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2002 Fiscal Year Final Research Report Summary

Statistical properties of weak Gibbs measures for complex systems with nonhyperbolic periodic orbits

Research Project

Project/Area Number 13640133
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field General mathematics (including Probability theory/Statistical mathematics)
Research InstitutionSapporo University

Principal Investigator

YURI Michiko  Sapporo University, Department of Business Administration, Professor, 経営学部, 教授 (70174836)

Project Period (FY) 2001 – 2002
KeywordsEquilibrium state / Weak Gibbs measure / Variational principle / Intermittency / Indifferent periodic point / Conformal measure / Multifractal formalism / Zeta function
Research Abstract

The purpose of this project is to construct mathematical models which show phase transition, failure of the Gibbs property and slow decay of correlations, that can be observed for many complex systems exhibiting Intermittency. For this purpose, we establish Thermodynamic formalism for non-Holder potentials in nonhyperbolic situation. More specifically, for countable to one transitive Markov systems we construct conformal measures ν which are weak Gibbs measures for potentials φ of weak bounded variation, and show the existence of equilibrium states μ for φ equivalent to the weak Gibbs measures ν ([1]). We see that generalized indifferent periodic orbits cause phase transition, non-Gibbsianness and force the decay of correlations to be slow ([1], [4]). Applying these results allow us to establish a multifractal formalism of weak Gibbs measures associated to potentials of weak bounded variation ([3]). Furthermore, we establish a version of the local product structure (weak local product structure) for the invertible extension of ergodic weak Gibbs measures μ ([2]). About the decay of correlations of weak Gibbs equilibrium measures, polynomial upper bounds can be obtained for a class of functions which contains all piecewise Lipschitz functions ([4]), by clarifying the speed of uniform convergence of iterated transfer operators on compact sets excluding indifferent periodic points.

  • Research Products

    (12 results)

All Other

All Publications (12 results)

  • [Publications] Michiko Yuri: "Thermodynamic formalism for countable to one Markov systems"Transactions of the American Mathematical Society. (発表予定)(印刷中).

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Michiko Yuri: "Weak Gibbs measures and the local product structure"Ergodic Theory and Dynamical Systems. 22. 1933-1954 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Michiko Yuri: "Multifractal Analysis of Weak Gibbs measures for Intermittent systems"Communications in Mathematical Physics. 230. 365-388 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Michiko Yuri: "On the speed of convergence to equilibrium states for multi-dimensional maps with indifferent periodic points"Nonlinearity. 15. 429-445 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] M.Pollicott, M.Yuri: "Statistical properties of maps with indifferent periodic points"Communications in Mathematical physics. 217. 503-520 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] M.Pollicott, M.Yuri: "Zeta functions for certain multi-dimensional non-hyperbolic maps"Nonlinearity. 14. 1265-1278 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Michiko Yuri: "Thermodynamic formalism for countable to one Markov systems"Transactions of the American Mathematical Society. (To appear).

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Michiko Yuri: "Weak Gibbs measures and the local product structure"Ergodic Theory and Dynamical Systems. 22. 1933-1954 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Michiko Yuri: "Multifractal Analysis of weak Gibbs measures for Intermittent systems"Communications in Mathematical Physics. 230. 365-388 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Michiko Yuri: "On the speed of convergence to equilibrium states for multi-dimensional maps with indifferent periodic points"Nonlinearity. 15. 429-445 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Michiko Yuri (with M. Pollicott): "Statistical properties of maps with indifferent periodic points"Communications in Mathematical Physics. 217. 503-520 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Michiko Yuri (with M. Pollicott): "Zeta functions for certain multi-dimensional nonhyperbolic maps"Nonlinearity. 14. 1265-1278 (2001)

    • Description
      「研究成果報告書概要(欧文)」より

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Published: 2004-04-14  

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