# American Institute of Mathematical Sciences

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 & Continuous Dynamical Systems, 2012, 32 (7) : 2485-2502. doi: 10.3934/dcds.2012.32.2485
##### References:
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Falconer, The Hausdorff dimension of some fractals and attractors of overlapping construction, J. Stat. Physics, 47 (1987), 123-132. doi: 10.1007/BF01009037.  Google Scholar [8] J. E. Fornaess and N. Sibony, Hyperbolic maps on $\mathbb P^2$, Math. Ann., 311 (1998), 305-333. doi: 10.1007/s002080050189.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [12] P.-D. Liu, Pesin's entropy formula for endomorphisms, Nagoya Math. J., 150 (1998), 197-209.  Google Scholar [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.  Google Scholar [14] A. Manning and H. McCluskey, Hausdorff dimension for horseshoes, Ergodic Th. and Dynam. Syst., 3 (1983), 251-260.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [18] E. Mihailescu, Higher dimensional expanding maps and toral extensions, to appear Proceed. Amer. Math. Soc., 2012. Available from: http://www.imar.ro/~mihailes. Google Scholar [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.  Google Scholar [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.  Google Scholar [21] E. Mihailescu, On a class of stable conditional measures, Ergodic Th. and Dynam. Syst., 31 (2011), 1499-1515. doi: 10.1017/S0143385710000477.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [25] E. Mihailescu, The set $K^-$ for hyperbolic non-invertible maps, Ergodic Th. and Dynam. Syst., 22 (2002), 873-887.  Google Scholar [26] E. Mihailescu, Periodic points for actions of tori in Stein manifolds, Math. Ann., 314 (1999), 39-52. doi: 10.1007/s002080050285.  Google Scholar [27] E. Mihailescu, Inverse limits and statistical properties for chaotic implicitly defined economic models, arXiv:1111.3482v1, 2011. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [31] E. Mihailescu and M. Urbański, Estimates for the stable dimension for holomorphic maps, Houston J. Math., 31 (2005), 367-389.  Google Scholar [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.  Google Scholar [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.  Google Scholar [34] Z. Nitecki, Topological entropy and the preimage structure of maps,, Real An. Exchange, 29 (): 9.   Google Scholar [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.  Google Scholar [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.  Google Scholar [37] Y. Peres and B. Solomyak, Absolute continuity of Bernoulli convolutions, a simple proof, Math. Res. Letters, 3 (1996), 231-239.  Google Scholar [38] Y. Pesin, "Dimension Theory in Dynamical Systems. Contemporary Views and Applications," Chicago Lectures in Mathematics, Univ. Chicago Press, Chicago, IL, 1997.  Google Scholar [39] F. Przytycki, Anosov endomorphisms, Studia Math., 58 (1976), 249-285.  Google Scholar [40] M. Qian and Z. S. Zhang, Ergodic theory for Axiom A endomorphisms, Ergodic Th. and Dynam. Syst., 15 (1995), 161-174.  Google Scholar [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.  Google Scholar [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.  Google Scholar [43] D. Ruelle, The thermodynamic formalism for expanding maps, Commun. in Math. Physics, 125 (1989), 239-262. doi: 10.1007/BF01217908.  Google Scholar [44] D. Ruelle, "Elements of Differentiable Dynamics and Bifurcation Theory," Academic Press, Inc., Boston, MA, 1989.  Google Scholar [45] D. Ruelle, Repellers for real-analytic maps, Ergodic Th. and Dynam. Syst., 2 (1982), 99-107. doi: 10.1017/S0143385700009603.  Google Scholar [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.  Google Scholar [47] Y. Sinai, Gibbs measures in ergodic theory, Uspehi Mat. Nauk, 27 (1972), 21-64.  Google Scholar [48] B. Solomyak, Measure and dimension for some fractal families, Math. Proceed. Cambridge Phil. Soc., 124 (1998), 531-546. doi: 10.1017/S0305004198002680.  Google Scholar [49] M. Tsujii, Fat solenoidal attractors, Nonlinearity, 14 (2001), 1011-1027. doi: 10.1088/0951-7715/14/5/306.  Google Scholar [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.  Google Scholar

show all references

##### References:
 [1] L. Barreira, "Dimension and Recurrence in Hyperbolic Dynamics," Progress in Mathematics, 272, Birkhäuser Verlag, Basel, 2008.  Google Scholar [2] H. G. Bothe, Shift spaces and attractors in noninvertible horseshoes, Fundamenta Math., 152 (1997), 267-289.  Google Scholar [3] R. Bowen, "Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms," Lecture Notes in Mathematics, 470, Springer-Verlag, Berlin-New York, 1975.  Google Scholar [4] H. Bruin and J. Hawkins, Rigidity of smooth one-sided Bernoulli endomorphisms, New York J. Math., 15 (2009), 451-483.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [8] J. E. Fornaess and N. Sibony, Hyperbolic maps on $\mathbb P^2$, Math. Ann., 311 (1998), 305-333. doi: 10.1007/s002080050189.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [12] P.-D. Liu, Pesin's entropy formula for endomorphisms, Nagoya Math. J., 150 (1998), 197-209.  Google Scholar [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.  Google Scholar [14] A. Manning and H. McCluskey, Hausdorff dimension for horseshoes, Ergodic Th. and Dynam. Syst., 3 (1983), 251-260.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [18] E. Mihailescu, Higher dimensional expanding maps and toral extensions, to appear Proceed. Amer. Math. Soc., 2012. Available from: http://www.imar.ro/~mihailes. Google Scholar [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.  Google Scholar [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.  Google Scholar [21] E. Mihailescu, On a class of stable conditional measures, Ergodic Th. and Dynam. Syst., 31 (2011), 1499-1515. doi: 10.1017/S0143385710000477.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [25] E. Mihailescu, The set $K^-$ for hyperbolic non-invertible maps, Ergodic Th. and Dynam. Syst., 22 (2002), 873-887.  Google Scholar [26] E. Mihailescu, Periodic points for actions of tori in Stein manifolds, Math. Ann., 314 (1999), 39-52. doi: 10.1007/s002080050285.  Google Scholar [27] E. Mihailescu, Inverse limits and statistical properties for chaotic implicitly defined economic models, arXiv:1111.3482v1, 2011. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [31] E. Mihailescu and M. Urbański, Estimates for the stable dimension for holomorphic maps, Houston J. Math., 31 (2005), 367-389.  Google Scholar [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.  Google Scholar [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.  Google Scholar [34] Z. Nitecki, Topological entropy and the preimage structure of maps,, Real An. Exchange, 29 (): 9.   Google Scholar [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.  Google Scholar [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.  Google Scholar [37] Y. Peres and B. Solomyak, Absolute continuity of Bernoulli convolutions, a simple proof, Math. Res. Letters, 3 (1996), 231-239.  Google Scholar [38] Y. Pesin, "Dimension Theory in Dynamical Systems. Contemporary Views and Applications," Chicago Lectures in Mathematics, Univ. Chicago Press, Chicago, IL, 1997.  Google Scholar [39] F. Przytycki, Anosov endomorphisms, Studia Math., 58 (1976), 249-285.  Google Scholar [40] M. Qian and Z. S. Zhang, Ergodic theory for Axiom A endomorphisms, Ergodic Th. and Dynam. Syst., 15 (1995), 161-174.  Google Scholar [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.  Google Scholar [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.  Google Scholar [43] D. Ruelle, The thermodynamic formalism for expanding maps, Commun. in Math. Physics, 125 (1989), 239-262. doi: 10.1007/BF01217908.  Google Scholar [44] D. Ruelle, "Elements of Differentiable Dynamics and Bifurcation Theory," Academic Press, Inc., Boston, MA, 1989.  Google Scholar [45] D. Ruelle, Repellers for real-analytic maps, Ergodic Th. and Dynam. Syst., 2 (1982), 99-107. doi: 10.1017/S0143385700009603.  Google Scholar [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.  Google Scholar [47] Y. Sinai, Gibbs measures in ergodic theory, Uspehi Mat. Nauk, 27 (1972), 21-64.  Google Scholar [48] B. Solomyak, Measure and dimension for some fractal families, Math. Proceed. Cambridge Phil. Soc., 124 (1998), 531-546. doi: 10.1017/S0305004198002680.  Google Scholar [49] M. Tsujii, Fat solenoidal attractors, Nonlinearity, 14 (2001), 1011-1027. doi: 10.1088/0951-7715/14/5/306.  Google Scholar [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.  Google Scholar
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