Approximations for Gibbs states of arbitrary Hölder potentials on hyperbolic folded sets

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March 2012, 32(3): 961-975. doi: 10.3934/dcds.2012.32.961

Approximations for Gibbs states of arbitrary Hölder potentials on hyperbolic folded sets

Eugen Mihailescu ^1,

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

Received October 2010 Revised January 2011 Published October 2011

Abstract
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We give approximations for the Gibbs states of arbitrary Hölder potentials $\phi$, with the help of weighted sums of atomic measures on preimage sets, in the case of smooth non-invertible maps hyperbolic on folded basic sets $\Lambda$. The endomorphism may have also stable directions on $\Lambda$ and is non-expanding in general. Folding of the phase space means that we do not have a foliation structure for the local unstable manifolds (instead they depend on the whole past and may intersect each other both inside and outside $\Lambda$). We consider here simultaneously all $n$-preimages in $\Lambda$ of a point, instead of the usual way of taking only the consecutive preimages from some given prehistory. We thus obtain the weighted distribution of consecutive preimage sets, with respect to various equilibrium measures on the saddle-type folded set $\Lambda$. In particular we obtain the distribution of preimage sets on $\Lambda$, with respect to the measure of maximal entropy. Our result is not a direct application of Birkhoff Ergodic Theorem on the inverse limit $\hat \Lambda$, since the set of prehistories of a point is uncountable in general, and the speed of convergence may vary for different prehistories in $\hat \Lambda$. For hyperbolic toral endomorphisms, we obtain the distribution of the consecutive preimage sets towards an inverse SRB measure, for Lebesgue-almost all points.

Keywords: equilibrium measures, Thermodynamic formalism for non-invertible maps, hyperbolic endomorphisms on folded basic sets, chaotic dynamics on fractals, inverse limits..

Mathematics Subject Classification: Primary: 37D35, 37D20; Secondary: 37A30, 37C4.

Citation: Eugen Mihailescu. Approximations for Gibbs states of arbitrary Hölder potentials on hyperbolic folded sets. Discrete and Continuous Dynamical Systems, 2012, 32 (3) : 961-975. doi: 10.3934/dcds.2012.32.961

References:

[1]	H. G. Bothe, Shift spaces and attractors in noninvertible horseshoes, Fundamenta Math., 152 (1997), 267-289.
[2]	R. Bowen, "Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms," Lecture Notes in Mathematics, 470, Springer-Verlag, Berlin, 1975.
[3]	A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems," With a supplementary chapter by A. Katok and L. Mendoza, Encyclopedia of Mathematics and its Applications, 54, Cambridge Univ. Press, Cambridge, 1995.
[4]	R. Mane, "Ergodic Theory and Differentiable Dynamics," Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 8, Springer-Verlag, Berlin, 1987.
[5]	E. Mihailescu, Unstable directions and fractal dimensions for skew products with overlaps in fibers, Math. Zeitschrift, 2011. doi: 10.1007/s00209-010-0761-y.
[6]	E. Mihailescu, Metric properties of some fractals and applications of inverse pressure, Math. Proceed. Cambridge Phil. Soc., 148 (2010), 553-572. doi: 10.1017/S0305004109990326.
[7]	E. Mihailescu, On a class of stable conditional measures, Ergodic Th. and Dynam. Syst., 31 (2011), 1499-1515. doi: 10.1017/S0143385710000477.
[8]	E. Mihailescu, Unstable manifolds and Hölder structures associated with noninvertible maps, Discrete and Cont. Dynam. Syst., 14 (2006), 419-446. doi: 10.3934/dcds.2006.14.419.
[9]	E. Mihailescu, The set $K^-$ for hyperbolic non-invertible maps, Ergodic Th. and Dynam. Syst., 22 (2002), 873-887.
[10]	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.
[11]	M. Qian and Z. S. Zhang, Ergodic theory for Axiom A endomorphisms, Ergodic Th. and Dyn. Syst., 15 (1995), 161-174.
[12]	M. Qian and Z. Shu, SRB measures and Pesin's entropy formula for endomorphisms, Trans. Amer. Math. Soc., 354 (2002), 1453-1471. doi: 10.1090/S0002-9947-01-02792-1.
[13]	D. Ruelle, The thermodynamic formalism for expanding maps, Commun. Math. Physics, 125 (1989), 239-262. doi: 10.1007/BF01217908.
[14]	D. Ruelle, "Elements of Differentiable Dynamics and Bifurcation Theory," Academic Press, Inc., Boston, MA, 1989.
[15]	D. Ruelle, Smooth dynamics and new theoretical ideas in nonequilibrium statistical mechanics, J. Statist. Phys., 95 (1999), 393-468. doi: 10.1023/A:1004593915069.
[16]	Y. Sinaǐ, Gibbs measures in ergodic theory, Uspehi Mat. Nauk, 27 (1972), 21-64.
[17]	P. Walters, "An Introduction to Ergodic Theory," Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982.
[18]	M. Zhang, "Anosov Endomorphisms: Shift Equivalence and Shift-Equivalence Classes," Ph.D Thesis, Peking Univ., 1989.

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References:

[1]	H. G. Bothe, Shift spaces and attractors in noninvertible horseshoes, Fundamenta Math., 152 (1997), 267-289.
[2]	R. Bowen, "Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms," Lecture Notes in Mathematics, 470, Springer-Verlag, Berlin, 1975.
[3]	A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems," With a supplementary chapter by A. Katok and L. Mendoza, Encyclopedia of Mathematics and its Applications, 54, Cambridge Univ. Press, Cambridge, 1995.
[4]	R. Mane, "Ergodic Theory and Differentiable Dynamics," Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 8, Springer-Verlag, Berlin, 1987.
[5]	E. Mihailescu, Unstable directions and fractal dimensions for skew products with overlaps in fibers, Math. Zeitschrift, 2011. doi: 10.1007/s00209-010-0761-y.
[6]	E. Mihailescu, Metric properties of some fractals and applications of inverse pressure, Math. Proceed. Cambridge Phil. Soc., 148 (2010), 553-572. doi: 10.1017/S0305004109990326.
[7]	E. Mihailescu, On a class of stable conditional measures, Ergodic Th. and Dynam. Syst., 31 (2011), 1499-1515. doi: 10.1017/S0143385710000477.
[8]	E. Mihailescu, Unstable manifolds and Hölder structures associated with noninvertible maps, Discrete and Cont. Dynam. Syst., 14 (2006), 419-446. doi: 10.3934/dcds.2006.14.419.
[9]	E. Mihailescu, The set $K^-$ for hyperbolic non-invertible maps, Ergodic Th. and Dynam. Syst., 22 (2002), 873-887.
[10]	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.
[11]	M. Qian and Z. S. Zhang, Ergodic theory for Axiom A endomorphisms, Ergodic Th. and Dyn. Syst., 15 (1995), 161-174.
[12]	M. Qian and Z. Shu, SRB measures and Pesin's entropy formula for endomorphisms, Trans. Amer. Math. Soc., 354 (2002), 1453-1471. doi: 10.1090/S0002-9947-01-02792-1.
[13]	D. Ruelle, The thermodynamic formalism for expanding maps, Commun. Math. Physics, 125 (1989), 239-262. doi: 10.1007/BF01217908.
[14]	D. Ruelle, "Elements of Differentiable Dynamics and Bifurcation Theory," Academic Press, Inc., Boston, MA, 1989.
[15]	D. Ruelle, Smooth dynamics and new theoretical ideas in nonequilibrium statistical mechanics, J. Statist. Phys., 95 (1999), 393-468. doi: 10.1023/A:1004593915069.
[16]	Y. Sinaǐ, Gibbs measures in ergodic theory, Uspehi Mat. Nauk, 27 (1972), 21-64.
[17]	P. Walters, "An Introduction to Ergodic Theory," Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982.
[18]	M. Zhang, "Anosov Endomorphisms: Shift Equivalence and Shift-Equivalence Classes," Ph.D Thesis, Peking Univ., 1989.

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