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

 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

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}