google Scholar [16] A. Katok, J. M. Strelcyn, A. Ledrappier and F. Przytycki, Invariant Manifolds, Entropy and Billiards; Smooth Maps with Singularities,, Lect. Notes. Math., (1222). Google Scholar [17] T. Kohda, Y. Ookubo and K. Ishii, A color image communication using YIQ signals by spread spectrum techniques,, Proc. IEEE Int. Symp. Spread Spectrum Techn. Appl., 3 (1998), 743. doi: 10.1109/ISSSTA.1998.722476. Google Scholar [18] A. Lasota and J. A. Yorke, On the existence of invariant measures for piecewise monotonic transformations,, Trans. Amer. Math. Soc., 186 (1973), 481. doi: 10.1090/S0002-9947-1973-0335758-1. Google Scholar [19] C. Liverani, Multidimensional expanding maps with singularities: A pedestrian approach,, Ergodic Theory Dynam. Systems, 33 (2013), 168. doi: 10.1017/S0143385711000939. Google Scholar [20] R. May, Simple mathematical models with very complicated dynamics,, Chapter: The Theory of Chaotic Attractors, (2004), 85. doi: 10.1007/978-0-387-21830-4_7. Google Scholar [21] Ya. B. Pesin, Dynamical systems with generalized hyperbolic attractors: Hyperbolic, ergodic and topological properties,, Ergodic Theory Dynam. Systems, 12 (1992), 123. doi: 10.1017/S0143385700006635. Google Scholar [22] Ya. B. Pesin and Ya. G. Sinai, Gibbs measures for partially hyperbolic attractors,, Ergodic Theory Dynam. Systems, 2 (1982), 417. doi: 10.1017/S014338570000170X. Google Scholar [23] C. Robinson, Dynamical Systems: Stability, Symbolic Dynamics and Chaos,, CRC Press, (1999). Google Scholar [24] F. Sánchez-Salas, Sinai-Ruelle-Bowen measures for piecewise hyperbolic transformations,, Divulg. Mat., 9 (2001), 35. Google Scholar [25] O. M. Sarig, Subexponential decay of corrlations,, Invent. Math., 150 (2002), 629. doi: 10.1007/s00222-002-0248-5. Google Scholar [26] Ya. G. Sinai, Gibbs measures in ergodic theory (Russian),, Uspehi Mat. Nauk, 27 (1972), 21. Google Scholar [27] L. S. Young, Bowen-Ruelle measures for certain piecewise hyperbolic maps,, Trans. Amer. Math. Soc., 287 (1985), 41. doi: 10.1090/S0002-9947-1985-0766205-1. Google Scholar [28] L. S. Young, Ergodic theory of differentiable dynamical systems,, in Real and Complex Dynamical Systems (eds. B. Branner and P. Hjorth), (1995), 293. Google Scholar [29] L. S. Young, Statistical properties of dynamical systems with some hyperbolicity,, Ann. of Math., 147 (1998), 585. doi: 10.2307/120960. Google Scholar [30] L. S. Young, Recurrence times and rates of mixing,, Isr. J. Math., 110 (1999), 153. doi: 10.1007/BF02808180. Google Scholar [31] L. S. Young, What are SRB measures, and which dynamical systems have them?, J. Statist. Phys., 108 (2002), 733. doi: 10.1023/A:1019762724717. Google Scholar
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