View in EDS HTML Full Text PDF Full Text

A note on Hausdorff dimension of invariant measures of circle diffeomorphisms with breaks.

Saved in:
Bibliographic Details
Title: A note on Hausdorff dimension of invariant measures of circle diffeomorphisms with breaks.
Authors: Marzougui, Habib1 (AUTHOR) habib.marzougui@fsb.rnu.tn
Source: Dynamical Systems: An International Journal. Sep2025, Vol. 40 Issue 3, p447-454. 8p.
Subjects: Invariant measures, Diffeomorphisms, Dynamical systems, Dimension theory (Topology), Homeomorphisms, Functions of bounded variation, Ergodic theory, Critical point theory
Abstract: Let f be a class P-homeomorphism of the circle $ S^{1} $ S 1 with several break points singularities, that are differentiable maps except at some singular points where the derivative has a jump, and such that $ \log \rm{Df} $ log ⁡ Df has bounded variation on $ S^{1} $ S 1 . Assume that f has an irrational rotation number α of bounded density $ d_{\alpha } $ d α . Then the Hausdorff dimension $ \mathrm {dim}_{H}(\mu _{f}) $ dim H (μ f) of its unique f-invariant measure $ \mu _{f} $ μ f is bounded away from zero, i.e. such that $ \mathrm {dim}_{H}(\mu _{f})\geq C $ dim H (μ f) ≥ C , for some constant C>0 (depending only on $ d_{\alpha } $ d α and the total variation of $ \log \rm{Df} $ log ⁡ Df ). This result generalizes and improves with a simple proof some results of Dzhalilov (Piecewise smoothness of conjugate homeomorphisms of a circle with corners, Theor. Math. Phys. 120 (1999), pp. 961–972) and Khanin and Kocić (Hausdorff dimension of invariant measure of circle diffeomorphisms with a break point, Ergodic Theory Dyn. Syst. 39 (2017), pp. 1–9). [ABSTRACT FROM AUTHOR]
Copyright of Dynamical Systems: An International Journal is the property of Taylor & Francis Ltd and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
Database: Engineering Source
Full text is not displayed to guests.
FullText Links:
  – Type: pdflink
Text:
  Availability: 1
Header DbId: egs
DbLabel: Engineering Source
An: 187348020
AccessLevel: 6
PubType: Academic Journal
PubTypeId: academicJournal
PreciseRelevancyScore: 0
IllustrationInfo
Items – Name: Title
  Label: Title
  Group: Ti
  Data: A note on Hausdorff dimension of invariant measures of circle diffeomorphisms with breaks.
– Name: Author
  Label: Authors
  Group: Au
  Data: <searchLink fieldCode="AR" term="%22Marzougui%2C+Habib%22">Marzougui, Habib</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> habib.marzougui@fsb.rnu.tn</i>
– Name: TitleSource
  Label: Source
  Group: Src
  Data: <searchLink fieldCode="JN" term="%22Dynamical+Systems%3A+An+International+Journal%22">Dynamical Systems: An International Journal</searchLink>. Sep2025, Vol. 40 Issue 3, p447-454. 8p.
– Name: Subject
  Label: Subjects
  Group: Su
  Data: <searchLink fieldCode="DE" term="%22Invariant+measures%22">Invariant measures</searchLink><br /><searchLink fieldCode="DE" term="%22Diffeomorphisms%22">Diffeomorphisms</searchLink><br /><searchLink fieldCode="DE" term="%22Dynamical+systems%22">Dynamical systems</searchLink><br /><searchLink fieldCode="DE" term="%22Dimension+theory+%28Topology%29%22">Dimension theory (Topology)</searchLink><br /><searchLink fieldCode="DE" term="%22Homeomorphisms%22">Homeomorphisms</searchLink><br /><searchLink fieldCode="DE" term="%22Functions+of+bounded+variation%22">Functions of bounded variation</searchLink><br /><searchLink fieldCode="DE" term="%22Ergodic+theory%22">Ergodic theory</searchLink><br /><searchLink fieldCode="DE" term="%22Critical+point+theory%22">Critical point theory</searchLink>
– Name: Abstract
  Label: Abstract
  Group: Ab
  Data: Let f be a class P-homeomorphism of the circle $ S^{1} $ S 1 with several break points singularities, that are differentiable maps except at some singular points where the derivative has a jump, and such that $ \log \rm{Df} $ log ⁡ Df has bounded variation on $ S^{1} $ S 1 . Assume that f has an irrational rotation number α of bounded density $ d_{\alpha } $ d α . Then the Hausdorff dimension $ \mathrm {dim}_{H}(\mu _{f}) $ dim H (μ f) of its unique f-invariant measure $ \mu _{f} $ μ f is bounded away from zero, i.e. such that $ \mathrm {dim}_{H}(\mu _{f})\geq C $ dim H (μ f) ≥ C , for some constant C>0 (depending only on $ d_{\alpha } $ d α and the total variation of $ \log \rm{Df} $ log ⁡ Df ). This result generalizes and improves with a simple proof some results of Dzhalilov (Piecewise smoothness of conjugate homeomorphisms of a circle with corners, Theor. Math. Phys. 120 (1999), pp. 961–972) and Khanin and Kocić (Hausdorff dimension of invariant measure of circle diffeomorphisms with a break point, Ergodic Theory Dyn. Syst. 39 (2017), pp. 1–9). [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
  Label:
  Group: Ab
  Data: <i>Copyright of Dynamical Systems: An International Journal is the property of Taylor & Francis Ltd and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
PLink https://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=egs&AN=187348020
RecordInfo BibRecord:
  BibEntity:
    Identifiers:
      – Type: doi
        Value: 10.1080/14689367.2025.2472011
    Languages:
      – Code: eng
        Text: English
    PhysicalDescription:
      Pagination:
        PageCount: 8
        StartPage: 447
    Subjects:
      – SubjectFull: Invariant measures
        Type: general
      – SubjectFull: Diffeomorphisms
        Type: general
      – SubjectFull: Dynamical systems
        Type: general
      – SubjectFull: Dimension theory (Topology)
        Type: general
      – SubjectFull: Homeomorphisms
        Type: general
      – SubjectFull: Functions of bounded variation
        Type: general
      – SubjectFull: Ergodic theory
        Type: general
      – SubjectFull: Critical point theory
        Type: general
    Titles:
      – TitleFull: A note on Hausdorff dimension of invariant measures of circle diffeomorphisms with breaks.
        Type: main
  BibRelationships:
    HasContributorRelationships:
      – PersonEntity:
          Name:
            NameFull: Marzougui, Habib
    IsPartOfRelationships:
      – BibEntity:
          Dates:
            – D: 01
              M: 09
              Text: Sep2025
              Type: published
              Y: 2025
          Identifiers:
            – Type: issn-print
              Value: 14689367
          Numbering:
            – Type: volume
              Value: 40
            – Type: issue
              Value: 3
          Titles:
            – TitleFull: Dynamical Systems: An International Journal
              Type: main
ResultId 1