RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
General information
Latest issue
Forthcoming papers
Archive
Impact factor
Subscription
Guidelines for authors
License agreement
Submit a manuscript

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Izv. RAN. Ser. Mat.:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


Izv. RAN. Ser. Mat., 2012, Volume 76, Issue 3, Pages 3–18 (Mi izv7332)  

The thermodynamic formalism for the de Rham function: increment method

M. Ben Slimane

College of Science, King Saud University

Abstract: We study the de Rham function: the unique continuous (nowhere differentiable) function $F \in L^1(\mathbb{R})$ with $\int F(x) dx=1$ satisfying the functional equation $F(x)=F(3x)+\frac{1}{3}(F(3x-1)+F(3x+1) )+\frac{2}{3}(F(3x-2)+F(3x+2))$. We show that its pointwise Hölder regularity $\alpha(x)=\liminf_{h\to 0}\frac{\log(|F(x+h)-F(x)|)}{\log |h|}$ differs widely from point to point, and the values of $\alpha(x)$ fill an interval parametrizing the fractal sets $E^{(\alpha)}$, where $E^{(\alpha)}$ is the set of points $x$ with Hölder exponent $\alpha(x)=\alpha$. We also prove that the thermodynamic formalism (increment method) holds for the de Rham function: we have a heuristic formula $d(\alpha)=\inf_{q >0}(\alpha q-\zeta(q)+1)$ relating the order of decay of $\int_{\mathbb{R}}|F(x+h)-F(x)|^{q} dx \sim |h|^{\zeta(q)}$ as $h \to 0$ with the Hausdorff dimension $d(\alpha)$ of $E^{(\alpha)}$.

Keywords: Hölder regularity, Hausdorff dimension, increments, thermodynamic formalism.

DOI: https://doi.org/10.4213/im7332

Full text: PDF file (527 kB)
References: PDF file   HTML file

English version:
Izvestiya: Mathematics, 2012, 76:3, 431–445

Bibliographic databases:

UDC: 517.589
MSC: 26A16, 26A30, 28A80, 42C15, 76F99
Received: 06.03.2011

Citation: M. Ben Slimane, “The thermodynamic formalism for the de Rham function: increment method”, Izv. RAN. Ser. Mat., 76:3 (2012), 3–18; Izv. Math., 76:3 (2012), 431–445

Citation in format AMSBIB
\Bibitem{Ben12}
\by M.~Ben Slimane
\paper The thermodynamic formalism for the de~Rham function: increment method
\jour Izv. RAN. Ser. Mat.
\yr 2012
\vol 76
\issue 3
\pages 3--18
\mathnet{http://mi.mathnet.ru/izv7332}
\crossref{https://doi.org/10.4213/im7332}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2978106}
\zmath{https://zbmath.org/?q=an:06062357}
\adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?2012IzMat..76..431B}
\elib{http://elibrary.ru/item.asp?id=20358842}
\transl
\jour Izv. Math.
\yr 2012
\vol 76
\issue 3
\pages 431--445
\crossref{https://doi.org/10.1070/IM2012v076n03ABEH002590}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000305690000001}
\elib{http://elibrary.ru/item.asp?id=18348029}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84862688991}


Linking options:
  • http://mi.mathnet.ru/eng/izv7332
  • https://doi.org/10.4213/im7332
  • http://mi.mathnet.ru/eng/izv/v76/i3/p3

    SHARE: VKontakte.ru FaceBook Twitter Mail.ru Livejournal Memori.ru


    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
    Number of views:
    This page:455
    Full text:57
    References:56
    First page:16

     
    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2019