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On the thermodynamic formalism for multifractal functions. (English) Zbl 0843.58091

Summary: The thermodynamic formalism for “multifractal” functions \(\varphi (x)\) is a heuristic principle that states that the singularity spectrum \(f(\alpha)\) (defined as the Hausdorff dimension of the set \(S_\alpha\) of points where \(\varphi\) has Hölder exponent \(\alpha\)) and the moment scaling exponent \(\tau (q)\) (giving the power law behavior of \(\int|\varphi (x+ t)- \varphi (x) |^q dx\) for small \(|t|\)) should be related by the Legendre transform, \(\tau(q)= 1+ \inf_{\alpha\geq 0} [q\alpha- f(\alpha )]\). The range of validity of this heuristic principle is unknown. Here this principle is rigorously verified for a family of “toy examples” that are solutions of refinement equations. These example functions exhibit oscillations on all scales, and correspond to multifractal signed measures rather than multifractal measures; moreover, their singularity spectra \(f(\alpha)\) are not concave.

MSC:

37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37A30 Ergodic theorems, spectral theory, Markov operators
28A80 Fractals
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