On measures resisting multifractal analysis

TL;DR

This paper explores the limitations of multifractal analysis by constructing examples where invariant measures under linear circle maps deviate drastically from typical multifractal spectra, challenging existing assumptions.

Contribution

It introduces specific examples of invariant measures under linear circle maps that violate the usual multifractal spectrum properties.

Findings

Invariant measures can have non-standard multifractal spectra.

Constructed examples show drastic failure of typical multifractal behavior.

Challenges assumptions about ergodic measures on smooth maps.

Abstract

Any ergodic measure of a smooth map on a compact manifold has a multifractal spectrum with one point - the dimension of the measure itself - at the diagonal. We will construct examples where this fails in the most drastic way for invariant measures invariant under linear maps of the circle.