Scaling properties of the thue–morse measure

Abstract

The classic Thue-Morse measure is a paradigmatic example of a purely singular continuous probability measure on the unit interval. Since it has a representation as an infinite Riesz product, many aspects of this measure have been studied in the past, including various scaling properties and a partly heuristic multifractal analysis. Some of the difficulties emerge from the appearance of an unbounded potential in the thermodynamic formalism. It is the purpose of this article to review and prove some of the observations that were previously established via numerical or scaling arguments.2010 Mathematics Subject Classification. 37D35, 37C45, 52C23.

Eberlein decomposition for PV inflation systems

“…holds for all 1 ≤ i, j ≤ N , where α i = dens(Λ i )/dens (W j ) . This proportionality of measures matches nicely with the density formula δ P δ Q {0} = dens(P) dens(Q) (7) for Meyer sets P and Q that define uniquely ergodic Delone dynamical systems. With μ = δ P and ν = δ Q , this relation follows from the complex polarisation identity…”

“…to be interpreted as the vague limit of a sequence of absolutely continuous measures; see [7] for a recent, detailed analysis of this measure.…”

Scaling of diffraction intensities near the origin: some rigorous results

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Inflation versus projection sets in aperiodic systems: the role of the window in averaging and diffraction

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