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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 and 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.

[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.

[3]

L. Barreira and Ya. Pesin, Lyapunov Exponents and Smooth Ergodic Theory, Univ. Lect. Series, 23, Amer. Math. Soc., Providence, RI, 2002.

[4]

M. Benedicks and L. Carleson, The dynamics of the Hénon map, Ann. of Math., 133 (1991), 73-169. doi: 10.2307/2944326.

[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.

[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.

[7]

R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lect. Notes Math., Vol. 470, Springer-Verlag, Berlin-New York, 1975.

[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.

[9]

N. Chernov and R. Markarian, Chaotic Billiards, Mathematical Surveys and Monographs, 127, Amer. Math. Soc., Providence, RI, 2006. doi: 10.1090/surv/127.

[10]

P. Collet and Y. Levy, Ergodic properties of the Lozi mappings, Commun. Math. Phys., 93 (1984), 461-481. doi: 10.1007/BF01212290.

[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.

[12]

Z. Elhadj, Lozi Mappings. Theory and Applications, CRC Press, Boca Raton, FL, 2014.

[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.

[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.

[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.

[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.

[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.

[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.

[19]

C. Liverani, Multidimensional expanding maps with singularities: A pedestrian approach, Ergodic Theory Dynam. Systems, 33 (2013), 168-182. doi: 10.1017/S0143385711000939.

[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.

[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.

[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.

[23]

C. Robinson, Dynamical Systems: Stability, Symbolic Dynamics and Chaos, CRC Press, Boca Raton, FL, 1999.

[24]

F. Sánchez-Salas, Sinai-Ruelle-Bowen measures for piecewise hyperbolic transformations, Divulg. Mat., 9 (2001), 35-54.

[25]

O. M. Sarig, Subexponential decay of corrlations, Invent. Math., 150 (2002), 629-653, doi: 10.1007/s00222-002-0248-5.

[26]

Ya. G. Sinai, Gibbs measures in ergodic theory (Russian), Uspehi Mat. Nauk, 27 (1972), 21-64.

[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.

[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.

[29]

L. S. Young, Statistical properties of dynamical systems with some hyperbolicity, Ann. of Math., 147 (1998), 585-650. doi: 10.2307/120960.

[30]

L. S. Young, Recurrence times and rates of mixing, Isr. J. Math., 110 (1999), 153-188. doi: 10.1007/BF02808180.

[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.

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.

[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.

[3]

L. Barreira and Ya. Pesin, Lyapunov Exponents and Smooth Ergodic Theory, Univ. Lect. Series, 23, Amer. Math. Soc., Providence, RI, 2002.

[4]

M. Benedicks and L. Carleson, The dynamics of the Hénon map, Ann. of Math., 133 (1991), 73-169. doi: 10.2307/2944326.

[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.

[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.

[7]

R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lect. Notes Math., Vol. 470, Springer-Verlag, Berlin-New York, 1975.

[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.

[9]

N. Chernov and R. Markarian, Chaotic Billiards, Mathematical Surveys and Monographs, 127, Amer. Math. Soc., Providence, RI, 2006. doi: 10.1090/surv/127.

[10]

P. Collet and Y. Levy, Ergodic properties of the Lozi mappings, Commun. Math. Phys., 93 (1984), 461-481. doi: 10.1007/BF01212290.

[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.

[12]

Z. Elhadj, Lozi Mappings. Theory and Applications, CRC Press, Boca Raton, FL, 2014.

[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.

[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.

[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.

[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.

[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.

[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.

[19]

C. Liverani, Multidimensional expanding maps with singularities: A pedestrian approach, Ergodic Theory Dynam. Systems, 33 (2013), 168-182. doi: 10.1017/S0143385711000939.

[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.

[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.

[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.

[23]

C. Robinson, Dynamical Systems: Stability, Symbolic Dynamics and Chaos, CRC Press, Boca Raton, FL, 1999.

[24]

F. Sánchez-Salas, Sinai-Ruelle-Bowen measures for piecewise hyperbolic transformations, Divulg. Mat., 9 (2001), 35-54.

[25]

O. M. Sarig, Subexponential decay of corrlations, Invent. Math., 150 (2002), 629-653, doi: 10.1007/s00222-002-0248-5.

[26]

Ya. G. Sinai, Gibbs measures in ergodic theory (Russian), Uspehi Mat. Nauk, 27 (1972), 21-64.

[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.

[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.

[29]

L. S. Young, Statistical properties of dynamical systems with some hyperbolicity, Ann. of Math., 147 (1998), 585-650. doi: 10.2307/120960.

[30]

L. S. Young, Recurrence times and rates of mixing, Isr. J. Math., 110 (1999), 153-188. doi: 10.1007/BF02808180.

[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.

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