July  2012, 32(7): 2485-2502. doi: 10.3934/dcds.2012.32.2485

Equilibrium measures, prehistories distributions and fractal dimensions for endomorphisms

1. 

Institute of Mathematics "Simion Stoilow" of the Romanian Academy, P.O Box 1-764, RO 014-700, Bucharest

Received  December 2009 Revised  January 2012 Published  March 2012

The dynamics of endomorphisms (i.e non-invertible smooth maps) presents many significant differences from that of diffeomorphisms, as well as from the dynamics of expanding maps. There are numerous concrete examples of hyperbolic endomorphisms. Many methods cannot be used here due to overlappings in the fractal set and to the existence of (possibly infinitely) many local unstable manifolds going through the same point. First we will present the general problems and explain how to construct certain useful limit measures for atomic measures supported on various prehistories. These limit measures are in many cases shown to be equal to certain equilibrium measures for Hölder potentials. We obtain thus an analogue of the SRB measure, namely an inverse SRB measure in the case of a hyperbolic repeller, or of an Anosov endomorphism. We study then the 1-sided Bernoullicity (or lack of it) for certain measures invariant to endomorphisms, and give a Classification Theorem for the ergodic and metric types of behaviour of perturbations of a class of maps on their respective basic sets, in terms of the values of the stable dimension. We give also relations between thermodynamic formalism and fractal dimensions (Hausdorff dimension of stable/unstable intersections with basic sets, stable/unstable box dimensions, dimension of the global unstable set for endomorphisms). Applications to certain nonlinear evolution models are also given in the end.
Citation: Eugen Mihailescu. Equilibrium measures, prehistories distributions and fractal dimensions for endomorphisms. Discrete and Continuous Dynamical Systems, 2012, 32 (7) : 2485-2502. doi: 10.3934/dcds.2012.32.2485
References:
[1]

L. Barreira, "Dimension and Recurrence in Hyperbolic Dynamics," Progress in Mathematics, 272, Birkhäuser Verlag, Basel, 2008.

[2]

H. G. Bothe, Shift spaces and attractors in noninvertible horseshoes, Fundamenta Math., 152 (1997), 267-289.

[3]

R. Bowen, "Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms," Lecture Notes in Mathematics, 470, Springer-Verlag, Berlin-New York, 1975.

[4]

H. Bruin and J. Hawkins, Rigidity of smooth one-sided Bernoulli endomorphisms, New York J. Math., 15 (2009), 451-483.

[5]

K. Dajani and J. Hawkins, Rohlin factors, product factors and joinings for n-to-1 maps, Indiana Univ. Math. J., 42 (1993), 237-258. doi: 10.1512/iumj.1993.42.42012.

[6]

J.-P. Eckmann and D. Ruelle, Ergodic theory of chaos and strange attractors, Rev. Modern Physics, 57 (1985), 617-656. doi: 10.1103/RevModPhys.57.617.

[7]

K. Falconer, The Hausdorff dimension of some fractals and attractors of overlapping construction, J. Stat. Physics, 47 (1987), 123-132. doi: 10.1007/BF01009037.

[8]

J. E. Fornaess and N. Sibony, Hyperbolic maps on $\mathbb P^2$, Math. Ann., 311 (1998), 305-333. doi: 10.1007/s002080050189.

[9]

A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems," Encyclopedia of Mathematics and its Applications, 54, Cambridge Univ. Press, Cambridge, 1995.

[10]

J. A. Kennedy and D. R. Stockman, Chaotic equilibria in models with backward dynamics, J. Economic Dynamics and Control, 32 (2008), 939-955. doi: 10.1016/j.jedc.2007.04.004.

[11]

F. Ledrappier, L.-S. Young, The metric entropy of diffeomorphisms. II. Relations between entropy, exponents and dimension, Ann. of Math. (2), 122 (1985), 540-574. doi: 10.2307/1971329.

[12]

P.-D. Liu, Pesin's entropy formula for endomorphisms, Nagoya Math. J., 150 (1998), 197-209.

[13]

P.-D. Liu, Invariant measures satisfying an equality relating entropy, folding entropy and negative Lyapunov exponents, Commun. Math. Physics, 284 (2008), 391-406. doi: 10.1007/s00220-008-0568-4.

[14]

A. Manning and H. McCluskey, Hausdorff dimension for horseshoes, Ergodic Th. and Dynam. Syst., 3 (1983), 251-260.

[15]

R. Mañé, "Ergodic Theory and Differentiable Dynamics," Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 8, Springer Verlag, Berlin, 1987.

[16]

E. Mihailescu, Unstable directions and fractal dimension for a class of skew products with overlaps in fibers, Math. Zeitschrift, 269 (2011), 733-750. doi: 10.1007/s00209-010-0761-y.

[17]

E. Mihailescu, On some coding and mixing properties for a class of chaotic systems, Monatshefte Math., online, 2011. doi: 10.1007/s00605-011-0347-8.

[18]

E. Mihailescu, Higher dimensional expanding maps and toral extensions, to appear Proceed. Amer. Math. Soc., 2012. Available from: http://www.imar.ro/~mihailes.

[19]

E. Mihailescu, Metric properties of some fractal sets and applications of inverse pressure, Math. Proc. Cambridge Phil. Soc., 148 (2010), 553-572. doi: 10.1017/S0305004109990326.

[20]

E. Mihailescu, Physical measures for multivalued inverse iterates near hyperbolic repellors, J. Stat. Physics, 139 (2010), 800-819. doi: 10.1007/s10955-010-9960-5.

[21]

E. Mihailescu, On a class of stable conditional measures, Ergodic Th. and Dynam. Syst., 31 (2011), 1499-1515. doi: 10.1017/S0143385710000477.

[22]

E. Mihailescu, Approximations of Gibbs states for Holder potentials on hyperbolic folded sets, Discrete and Cont. Dynam. Syst., 32 (2012), 961-975. doi: 10.3934/dcds.2012.32.961.

[23]

E. Mihailescu, Local geometry and dynamical behavior on folded basic sets, J. Statistical Physics, 142 (2011), 154-167. doi: 10.1007/s10955-010-0097-3.

[24]

E. Mihailescu, Unstable manifolds and Hölder structures for noninvertible maps, Discrete and Cont. Dynam. Syst., 14 (2006), 419-446. doi: 10.3934/dcds.2006.14.419.

[25]

E. Mihailescu, The set $K^-$ for hyperbolic non-invertible maps, Ergodic Th. and Dynam. Syst., 22 (2002), 873-887.

[26]

E. Mihailescu, Periodic points for actions of tori in Stein manifolds, Math. Ann., 314 (1999), 39-52. doi: 10.1007/s002080050285.

[27]

E. Mihailescu, Inverse limits and statistical properties for chaotic implicitly defined economic models, arXiv:1111.3482v1, 2011.

[28]

E. Mihailescu and M. Urbański, Relations between stable dimension and the preimage counting function on basic sets with overlaps, Bull. London Math. Soc., 42 (2010), 15-27. doi: 10.1112/blms/bdp092.

[29]

E. Mihailescu and M. Urbański, Inverse pressure estimates and the independence of stable dimension for non-invertible maps, Canadian J. Math., 60 (2008), 658-684. doi: 10.4153/CJM-2008-029-2.

[30]

E. Mihailescu and M. Urbański, Transversal families of hyperbolic skew-products, Discrete and Cont. Dynam. Syst., 21 (2008), 907-928. doi: 10.3934/dcds.2008.21.907.

[31]

E. Mihailescu and M. Urbański, Estimates for the stable dimension for holomorphic maps, Houston J. Math., 31 (2005), 367-389.

[32]

E. Mihailescu and M. Urbański, Inverse topological pressure with applications to holomorphic dynamics in several variables, Commun. Contemp. Math., 6 (2004), 653-679. doi: 10.1142/S0219199704001446.

[33]

E. Mihailescu and M. Urbański, Hausdorff dimension of the limit set of conformal iterated function systems with overlaps, Proceed. Amer. Math. Soc., 139 (2011), 2767-2775. doi: 10.1090/S0002-9939-2011-10704-4.

[34]

Z. Nitecki, Topological entropy and the preimage structure of maps,, Real An. Exchange, 29 (): 9. 

[35]

W. Parry and P. Walters, Endomorphisms of a Lebesgue space, Bull. AMS, 78 (1972), 272-276. doi: 10.1090/S0002-9904-1972-12954-9.

[36]

W. Parry and M. Pollicott, Skew products and Livsic theory, in "Representation Theory, Dynamical Systems, and Asymptotic Combinatorics," Amer. Math. Soc. Transl. Ser. 2, 217, Amer. Math. Soc., Providence, RI, (2006), 139-165.

[37]

Y. Peres and B. Solomyak, Absolute continuity of Bernoulli convolutions, a simple proof, Math. Res. Letters, 3 (1996), 231-239.

[38]

Y. Pesin, "Dimension Theory in Dynamical Systems. Contemporary Views and Applications," Chicago Lectures in Mathematics, Univ. Chicago Press, Chicago, IL, 1997.

[39]

F. Przytycki, Anosov endomorphisms, Studia Math., 58 (1976), 249-285.

[40]

M. Qian and Z. S. Zhang, Ergodic theory for Axiom A endomorphisms, Ergodic Th. and Dynam. Syst., 15 (1995), 161-174.

[41]

M. Qian and S. Zhu, SRB measures and Pesin's entropy formula for endomorphisms, Trans. AMS, 354 (2002), 1453-1471. doi: 10.1090/S0002-9947-01-02792-1.

[42]

V. A. Rokhlin, Lectures on the theory of entropy of transformations with invariant measures, Russ. Math. Surveys, 22 (1967), 1-54. doi: 10.1070/RM1967v022n05ABEH001224.

[43]

D. Ruelle, The thermodynamic formalism for expanding maps, Commun. in Math. Physics, 125 (1989), 239-262. doi: 10.1007/BF01217908.

[44]

D. Ruelle, "Elements of Differentiable Dynamics and Bifurcation Theory," Academic Press, Inc., Boston, MA, 1989.

[45]

D. Ruelle, Repellers for real-analytic maps, Ergodic Th. and Dynam. Syst., 2 (1982), 99-107. doi: 10.1017/S0143385700009603.

[46]

K. Simon, B. Solomyak and M. Urbański, Hausdorff dimension of limit sets for parabolic IFS with overlaps, Pacific J. Math., 201 (2001), 441-478. doi: 10.2140/pjm.2001.201.441.

[47]

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

[48]

B. Solomyak, Measure and dimension for some fractal families, Math. Proceed. Cambridge Phil. Soc., 124 (1998), 531-546. doi: 10.1017/S0305004198002680.

[49]

M. Tsujii, Fat solenoidal attractors, Nonlinearity, 14 (2001), 1011-1027. doi: 10.1088/0951-7715/14/5/306.

[50]

L.-S. Young, What are SRB measures and which dynamical systems have them?, J. Statistical Physics, 108 (2002), 733-754. doi: 10.1023/A:1019762724717.

show all references

References:
[1]

L. Barreira, "Dimension and Recurrence in Hyperbolic Dynamics," Progress in Mathematics, 272, Birkhäuser Verlag, Basel, 2008.

[2]

H. G. Bothe, Shift spaces and attractors in noninvertible horseshoes, Fundamenta Math., 152 (1997), 267-289.

[3]

R. Bowen, "Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms," Lecture Notes in Mathematics, 470, Springer-Verlag, Berlin-New York, 1975.

[4]

H. Bruin and J. Hawkins, Rigidity of smooth one-sided Bernoulli endomorphisms, New York J. Math., 15 (2009), 451-483.

[5]

K. Dajani and J. Hawkins, Rohlin factors, product factors and joinings for n-to-1 maps, Indiana Univ. Math. J., 42 (1993), 237-258. doi: 10.1512/iumj.1993.42.42012.

[6]

J.-P. Eckmann and D. Ruelle, Ergodic theory of chaos and strange attractors, Rev. Modern Physics, 57 (1985), 617-656. doi: 10.1103/RevModPhys.57.617.

[7]

K. Falconer, The Hausdorff dimension of some fractals and attractors of overlapping construction, J. Stat. Physics, 47 (1987), 123-132. doi: 10.1007/BF01009037.

[8]

J. E. Fornaess and N. Sibony, Hyperbolic maps on $\mathbb P^2$, Math. Ann., 311 (1998), 305-333. doi: 10.1007/s002080050189.

[9]

A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems," Encyclopedia of Mathematics and its Applications, 54, Cambridge Univ. Press, Cambridge, 1995.

[10]

J. A. Kennedy and D. R. Stockman, Chaotic equilibria in models with backward dynamics, J. Economic Dynamics and Control, 32 (2008), 939-955. doi: 10.1016/j.jedc.2007.04.004.

[11]

F. Ledrappier, L.-S. Young, The metric entropy of diffeomorphisms. II. Relations between entropy, exponents and dimension, Ann. of Math. (2), 122 (1985), 540-574. doi: 10.2307/1971329.

[12]

P.-D. Liu, Pesin's entropy formula for endomorphisms, Nagoya Math. J., 150 (1998), 197-209.

[13]

P.-D. Liu, Invariant measures satisfying an equality relating entropy, folding entropy and negative Lyapunov exponents, Commun. Math. Physics, 284 (2008), 391-406. doi: 10.1007/s00220-008-0568-4.

[14]

A. Manning and H. McCluskey, Hausdorff dimension for horseshoes, Ergodic Th. and Dynam. Syst., 3 (1983), 251-260.

[15]

R. Mañé, "Ergodic Theory and Differentiable Dynamics," Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 8, Springer Verlag, Berlin, 1987.

[16]

E. Mihailescu, Unstable directions and fractal dimension for a class of skew products with overlaps in fibers, Math. Zeitschrift, 269 (2011), 733-750. doi: 10.1007/s00209-010-0761-y.

[17]

E. Mihailescu, On some coding and mixing properties for a class of chaotic systems, Monatshefte Math., online, 2011. doi: 10.1007/s00605-011-0347-8.

[18]

E. Mihailescu, Higher dimensional expanding maps and toral extensions, to appear Proceed. Amer. Math. Soc., 2012. Available from: http://www.imar.ro/~mihailes.

[19]

E. Mihailescu, Metric properties of some fractal sets and applications of inverse pressure, Math. Proc. Cambridge Phil. Soc., 148 (2010), 553-572. doi: 10.1017/S0305004109990326.

[20]

E. Mihailescu, Physical measures for multivalued inverse iterates near hyperbolic repellors, J. Stat. Physics, 139 (2010), 800-819. doi: 10.1007/s10955-010-9960-5.

[21]

E. Mihailescu, On a class of stable conditional measures, Ergodic Th. and Dynam. Syst., 31 (2011), 1499-1515. doi: 10.1017/S0143385710000477.

[22]

E. Mihailescu, Approximations of Gibbs states for Holder potentials on hyperbolic folded sets, Discrete and Cont. Dynam. Syst., 32 (2012), 961-975. doi: 10.3934/dcds.2012.32.961.

[23]

E. Mihailescu, Local geometry and dynamical behavior on folded basic sets, J. Statistical Physics, 142 (2011), 154-167. doi: 10.1007/s10955-010-0097-3.

[24]

E. Mihailescu, Unstable manifolds and Hölder structures for noninvertible maps, Discrete and Cont. Dynam. Syst., 14 (2006), 419-446. doi: 10.3934/dcds.2006.14.419.

[25]

E. Mihailescu, The set $K^-$ for hyperbolic non-invertible maps, Ergodic Th. and Dynam. Syst., 22 (2002), 873-887.

[26]

E. Mihailescu, Periodic points for actions of tori in Stein manifolds, Math. Ann., 314 (1999), 39-52. doi: 10.1007/s002080050285.

[27]

E. Mihailescu, Inverse limits and statistical properties for chaotic implicitly defined economic models, arXiv:1111.3482v1, 2011.

[28]

E. Mihailescu and M. Urbański, Relations between stable dimension and the preimage counting function on basic sets with overlaps, Bull. London Math. Soc., 42 (2010), 15-27. doi: 10.1112/blms/bdp092.

[29]

E. Mihailescu and M. Urbański, Inverse pressure estimates and the independence of stable dimension for non-invertible maps, Canadian J. Math., 60 (2008), 658-684. doi: 10.4153/CJM-2008-029-2.

[30]

E. Mihailescu and M. Urbański, Transversal families of hyperbolic skew-products, Discrete and Cont. Dynam. Syst., 21 (2008), 907-928. doi: 10.3934/dcds.2008.21.907.

[31]

E. Mihailescu and M. Urbański, Estimates for the stable dimension for holomorphic maps, Houston J. Math., 31 (2005), 367-389.

[32]

E. Mihailescu and M. Urbański, Inverse topological pressure with applications to holomorphic dynamics in several variables, Commun. Contemp. Math., 6 (2004), 653-679. doi: 10.1142/S0219199704001446.

[33]

E. Mihailescu and M. Urbański, Hausdorff dimension of the limit set of conformal iterated function systems with overlaps, Proceed. Amer. Math. Soc., 139 (2011), 2767-2775. doi: 10.1090/S0002-9939-2011-10704-4.

[34]

Z. Nitecki, Topological entropy and the preimage structure of maps,, Real An. Exchange, 29 (): 9. 

[35]

W. Parry and P. Walters, Endomorphisms of a Lebesgue space, Bull. AMS, 78 (1972), 272-276. doi: 10.1090/S0002-9904-1972-12954-9.

[36]

W. Parry and M. Pollicott, Skew products and Livsic theory, in "Representation Theory, Dynamical Systems, and Asymptotic Combinatorics," Amer. Math. Soc. Transl. Ser. 2, 217, Amer. Math. Soc., Providence, RI, (2006), 139-165.

[37]

Y. Peres and B. Solomyak, Absolute continuity of Bernoulli convolutions, a simple proof, Math. Res. Letters, 3 (1996), 231-239.

[38]

Y. Pesin, "Dimension Theory in Dynamical Systems. Contemporary Views and Applications," Chicago Lectures in Mathematics, Univ. Chicago Press, Chicago, IL, 1997.

[39]

F. Przytycki, Anosov endomorphisms, Studia Math., 58 (1976), 249-285.

[40]

M. Qian and Z. S. Zhang, Ergodic theory for Axiom A endomorphisms, Ergodic Th. and Dynam. Syst., 15 (1995), 161-174.

[41]

M. Qian and S. Zhu, SRB measures and Pesin's entropy formula for endomorphisms, Trans. AMS, 354 (2002), 1453-1471. doi: 10.1090/S0002-9947-01-02792-1.

[42]

V. A. Rokhlin, Lectures on the theory of entropy of transformations with invariant measures, Russ. Math. Surveys, 22 (1967), 1-54. doi: 10.1070/RM1967v022n05ABEH001224.

[43]

D. Ruelle, The thermodynamic formalism for expanding maps, Commun. in Math. Physics, 125 (1989), 239-262. doi: 10.1007/BF01217908.

[44]

D. Ruelle, "Elements of Differentiable Dynamics and Bifurcation Theory," Academic Press, Inc., Boston, MA, 1989.

[45]

D. Ruelle, Repellers for real-analytic maps, Ergodic Th. and Dynam. Syst., 2 (1982), 99-107. doi: 10.1017/S0143385700009603.

[46]

K. Simon, B. Solomyak and M. Urbański, Hausdorff dimension of limit sets for parabolic IFS with overlaps, Pacific J. Math., 201 (2001), 441-478. doi: 10.2140/pjm.2001.201.441.

[47]

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

[48]

B. Solomyak, Measure and dimension for some fractal families, Math. Proceed. Cambridge Phil. Soc., 124 (1998), 531-546. doi: 10.1017/S0305004198002680.

[49]

M. Tsujii, Fat solenoidal attractors, Nonlinearity, 14 (2001), 1011-1027. doi: 10.1088/0951-7715/14/5/306.

[50]

L.-S. Young, What are SRB measures and which dynamical systems have them?, J. Statistical Physics, 108 (2002), 733-754. doi: 10.1023/A:1019762724717.

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