Abstract
We compute numerically the Hausdorff dimensions of the Gibbs measures on the invariant sets of Axiom A systems. In particular, we stress the existence of a measure which has maximal dimension and can be relevant for the ergodic properties of the system. For hyperbolic maps of the plane with constant Jacobianj, we apply the Bowen-Ruelle formula, using the relationF(β=d H−1)=lnj, which links the Hausdorff dimensiond H of an attractor to a free energy functionalF(β) defined in the thermodynamic formalism. We provide numerical evidence that this relation remains valid for some nonhyperbolic maps, such as the Hénon map.
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References
R. Bowen,Publ. Math. IHES 50:11 (1979); D. Ruelle,Prog. Phys. 7:351 (1983).
D. Ruelle,Thermodynamic Formalism (Addison-Wesley, Reading, Massachusetts, 1978).
H. Fujsaka,Prog. Theor. Phys. 70:1264 (1983); R. Benzi, G. Paladin, G. Parisi, and A. Vulpiani,J. Phys. A 18:2157 (1985).
J. L. Kaplan and J. A. Yorke, inLecture Notes in Mathematics, Vol. 730 (Springer, Berlin, 1978), p. 204.
G. Paladin and A. Vulpiani,Phys. Rep. 186:147 (1987), and references therein.
M. J. Feigenbaum,J. Stat. Phys. 46:919 (1987).
P. Collet, J. L. Lebowitz, and A. Porzio,J. Stat. Phys. 47:609 (1987).
D. Bessis, G. Paladin, G. Turchetti, and S. Vaienti,J. Stat. Phys. 51:109 (1988).
S. Vaienti,J. Stat. Phys. 56:403 (1989).
T. Bohr and T. Tel, Thermodynamics of fractals, inDirection of Chaos, Vol. 2 (World Scientific, Singapore, to appear).
G. Paladin, inUniversalities in Condensed Matter Physics, R. Jullien, L. Peliti, and R. Rammal, eds. (Springer-Verlag, 1988).
G. Paladin and S. Vaienti,J. Phys. A 21:4609 (1988).
J. P. Eckmann and D. Ruelle,Rev. Mod. Phys. 57:617 (1985).
D. Auerbach, P. Cvitanović, J. P. Eckmann, G. H. Gunaratne, and I. Procaccia,Phys. Rev. Lett. 58:2387 (1987); G. H. Gunaratne and I. Procaccia,Phys. Rev. Lett. 59:1377 (1987).
P. Cvitanović, Invariant measurements of strange set in terms of cycles. Preprint (1988).
L. Young,Ergodic Theory Dynam. Syst. 2:109 (1982).
Ya. B. Pesin,Ergodic Theory Dynam. Syst. 5:595 (1985).
L. P. Kadanoff and C. Tang,Proc. Natl. Acad. Sci. USA 81:1276 (1984); H. Kantz and P. Grassberger,Physica D 17:75 (1985).
P. Collet and Y. Levry,Commun. Math. Phys. 93:461 (1984).
P. Grassberger, A. Politi, and R. Badii,J. Stat. Phys. 51:135 (1988).
A. Politi, R. Badii, and P. Grassberger,J. Phys. A 21:L763 (1988).
P. Walters,Am. J. Math. 97:937 (1975).
H. McCluskey and A. Manning,Ergodic Theory Dynam. Syst. 3:251 (1983).
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Paladin, G., Vaienti, S. Hausdorff dimensions in two-dimensional maps and thermodynamic formalism. J Stat Phys 57, 289–299 (1989). https://doi.org/10.1007/BF01023644
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DOI: https://doi.org/10.1007/BF01023644