Fixed Points for the Multifractal Spectrum Map
- 120 Downloads
For all continuous function g having a specific form that we call with increasing visibility, we construct a function f whose multifractal spectrum is such that \( d_f =g\circ f\). The function f is obtained as an infinite superposition of piecewise \(C^1\) functions, is also with increasing visibility, and is homogeneously multifractal; i.e., its restriction on any subinterval of \([0,1]\) has the same multifractal spectrum as the function f itself. In particular, we prove the existence of a function f which is its own multifractal spectrum; i.e., \(f=d_f\).
KeywordsLocal regularity Hausdorff dimension and measure Multifractals Hölder exponent
Mathematics Subject Classification26A16 28A80 28C15
- 1.Barral, J.: Inverse problems in multifractal analysis. Ann. ENS Paris (2015)Google Scholar
- 2.Barral, J., Durand, A., Jaffard, S., Seuret, S.: Local multifractal analysis, applications of fractals and dynamical systems in science and economics. In: Carfi, D., Lapidus, M.L., Pearse, E.J., van Frankenhuijsen, M. (eds.) Contemporary Mathematics. AMS, Providence (2013)Google Scholar