Abstract
We examine the dimension of the invariant measure for some singular circle homeomorphisms for a variety of rotation numbers, through both the thermodynamic formalism and numerical computation. The maps we consider include those induced by the action of the standard map on an invariant curve at the critical parameter value beyond which the curve is destroyed. Our results indicate that the dimension is universal for a given type of singularity and rotation number, and that among all rotation numbers, the golden mean produces the largest dimension.
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References
R. Artuso, E. Aurell and P. Cvitanovic, Recycling of strange sets: I. Circle expansion, II. Applications,Nonlinearity 3:325–386 (1990).
E. Aurell, On the metric properties of the Feigenbaum attractor,J. Stat. Phys. 47:439–458 (1987).
J. Balatoni and A. Rényi, Remarks on entropy,Publ. Math. Inst. Hung. Acad. Sci. 1:9–40 (1956) [in Hungarian with English summary]; English translation inSelected Papers of Alfréd Rényi, Vol. 1 (Akadémiai Kiadó, Budapest, 1976), pp. 558–586.
G. D. Birkhoff, Sur quelques courbes fermees remarquable,Bull. Soc. Math. France 60 (1932); also inCollected Papers, Vol. II (American Mathematical Society, Providence, Rhode Island, 1950), pp. 444–461.
P. Collet, J. L. Lebowitz, and A. Porzio, The dimension spectrum of some dynamical systems,J. Stat. Phys. 47:609–644 (1987).
J. D. Farmer, E. Ott, and J. A. Yorke, The dimension of chaotic attractors,Physica 7D:153–180 (1983).
M. J. Feigenbaum, Quantitative universality for a class of nonlinear transformations,J. Stat. Phys. 19:25–52 (1978).
M. J. Feigenbaum, The universal metric properties of nonlinear transformations,J. Stat. Phys. 21:669–706 (1979).
M. J. Feigenbaum, L. P. Kadanoff, and S. J Shenker, Quasiperiodicity in dissipative systems: A renormalization group analysis,Physica 5D:370–386 (1982).
J. M. Greene A method for determining a stochastic transition,J. Math. Phys. 20:1183–1201 (1979).
J. Graczyk and G. Świątek, Critical circle maps near bifurcation, SUNY Stony Brook IMS preprint #1991/8.
J. Graczyk and G. Świątek, Singular measures in circle dynamics,Commun. Math. Phys. 157:213–230 (1993).
T. Halsey, M. Jensen, L. Kadanoff, I. Procaccia, and B. Shraiman, Fractal measures and their singularities: The characterization of strange sets,Phys. Rev. A 33:1141–1151 (1986).
M. Herman, Sur la conjugasion différentiable des difféomorphismes du cercle à des rotations,Pub. Math. IHES 49:5–233 (1979).
M. Herman, Sur les Courbes Invariantes par les Difféomorphismes de l'Anneau, Vol. 1,Astérisque 103–104 (1983).
Y. Katznelson and D. Ornstein, The differentiability of the conjugation of certain diffeomorphisms of the circle,Ergodic Theory Dynam. Syst. 9:643–680 (1989).
Y. Katznelson and D. Ornstein, A new method for twist theorems,J. Analyse 60:157–208 (1993).
K. M. Khanin, Universal estimates for critical circle mappings,Chaos 1:181–186 (1991).
K. M. Khanin, Rigidity for circle homeomorphisms with the break singularity,Russ. Acad. Sci. Dokl. Math., to appear.
K. M. Khanin and E. B. Vul, Circle homeomorphisms with weak discontinuities,Adv. Sov. Math. 3:57–98 (1991).
R. S. Mackay,Renormalization in Area-Preserving Maps (World Scientific, Singapore, 1993).
A. Osbaldestin and M. Sarkis, Singularity spectrum of a critical KAM torus,J. Phys. A: Math. Gen. 20:L953–958 (1987).
S. Ostlund, D. Rand, J. Sethna, and E. Siggia, Universal properties of the transition from quasi-periodicity to chaos in dissipative systems,Physica 8D:303–342 (1983).
Ya. G. Sinai,Topics in Ergodic Theory (Princeton University Press, Princeton, New Jersey, 1994).
A. Stirnemann, Renormalization for golden circles,Commun. Math. Phys. 152:369–431 (1993).
D. Sullivan, Private communication.
G. Świątek, Rational rotation numbers for maps of the circle,Commun. Math. Phys. 119:109–128 (1988).
E. B. Vul, Ya. G. Sinai, and K. M. Khanin, Feigenbaum universality and the thermodynamic formalism,Russ. Math. Surv. 39:3 (1984).
L.-S. Young, Dimension, entropy and Lyapunov exponents,Ergodic Theory Dynam. Syst. 2:109–124 (1982).
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Communicated by J. L. Lebowitz
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Hunt, B.R., Khanin, K.M., Sinai, Y.G. et al. Fractal properties of critical invariant curves. J Stat Phys 85, 261–276 (1996). https://doi.org/10.1007/BF02175565
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DOI: https://doi.org/10.1007/BF02175565