Linear parabolic maps on the torus

Karol Zyczkowski; Takashi Nishikawa

doi:10.1016/S0375-9601(99)00465-X

Linear parabolic maps on the torus

Karol Zyczkowski, Takashi Nishikawa^*

^*Corresponding author for this work

Physics and Astronomy

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

8 Scopus citations

Abstract

We investigate linear parabolic maps on the torus. In a generic case these maps are non-invertible and discontinuous. Although the metric entropy of these systems is equal to zero, their dynamics is non-trivial due to folding of the image of the unit square into the torus. We study the structure of the maximal invariant set, and in a generic case we prove the sensitive dependence on the initial conditions. We study the decay of correlations and the diffusion in the corresponding system on the plane. We also demonstrate how the rationality of the real numbers defining the map influences the dynamical properties of the system.

Original language	English (US)
Pages (from-to)	377-386
Number of pages	10
Journal	Physics Letters, Section A: General, Atomic and Solid State Physics
Volume	259
Issue number	5
DOIs	https://doi.org/10.1016/S0375-9601(99)00465-X
State	Published - 1999

Funding

We are indebted to P. Ashwin for many valuable remarks and his constant interest in the progress of this work. We also thank M. Arjunwadkar, U. Feudel, C. Grebogi, J. Meiss, E. Ott, J. Stark, M. Woitkowski and J. Yorke for helpful discussions and are grateful to the anonymous referee for the several useful comments. K.Ż. acknowledges the Fulbright Fellowship and a support by the Polish KBN grant no. P03B 060 13.

Keywords

Decay of correlations
Diffusion
Parabolic maps on the torus
Sensitive dependence on initial conditions

ASJC Scopus subject areas

General Physics and Astronomy

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10.1016/S0375-9601(99)00465-X

Cite this

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title = "Linear parabolic maps on the torus",

abstract = "We investigate linear parabolic maps on the torus. In a generic case these maps are non-invertible and discontinuous. Although the metric entropy of these systems is equal to zero, their dynamics is non-trivial due to folding of the image of the unit square into the torus. We study the structure of the maximal invariant set, and in a generic case we prove the sensitive dependence on the initial conditions. We study the decay of correlations and the diffusion in the corresponding system on the plane. We also demonstrate how the rationality of the real numbers defining the map influences the dynamical properties of the system.",

keywords = "Decay of correlations, Diffusion, Parabolic maps on the torus, Sensitive dependence on initial conditions",

author = "Karol Zyczkowski and Takashi Nishikawa",

note = "Funding Information: We are indebted to P. Ashwin for many valuable remarks and his constant interest in the progress of this work. We also thank M. Arjunwadkar, U. Feudel, C. Grebogi, J. Meiss, E. Ott, J. Stark, M. Woitkowski and J. Yorke for helpful discussions and are grateful to the anonymous referee for the several useful comments. K.{\.Z}. acknowledges the Fulbright Fellowship and a support by the Polish KBN grant no. P03B 060 13. ",

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doi = "10.1016/S0375-9601(99)00465-X",

language = "English (US)",

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pages = "377--386",

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TY - JOUR

T1 - Linear parabolic maps on the torus

AU - Zyczkowski, Karol

AU - Nishikawa, Takashi

N1 - Funding Information: We are indebted to P. Ashwin for many valuable remarks and his constant interest in the progress of this work. We also thank M. Arjunwadkar, U. Feudel, C. Grebogi, J. Meiss, E. Ott, J. Stark, M. Woitkowski and J. Yorke for helpful discussions and are grateful to the anonymous referee for the several useful comments. K.Ż. acknowledges the Fulbright Fellowship and a support by the Polish KBN grant no. P03B 060 13.

PY - 1999

Y1 - 1999

N2 - We investigate linear parabolic maps on the torus. In a generic case these maps are non-invertible and discontinuous. Although the metric entropy of these systems is equal to zero, their dynamics is non-trivial due to folding of the image of the unit square into the torus. We study the structure of the maximal invariant set, and in a generic case we prove the sensitive dependence on the initial conditions. We study the decay of correlations and the diffusion in the corresponding system on the plane. We also demonstrate how the rationality of the real numbers defining the map influences the dynamical properties of the system.

AB - We investigate linear parabolic maps on the torus. In a generic case these maps are non-invertible and discontinuous. Although the metric entropy of these systems is equal to zero, their dynamics is non-trivial due to folding of the image of the unit square into the torus. We study the structure of the maximal invariant set, and in a generic case we prove the sensitive dependence on the initial conditions. We study the decay of correlations and the diffusion in the corresponding system on the plane. We also demonstrate how the rationality of the real numbers defining the map influences the dynamical properties of the system.

KW - Decay of correlations

KW - Diffusion

KW - Parabolic maps on the torus

KW - Sensitive dependence on initial conditions

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DO - 10.1016/S0375-9601(99)00465-X

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VL - 259

SP - 377

EP - 386

JO - Physics Letters, Section A: General, Atomic and Solid State Physics

JF - Physics Letters, Section A: General, Atomic and Solid State Physics

IS - 5

ER -

Linear parabolic maps on the torus

Abstract

Funding

Keywords

ASJC Scopus subject areas

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