DOI: 10.1017/s0143385700006635. Dynamical systems with generalized hyperbolic attractors: hyperbolic, ergodic and topological properties

Dynamical systems with generalized hyperbolic attractors: hyperbolic, ergodic and topological properties

10.1017/s0143385700006635

Crossref journal-article

Cambridge University Press (CUP)

Ergodic Theory and Dynamical Systems (56)

Abstract

AbstractWe introduce a class of dynamical systems on a Riemannian manifold with singularities having attractors with strong hyperbolic behavior of trajectories. This class includes a number of famous examples such as the Lorenz type attractor, the Lozi attractor and some others which have been of great interest in recent years. We prove the existence of a special invariant measure which is an analog of the Bowen-Ruelle-Sinai measure for classical hyperbolic attractors and study the ergodic properties of the system with respect to this measure. We also describe some topological properties of the system on the attractor. Our results can be considered a dissipative version of the theory of systems with singularities preserving the smooth measure.

Bibliography

Pesin, Ya. B. (1992). Dynamical systems with generalized hyperbolic attractors: hyperbolic, ergodic and topological properties. Ergodic Theory and Dynamical Systems, 12(1), 123â151.

Authors 1

Ya. B. Pesin (first)

References 25 Referenced 83

10.1070/IM1976v010n06ABEH001835
{'key': 'S0143385700006635_ref016', 'volume-title': 'Modern Problems in Mathematics', 'volume': '2', 'author': 'Pesin', 'year': '1989'} / Modern Problems in Mathematics by Pesin (1989)
{'key': 'S0143385700006635_ref023', 'first-page': '94', 'volume-title': 'Structure of Lorenz attractors', 'author': 'Williams', 'year': '1977'} / Structure of Lorenz attractors by Williams (1977)
10.1007/BFb0099031
10.1070/RM1967v022n05ABEH001228
10.1070/RM1977v032n04ABEH001639
10.1007/BF02684769
{'key': 'S0143385700006635_ref008', 'volume-title': 'Modern Problems of Mathematics', 'volume': '2', 'author': 'Bunimovich', 'year': '1989'} / Modern Problems of Mathematics by Bunimovich (1989)
{'key': 'S0143385700006635_ref011', 'first-page': '593', 'article-title': 'Propriétés ergodiques des mesures de Sinai', 'volume': '294', 'author': 'Ledrappier', 'year': '1982', 'journal-title': 'C. R. Acad. Sci.'} / C. R. Acad. Sci. / Propriétés ergodiques des mesures de Sinai by Ledrappier (1982)
{'key': 'S0143385700006635_ref014', 'first-page': '398', 'volume-title': 'Nonlinear Dynamics', 'author': 'Misiurewicz', 'year': '1980'} / Nonlinear Dynamics by Misiurewicz (1980)
{'key': 'S0143385700006635_ref012', 'first-page': '103', 'volume-title': 'Ergodic properties of the Lozi map', 'author': 'Levy', 'year': '1985'} / Ergodic properties of the Lozi map by Levy (1985)
{'key': 'S0143385700006635_ref007', 'volume-title': 'Qualitative methods of the theory of nonlinear oscillations in point systems', 'author': 'Belykh', 'year': '1980'} / Qualitative methods of the theory of nonlinear oscillations in point systems by Belykh (1980)
{'key': 'S0143385700006635_ref021', 'volume-title': 'Statistical Irreversibility in Nonlinear Systems', 'author': 'Sinai', 'year': '1979'} / Statistical Irreversibility in Nonlinear Systems by Sinai (1979)
{'key': 'S0143385700006635_ref004', 'first-page': '336', 'article-title': 'On appearance and structure of Lorenz attractor', 'volume': '234', 'author': 'Afraimovich', 'year': '1977', 'journal-title': 'Dokl. Akad. Nauk USSR'} / Dokl. Akad. Nauk USSR / On appearance and structure of Lorenz attractor by Afraimovich (1977)
{'key': 'S0143385700006635_ref005', 'first-page': '150', 'article-title': 'On attracting nonstructurally stable limit sets of type of Lorenz attractor', 'volume': '44', 'author': 'Afraimovich', 'year': '1983', 'journal-title': 'Trudy Moskov. Mat. Obshch.'} / Trudy Moskov. Mat. Obshch. / On attracting nonstructurally stable limit sets of type of Lorenz attractor by Afraimovich (1983)
{'key': 'S0143385700006635_ref013', 'first-page': '9', 'article-title': 'Un attracteur étrange du type attracteur de Hénon', 'volume': '39', 'author': 'Lozi', 'year': '1978', 'journal-title': 'J. Phys.'} / J. Phys. / Un attracteur étrange du type attracteur de Hénon by Lozi (1978)
{'key': 'S0143385700006635_ref002', 'first-page': '39', 'volume-title': 'Trudy V Int. Conf. on Nonlinear Oscillations', 'volume': '2', 'author': 'Anosov', 'year': '1970'} / Trudy V Int. Conf. on Nonlinear Oscillations by Anosov (1970)
10.1070/SM1969v007n01ABEH001076
{'key': 'S0143385700006635_ref009', 'first-page': '212', 'volume-title': 'Nonlinear Waves', 'author': 'Bunimovich', 'year': '1980'} / Nonlinear Waves by Bunimovich (1980)
{'key': 'S0143385700006635_ref017', 'article-title': 'Ergodic properties and dimensionlike characteristics of strange attractors that are close to hyperbolic', 'author': 'Pesin', 'year': '1986', 'journal-title': 'Proc. Int. Cong. Math.'} / Proc. Int. Cong. Math. / Ergodic properties and dimensionlike characteristics of strange attractors that are close to hyperbolic by Pesin (1986)
10.1017/S014338570000170X / Ergod. Th. and Dynam. Sys. / Gibbs measures for partially hyperbolic attractors by Pesin (1982)
[25] Rychlik M. . Invariant measures and the variation principle for Lozi mappings. PhD dissertation. University of California, Berkeley, 1983.
10.1090/S0002-9947-1985-0766205-1
{'key': 'S0143385700006635_ref006', 'volume-title': 'The Dimension of Lorenz Type Attractors', 'volume': '6', 'author': 'Afraimovich', 'year': '1987'} / The Dimension of Lorenz Type Attractors by Afraimovich (1987)
10.1007/BF02684768

Dates

Type	When
Created	16 years, 11 months ago (Sept. 19, 2008, 11:17 a.m.)
Deposited	5 years, 3 months ago (May 8, 2020, 6:58 a.m.)
Indexed	3 weeks, 6 days ago (July 25, 2025, 6:10 a.m.)
Issued	33 years, 5 months ago (March 1, 1992)
Published	33 years, 5 months ago (March 1, 1992)
Published Online	16 years, 11 months ago (Sept. 19, 2008)
Published Print	33 years, 5 months ago (March 1, 1992)

Funders 0

None

BibTeX

@article{Pesin_1992, title={Dynamical systems with generalized hyperbolic attractors: hyperbolic, ergodic and topological properties}, volume={12}, ISSN={1469-4417}, url={http://dx.doi.org/10.1017/s0143385700006635}, DOI={10.1017/s0143385700006635}, number={1}, journal={Ergodic Theory and Dynamical Systems}, publisher={Cambridge University Press (CUP)}, author={Pesin, Ya. B.}, year={1992}, month=mar, pages={123–151} }

JSON

{
  "indexed": {
    "date-parts": [
      [
        2025,
        7,
        25
      ]
    ],
    "date-time": "2025-07-25T10:10:17Z",
    "timestamp": 1753438217691
  },
  "reference-count": 25,
  "publisher": "Cambridge University Press (CUP)",
  "issue": "1",
  "license": [
    {
      "start": {
        "date-parts": [
          [
            2008,
            9,
            19
          ]
        ],
        "date-time": "2008-09-19T00:00:00Z",
        "timestamp": 1221782400000
      },
      "content-version": "unspecified",
      "delay-in-days": 6046,
      "URL": "https://www.cambridge.org/core/terms"
    }
  ],
  "content-domain": {
    "domain": [],
    "crossmark-restriction": false
  },
  "published-print": {
    "date-parts": [
      [
        1992,
        3
      ]
    ]
  },
  "abstract": "<jats:title>Abstract</jats:title><jats:p>We introduce a class of dynamical systems on a Riemannian manifold with singularities having attractors with strong hyperbolic behavior of trajectories. This class includes a number of famous examples such as the Lorenz type attractor, the Lozi attractor and some others which have been of great interest in recent years. We prove the existence of a special invariant measure which is an analog of the Bowen-Ruelle-Sinai measure for classical hyperbolic attractors and study the ergodic properties of the system with respect to this measure. We also describe some topological properties of the system on the attractor. Our results can be considered a dissipative version of the theory of systems with singularities preserving the smooth measure.</jats:p>",
  "DOI": "10.1017/s0143385700006635",
  "type": "journal-article",
  "created": {
    "date-parts": [
      [
        2008,
        9,
        19
      ]
    ],
    "date-time": "2008-09-19T15:17:20Z",
    "timestamp": 1221837440000
  },
  "page": "123-151",
  "source": "Crossref",
  "is-referenced-by-count": 83,
  "title": "Dynamical systems with generalized hyperbolic attractors: hyperbolic, ergodic and topological properties",
  "prefix": "10.1017",
  "volume": "12",
  "author": [
    {
      "given": "Ya. B.",
      "family": "Pesin",
      "sequence": "first",
      "affiliation": []
    }
  ],
  "member": "56",
  "published-online": {
    "date-parts": [
      [
        2008,
        9,
        19
      ]
    ]
  },
  "reference": [
    {
      "key": "S0143385700006635_ref018",
      "doi-asserted-by": "publisher",
      "DOI": "10.1070/IM1976v010n06ABEH001835"
    },
    {
      "key": "S0143385700006635_ref016",
      "volume-title": "Modern Problems in Mathematics",
      "volume": "2",
      "author": "Pesin",
      "year": "1989"
    },
    {
      "key": "S0143385700006635_ref023",
      "first-page": "94",
      "volume-title": "Structure of Lorenz attractors",
      "author": "Williams",
      "year": "1977"
    },
    {
      "key": "S0143385700006635_ref010",
      "doi-asserted-by": "publisher",
      "DOI": "10.1007/BFb0099031"
    },
    {
      "key": "S0143385700006635_ref003",
      "doi-asserted-by": "publisher",
      "DOI": "10.1070/RM1967v022n05ABEH001228"
    },
    {
      "key": "S0143385700006635_ref015",
      "doi-asserted-by": "publisher",
      "DOI": "10.1070/RM1977v032n04ABEH001639"
    },
    {
      "key": "S0143385700006635_ref024",
      "doi-asserted-by": "publisher",
      "DOI": "10.1007/BF02684769"
    },
    {
      "key": "S0143385700006635_ref008",
      "volume-title": "Modern Problems of Mathematics",
      "volume": "2",
      "author": "Bunimovich",
      "year": "1989"
    },
    {
      "key": "S0143385700006635_ref011",
      "first-page": "593",
      "article-title": "Propri\u00e9t\u00e9s ergodiques des mesures de Sinai",
      "volume": "294",
      "author": "Ledrappier",
      "year": "1982",
      "journal-title": "C. R. Acad. Sci."
    },
    {
      "key": "S0143385700006635_ref014",
      "first-page": "398",
      "volume-title": "Nonlinear Dynamics",
      "author": "Misiurewicz",
      "year": "1980"
    },
    {
      "key": "S0143385700006635_ref012",
      "first-page": "103",
      "volume-title": "Ergodic properties of the Lozi map",
      "author": "Levy",
      "year": "1985"
    },
    {
      "key": "S0143385700006635_ref007",
      "volume-title": "Qualitative methods of the theory of nonlinear oscillations in point systems",
      "author": "Belykh",
      "year": "1980"
    },
    {
      "key": "S0143385700006635_ref021",
      "volume-title": "Statistical Irreversibility in Nonlinear Systems",
      "author": "Sinai",
      "year": "1979"
    },
    {
      "key": "S0143385700006635_ref004",
      "first-page": "336",
      "article-title": "On appearance and structure of Lorenz attractor",
      "volume": "234",
      "author": "Afraimovich",
      "year": "1977",
      "journal-title": "Dokl. Akad. Nauk USSR"
    },
    {
      "key": "S0143385700006635_ref005",
      "first-page": "150",
      "article-title": "On attracting nonstructurally stable limit sets of type of Lorenz attractor",
      "volume": "44",
      "author": "Afraimovich",
      "year": "1983",
      "journal-title": "Trudy Moskov. Mat. Obshch."
    },
    {
      "key": "S0143385700006635_ref013",
      "first-page": "9",
      "article-title": "Un attracteur \u00e9trange du type attracteur de H\u00e9non",
      "volume": "39",
      "author": "Lozi",
      "year": "1978",
      "journal-title": "J. Phys."
    },
    {
      "key": "S0143385700006635_ref002",
      "first-page": "39",
      "volume-title": "Trudy V Int. Conf. on Nonlinear Oscillations",
      "volume": "2",
      "author": "Anosov",
      "year": "1970"
    },
    {
      "key": "S0143385700006635_ref001",
      "doi-asserted-by": "publisher",
      "DOI": "10.1070/SM1969v007n01ABEH001076"
    },
    {
      "key": "S0143385700006635_ref009",
      "first-page": "212",
      "volume-title": "Nonlinear Waves",
      "author": "Bunimovich",
      "year": "1980"
    },
    {
      "key": "S0143385700006635_ref017",
      "article-title": "Ergodic properties and dimensionlike characteristics of strange attractors that are close to hyperbolic",
      "author": "Pesin",
      "year": "1986",
      "journal-title": "Proc. Int. Cong. Math."
    },
    {
      "key": "S0143385700006635_ref019",
      "doi-asserted-by": "crossref",
      "first-page": "417",
      "DOI": "10.1017/S014338570000170X",
      "article-title": "Gibbs measures for partially hyperbolic attractors",
      "volume": "9",
      "author": "Pesin",
      "year": "1982",
      "journal-title": "Ergod. Th. and Dynam. Sys."
    },
    {
      "key": "S0143385700006635_ref025",
      "unstructured": "[25] Rychlik M. . Invariant measures and the variation principle for Lozi mappings. PhD dissertation. University of California, Berkeley, 1983."
    },
    {
      "key": "S0143385700006635_ref022",
      "doi-asserted-by": "publisher",
      "DOI": "10.1090/S0002-9947-1985-0766205-1"
    },
    {
      "key": "S0143385700006635_ref006",
      "volume-title": "The Dimension of Lorenz Type Attractors",
      "volume": "6",
      "author": "Afraimovich",
      "year": "1987"
    },
    {
      "key": "S0143385700006635_ref020",
      "doi-asserted-by": "publisher",
      "DOI": "10.1007/BF02684768"
    }
  ],
  "container-title": "Ergodic Theory and Dynamical Systems",
  "original-title": [],
  "language": "en",
  "link": [
    {
      "URL": "https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0143385700006635",
      "content-type": "unspecified",
      "content-version": "vor",
      "intended-application": "similarity-checking"
    }
  ],
  "deposited": {
    "date-parts": [
      [
        2020,
        5,
        8
      ]
    ],
    "date-time": "2020-05-08T10:58:06Z",
    "timestamp": 1588935486000
  },
  "score": 1,
  "resource": {
    "primary": {
      "URL": "https://www.cambridge.org/core/product/identifier/S0143385700006635/type/journal_article"
    }
  },
  "subtitle": [],
  "short-title": [],
  "issued": {
    "date-parts": [
      [
        1992,
        3
      ]
    ]
  },
  "references-count": 25,
  "journal-issue": {
    "issue": "1",
    "published-print": {
      "date-parts": [
        [
          1992,
          3
        ]
      ]
    }
  },
  "alternative-id": [
    "S0143385700006635"
  ],
  "URL": "http://dx.doi.org/10.1017/S0143385700006635",
  "relation": {},
  "ISSN": [
    "0143-3857",
    "1469-4417"
  ],
  "subject": [],
  "container-title-short": "Ergod. Th. Dynam. Sys.",
  "published": {
    "date-parts": [
      [
        1992,
        3
      ]
    ]
  }
}