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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, 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

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