×

Multifractal formalism for self-similar functions under the action of nonlinear dynamical systems. (English) Zbl 0928.28002

Multifractal formalism for self-similar functions associated to contractive similitudes has been established by S. Jaffard [SIAM J. Math. Anal. 28, No. 4, 944-970 (1997; Zbl 0876.42021); ibid. 971-998 (1997; Zbl 0876.42022)]. In this paper, self-similar functions associated to some nonlinear contractions are studied. In particular, the author calculates the Hölder exponents \(\alpha(x)\) of such functions at any point \(x\), the spectrum of singularities and the Besov “smoothness” index and proves the multifractal formalism.

MSC:

28A80 Fractals
26A30 Singular functions, Cantor functions, functions with other special properties
26A18 Iteration of real functions in one variable
28A78 Hausdorff and packing measures
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
54H20 Topological dynamics (MSC2010)
37B30 Index theory for dynamical systems, Morse-Conley indices
PDFBibTeX XMLCite
Full Text: DOI