Fractal properties of critical invariant curves

Brian R. Hunt; Konstantin M. Khanin; Yakov G. Sinai; James A. Yorke

doi:10.1007/BF02175565

Fractal properties of critical invariant curves

Brian R. Hunt, Konstantin M. Khanin, Yakov G. Sinai, James A. Yorke

Research output: Contribution to journal › Article › peer-review

9 Scopus citations

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.

Original language	English (US)
Pages (from-to)	261-276
Number of pages	16
Journal	Journal of Statistical Physics
Volume	85
Issue number	1-2
DOIs	https://doi.org/10.1007/BF02175565
State	Published - Oct 1996

All Science Journal Classification (ASJC) codes

Statistical and Nonlinear Physics
Mathematical Physics

Keywords

Circle homeomorphism
Critical curve
Fractal dimension
Invariant measure
Renormalization
Rotation number
Thermodynamic formalism
Twist map

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10.1007/BF02175565

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title = "Fractal properties of critical invariant curves",

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.",

keywords = "Circle homeomorphism, Critical curve, Fractal dimension, Invariant measure, Renormalization, Rotation number, Thermodynamic formalism, Twist map",

author = "Hunt, {Brian R.} and Khanin, {Konstantin M.} and Sinai, {Yakov G.} and Yorke, {James A.}",

year = "1996",

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language = "English (US)",

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pages = "261--276",

journal = "Journal of Statistical Physics",

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TY - JOUR

T1 - Fractal properties of critical invariant curves

AU - Hunt, Brian R.

AU - Khanin, Konstantin M.

AU - Sinai, Yakov G.

AU - Yorke, James A.

PY - 1996/10

Y1 - 1996/10

N2 - 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.

AB - 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.

KW - Circle homeomorphism

KW - Critical curve

KW - Fractal dimension

KW - Invariant measure

KW - Renormalization

KW - Rotation number

KW - Thermodynamic formalism

KW - Twist map

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DO - 10.1007/BF02175565

M3 - Article

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VL - 85

SP - 261

EP - 276

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

IS - 1-2

ER -

Fractal properties of critical invariant curves

Abstract

All Science Journal Classification (ASJC) codes

Keywords

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