# American Institute of Mathematical Sciences

• Previous Article
Fractal dimension of random attractor for stochastic non-autonomous damped wave equation with linear multiplicative white noise
• DCDS Home
• This Issue
• Next Article
A note on quasilinear wave equations in two space dimensions
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, 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), 91-137. 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, WA, 1999), Proc. Sympos. Pure Math., 69, Amer. Math. Soc., Providence, RI, 2001, 3-106. doi: 10.1090/pspum/069/1858534.  Google Scholar [3] L. Barreira and Ya. Pesin, Lyapunov Exponents and Smooth Ergodic Theory, Univ. Lect. Series, 23, Amer. Math. Soc., Providence, RI, 2002.  Google Scholar [4] M. Benedicks and L. Carleson, The dynamics of the Hénon map, Ann. of Math., 133 (1991), 73-169. 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), 541-576. 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), 1881-1908. 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., Vol. 470, Springer-Verlag, Berlin-New York, 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, Boston, MA, 1997. doi: 10.1007/978-1-4612-2024-4.  Google Scholar [9] N. Chernov and R. Markarian, Chaotic Billiards, Mathematical Surveys and Monographs, 127, Amer. Math. Soc., Providence, RI, 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), 461-481. 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), 4777-4814. doi: 10.1090/S0002-9947-08-04464-4.  Google Scholar [12] Z. Elhadj, Lozi Mappings. Theory and Applications, CRC Press, Boca Raton, FL, 2014.  Google Scholar [13] L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 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, Amer. Math. Soc. Transl. Ser. 2, 171, Amer. Math. Soc., Providence, RI, 1996, 89-105.  Google Scholar [15] M. Jessa, Data encryption algorithms using one dimensional chaotic maps, IEEE Int. Symp. on Circuits and Systems, Vol. 1, May 28-31, Geneva, Switzerland, 2000, 711-714. 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., Vol. 1222, Springer-Verlag, Berlin, 1986.  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), 743-747. 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), 481-488. 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), 168-182. doi: 10.1017/S0143385711000939.  Google Scholar [20] R. May, Simple mathematical models with very complicated dynamics, Chapter: The Theory of Chaotic Attractors, (2004), 85-93. 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), 123-151. 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), 417-438. doi: 10.1017/S014338570000170X.  Google Scholar [23] C. Robinson, Dynamical Systems: Stability, Symbolic Dynamics and Chaos, CRC Press, Boca Raton, FL, 1999.  Google Scholar [24] F. Sánchez-Salas, Sinai-Ruelle-Bowen measures for piecewise hyperbolic transformations, Divulg. Mat., 9 (2001), 35-54.  Google Scholar [25] O. M. Sarig, Subexponential decay of corrlations, Invent. Math., 150 (2002), 629-653, doi: 10.1007/s00222-002-0248-5.  Google Scholar [26] Ya. G. Sinai, Gibbs measures in ergodic theory (Russian), Uspehi Mat. Nauk, 27 (1972), 21-64.  Google Scholar [27] L. S. Young, Bowen-Ruelle measures for certain piecewise hyperbolic maps, Trans. Amer. Math. Soc., 287 (1985), 41-48. 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), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 464, Kluwer Acad. Publ., Dordrecht, 1995, 293-336.  Google Scholar [29] L. S. Young, Statistical properties of dynamical systems with some hyperbolicity, Ann. of Math., 147 (1998), 585-650. doi: 10.2307/120960.  Google Scholar [30] L. S. Young, Recurrence times and rates of mixing, Isr. J. Math., 110 (1999), 153-188. 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), 733-754. 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), 91-137. 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, WA, 1999), Proc. Sympos. Pure Math., 69, Amer. Math. Soc., Providence, RI, 2001, 3-106. doi: 10.1090/pspum/069/1858534.  Google Scholar [3] L. Barreira and Ya. Pesin, Lyapunov Exponents and Smooth Ergodic Theory, Univ. Lect. Series, 23, Amer. Math. Soc., Providence, RI, 2002.  Google Scholar [4] M. Benedicks and L. Carleson, The dynamics of the Hénon map, Ann. of Math., 133 (1991), 73-169. 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), 541-576. 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), 1881-1908. 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., Vol. 470, Springer-Verlag, Berlin-New York, 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, Boston, MA, 1997. doi: 10.1007/978-1-4612-2024-4.  Google Scholar [9] N. Chernov and R. Markarian, Chaotic Billiards, Mathematical Surveys and Monographs, 127, Amer. Math. Soc., Providence, RI, 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), 461-481. 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), 4777-4814. doi: 10.1090/S0002-9947-08-04464-4.  Google Scholar [12] Z. Elhadj, Lozi Mappings. Theory and Applications, CRC Press, Boca Raton, FL, 2014.  Google Scholar [13] L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 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, Amer. Math. Soc. Transl. Ser. 2, 171, Amer. Math. Soc., Providence, RI, 1996, 89-105.  Google Scholar [15] M. Jessa, Data encryption algorithms using one dimensional chaotic maps, IEEE Int. Symp. on Circuits and Systems, Vol. 1, May 28-31, Geneva, Switzerland, 2000, 711-714. 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., Vol. 1222, Springer-Verlag, Berlin, 1986.  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), 743-747. 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), 481-488. 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), 168-182. doi: 10.1017/S0143385711000939.  Google Scholar [20] R. May, Simple mathematical models with very complicated dynamics, Chapter: The Theory of Chaotic Attractors, (2004), 85-93. 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), 123-151. 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), 417-438. doi: 10.1017/S014338570000170X.  Google Scholar [23] C. Robinson, Dynamical Systems: Stability, Symbolic Dynamics and Chaos, CRC Press, Boca Raton, FL, 1999.  Google Scholar [24] F. Sánchez-Salas, Sinai-Ruelle-Bowen measures for piecewise hyperbolic transformations, Divulg. Mat., 9 (2001), 35-54.  Google Scholar [25] O. M. Sarig, Subexponential decay of corrlations, Invent. Math., 150 (2002), 629-653, doi: 10.1007/s00222-002-0248-5.  Google Scholar [26] Ya. G. Sinai, Gibbs measures in ergodic theory (Russian), Uspehi Mat. Nauk, 27 (1972), 21-64.  Google Scholar [27] L. S. Young, Bowen-Ruelle measures for certain piecewise hyperbolic maps, Trans. Amer. Math. Soc., 287 (1985), 41-48. 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), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 464, Kluwer Acad. Publ., Dordrecht, 1995, 293-336.  Google Scholar [29] L. S. Young, Statistical properties of dynamical systems with some hyperbolicity, Ann. of Math., 147 (1998), 585-650. doi: 10.2307/120960.  Google Scholar [30] L. S. Young, Recurrence times and rates of mixing, Isr. J. Math., 110 (1999), 153-188. 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), 733-754. doi: 10.1023/A:1019762724717.  Google Scholar
 [1] Maria Pires De Carvalho. Persistence of Bowen-Ruelle-Sinai measures. Discrete & Continuous Dynamical Systems, 2007, 17 (1) : 213-221. doi: 10.3934/dcds.2007.17.213 [2] Sébastien Gouëzel. An interval map with a spectral gap on Lipschitz functions, but not on bounded variation functions. Discrete & Continuous Dynamical Systems, 2009, 24 (4) : 1205-1208. doi: 10.3934/dcds.2009.24.1205 [3] Mark F. Demers. Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps. Discrete & Continuous Dynamical Systems, 2021, 41 (1) : 217-256. doi: 10.3934/dcds.2020217 [4] Mario Roy. A new variation of Bowen's formula for graph directed Markov systems. Discrete & Continuous Dynamical Systems, 2012, 32 (7) : 2533-2551. doi: 10.3934/dcds.2012.32.2533 [5] Tomasz Downarowicz, Yonatan Gutman, Dawid Huczek. Rank as a function of measure. Discrete & Continuous Dynamical Systems, 2014, 34 (7) : 2741-2750. doi: 10.3934/dcds.2014.34.2741 [6] Harrison Bray. Ergodicity of Bowen–Margulis measure for the Benoist 3-manifolds. Journal of Modern Dynamics, 2020, 16: 305-329. doi: 10.3934/jmd.2020011 [7] Christian Wolf. A shift map with a discontinuous entropy function. Discrete & Continuous Dynamical Systems, 2020, 40 (1) : 319-329. doi: 10.3934/dcds.2020012 [8] Franco Obersnel, Pierpaolo Omari. Multiple bounded variation solutions of a capillarity problem. Conference Publications, 2011, 2011 (Special) : 1129-1137. doi: 10.3934/proc.2011.2011.1129 [9] Luis Barreira, Yakov Pesin and Jorg Schmeling. On the pointwise dimension of hyperbolic measures: a proof of the Eckmann-Ruelle conjecture. Electronic Research Announcements, 1996, 2: 69-72. [10] Wacław Marzantowicz, Justyna Signerska. Firing map of an almost periodic input function. Conference Publications, 2011, 2011 (Special) : 1032-1041. doi: 10.3934/proc.2011.2011.1032 [11] Rinaldo M. Colombo, Francesca Monti. Solutions with large total variation to nonconservative hyperbolic systems. Communications on Pure & Applied Analysis, 2010, 9 (1) : 47-60. doi: 10.3934/cpaa.2010.9.47 [12] Michal Málek, Peter Raith. Stability of the distribution function for piecewise monotonic maps on the interval. Discrete & Continuous Dynamical Systems, 2018, 38 (5) : 2527-2539. doi: 10.3934/dcds.2018105 [13] Magnus Aspenberg, Viviane Baladi, Juho Leppänen, Tomas Persson. On the fractional susceptibility function of piecewise expanding maps. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021133 [14] Ugo Bessi. The stochastic value function in metric measure spaces. Discrete & Continuous Dynamical Systems, 2017, 37 (4) : 1819-1839. doi: 10.3934/dcds.2017076 [15] Yunho Kim, Luminita A. Vese. Image recovery using functions of bounded variation and Sobolev spaces of negative differentiability. Inverse Problems & Imaging, 2009, 3 (1) : 43-68. doi: 10.3934/ipi.2009.3.43 [16] Hun Ki Baek, Younghae Do. Dangerous Border-Collision bifurcations of a piecewise-smooth map. Communications on Pure & Applied Analysis, 2006, 5 (3) : 493-503. doi: 10.3934/cpaa.2006.5.493 [17] Zhiying Qin, Jichen Yang, Soumitro Banerjee, Guirong Jiang. Border-collision bifurcations in a generalized piecewise linear-power map. Discrete & Continuous Dynamical Systems - B, 2011, 16 (2) : 547-567. doi: 10.3934/dcdsb.2011.16.547 [18] Viviane Baladi, Sébastien Gouëzel. Banach spaces for piecewise cone-hyperbolic maps. Journal of Modern Dynamics, 2010, 4 (1) : 91-137. doi: 10.3934/jmd.2010.4.91 [19] Anja Randecker, Giulio Tiozzo. Cusp excursion in hyperbolic manifolds and singularity of harmonic measure. Journal of Modern Dynamics, 2021, 17: 183-211. doi: 10.3934/jmd.2021006 [20] Pengfei Zhang. Partially hyperbolic sets with positive measure and $ACIP$ for partially hyperbolic systems. Discrete & Continuous Dynamical Systems, 2012, 32 (4) : 1435-1447. doi: 10.3934/dcds.2012.32.1435

2020 Impact Factor: 1.392