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

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

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.
Citation: Eugen Mihailescu. Approximations for Gibbs states of arbitrary Hölder potentials on hyperbolic folded sets. Discrete & Continuous Dynamical Systems - A, 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.   Google Scholar

[2]

R. Bowen, "Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms,", Lecture Notes in Mathematics, 470 (1975).   Google Scholar

[3]

A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems,", With a supplementary chapter by A. Katok and L. Mendoza, 54 (1995).   Google Scholar

[4]

R. Mane, "Ergodic Theory and Differentiable Dynamics,", Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 8 (1987).   Google Scholar

[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.  Google Scholar

[6]

E. Mihailescu, Metric properties of some fractals and applications of inverse pressure,, Math. Proceed. Cambridge Phil. Soc., 148 (2010), 553.  doi: 10.1017/S0305004109990326.  Google Scholar

[7]

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

[8]

E. Mihailescu, Unstable manifolds and Hölder structures associated with noninvertible maps,, Discrete and Cont. Dynam. Syst., 14 (2006), 419.  doi: 10.3934/dcds.2006.14.419.  Google Scholar

[9]

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

[10]

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

[11]

M. Qian and Z. S. Zhang, Ergodic theory for Axiom A endomorphisms,, Ergodic Th. and Dyn. Syst., 15 (1995), 161.   Google Scholar

[12]

M. Qian and Z. Shu, SRB measures and Pesin's entropy formula for endomorphisms,, Trans. Amer. Math. Soc., 354 (2002), 1453.  doi: 10.1090/S0002-9947-01-02792-1.  Google Scholar

[13]

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

[14]

D. Ruelle, "Elements of Differentiable Dynamics and Bifurcation Theory,", Academic Press, (1989).   Google Scholar

[15]

D. Ruelle, Smooth dynamics and new theoretical ideas in nonequilibrium statistical mechanics,, J. Statist. Phys., 95 (1999), 393.  doi: 10.1023/A:1004593915069.  Google Scholar

[16]

Y. Sinaǐ, Gibbs measures in ergodic theory,, Uspehi Mat. Nauk, 27 (1972), 21.   Google Scholar

[17]

P. Walters, "An Introduction to Ergodic Theory,", Graduate Texts in Mathematics, 79 (1982).   Google Scholar

[18]

M. Zhang, "Anosov Endomorphisms: Shift Equivalence and Shift-Equivalence Classes,", Ph.D Thesis, (1989).   Google Scholar

show all references

References:
[1]

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

[2]

R. Bowen, "Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms,", Lecture Notes in Mathematics, 470 (1975).   Google Scholar

[3]

A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems,", With a supplementary chapter by A. Katok and L. Mendoza, 54 (1995).   Google Scholar

[4]

R. Mane, "Ergodic Theory and Differentiable Dynamics,", Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 8 (1987).   Google Scholar

[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.  Google Scholar

[6]

E. Mihailescu, Metric properties of some fractals and applications of inverse pressure,, Math. Proceed. Cambridge Phil. Soc., 148 (2010), 553.  doi: 10.1017/S0305004109990326.  Google Scholar

[7]

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

[8]

E. Mihailescu, Unstable manifolds and Hölder structures associated with noninvertible maps,, Discrete and Cont. Dynam. Syst., 14 (2006), 419.  doi: 10.3934/dcds.2006.14.419.  Google Scholar

[9]

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

[10]

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

[11]

M. Qian and Z. S. Zhang, Ergodic theory for Axiom A endomorphisms,, Ergodic Th. and Dyn. Syst., 15 (1995), 161.   Google Scholar

[12]

M. Qian and Z. Shu, SRB measures and Pesin's entropy formula for endomorphisms,, Trans. Amer. Math. Soc., 354 (2002), 1453.  doi: 10.1090/S0002-9947-01-02792-1.  Google Scholar

[13]

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

[14]

D. Ruelle, "Elements of Differentiable Dynamics and Bifurcation Theory,", Academic Press, (1989).   Google Scholar

[15]

D. Ruelle, Smooth dynamics and new theoretical ideas in nonequilibrium statistical mechanics,, J. Statist. Phys., 95 (1999), 393.  doi: 10.1023/A:1004593915069.  Google Scholar

[16]

Y. Sinaǐ, Gibbs measures in ergodic theory,, Uspehi Mat. Nauk, 27 (1972), 21.   Google Scholar

[17]

P. Walters, "An Introduction to Ergodic Theory,", Graduate Texts in Mathematics, 79 (1982).   Google Scholar

[18]

M. Zhang, "Anosov Endomorphisms: Shift Equivalence and Shift-Equivalence Classes,", Ph.D Thesis, (1989).   Google Scholar

[1]

Gunther Uhlmann, Jian Zhai. Inverse problems for nonlinear hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 455-469. doi: 10.3934/dcds.2020380

[2]

Mark F. Demers. Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 217-256. doi: 10.3934/dcds.2020217

[3]

Héctor Barge. Čech cohomology, homoclinic trajectories and robustness of non-saddle sets. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020381

[4]

Giuseppina Guatteri, Federica Masiero. Stochastic maximum principle for problems with delay with dependence on the past through general measures. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020048

[5]

Ilyasse Lamrani, Imad El Harraki, Ali Boutoulout, Fatima-Zahrae El Alaoui. Feedback stabilization of bilinear coupled hyperbolic systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020434

[6]

Shasha Hu, Yihong Xu, Yuhan Zhang. Second-Order characterizations for set-valued equilibrium problems with variable ordering structures. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020164

[7]

Yi-Hsuan Lin, Gen Nakamura, Roland Potthast, Haibing Wang. Duality between range and no-response tests and its application for inverse problems. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020072

[8]

Kha Van Huynh, Barbara Kaltenbacher. Some application examples of minimization based formulations of inverse problems and their regularization. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020074

[9]

Weiwei Liu, Jinliang Wang, Yuming Chen. Threshold dynamics of a delayed nonlocal reaction-diffusion cholera model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020316

[10]

Manil T. Mohan. First order necessary conditions of optimality for the two dimensional tidal dynamics system. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020045

[11]

Vieri Benci, Marco Cococcioni. The algorithmic numbers in non-archimedean numerical computing environments. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020449

[12]

Cuicui Li, Lin Zhou, Zhidong Teng, Buyu Wen. The threshold dynamics of a discrete-time echinococcosis transmission model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020339

[13]

Ebraheem O. Alzahrani, Muhammad Altaf Khan. Androgen driven evolutionary population dynamics in prostate cancer growth. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020426

[14]

Shao-Xia Qiao, Li-Jun Du. Propagation dynamics of nonlocal dispersal equations with inhomogeneous bistable nonlinearity. Electronic Research Archive, , () : -. doi: 10.3934/era.2020116

[15]

Antoine Benoit. Weak well-posedness of hyperbolic boundary value problems in a strip: when instabilities do not reflect the geometry. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5475-5486. doi: 10.3934/cpaa.2020248

[16]

Wenbin Li, Jianliang Qian. Simultaneously recovering both domain and varying density in inverse gravimetry by efficient level-set methods. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020073

[17]

Yangrong Li, Shuang Yang, Qiangheng Zhang. Odd random attractors for stochastic non-autonomous Kuramoto-Sivashinsky equations without dissipation. Electronic Research Archive, 2020, 28 (4) : 1529-1544. doi: 10.3934/era.2020080

[18]

José Luis López. A quantum approach to Keller-Segel dynamics via a dissipative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020376

[19]

Pengyu Chen. Non-autonomous stochastic evolution equations with nonlinear noise and nonlocal conditions governed by noncompact evolution families. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020383

[20]

Lin Shi, Xuemin Wang, Dingshi Li. Limiting behavior of non-autonomous stochastic reaction-diffusion equations with colored noise on unbounded thin domains. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5367-5386. doi: 10.3934/cpaa.2020242

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (25)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]