Spectral Hausdorff dimensions for a class of Schrödinger operators in bounded intervals

Lower bounds for fractal dimensions of spectral measures of the period doubling Schrödinger operator

“…there is a generic (i.e., dense G δ ) set G ⊂ Ω ̟ , with ̟ ∈ G, so that dim + P (µ ω φ ) ≥ α, for all ω ∈ G. For one-dimensional Schrödinger operators with singular continuous spectrum, this adds to a small list of results on α-continuity outside the scope of (quasi) Sturmian [13,14] and sparse potentials [22,31,30,9,11,7]; see also [23,3] (note that most of these results were obtained around 20 years ago). We underline that we also know explicitly elements of the generic set G.…”

Forward boundedness of the trace map for the period doubling substitution

“…The result has further applications to packing continuity properties [1,2,6,7], which are subject of current research. Bounding forward orbits by a constant is an improvement on the weak bound for the trace map in [4] (lemma 5), which is the only known bound since the interest on the period doubling sequence in the context of Schrödinger operators began.…”

“…A natural question is about fractal properties of such spectra, with prominent roles played by Hausdorff and packing measures on the line. Usually such fractal properties are not easy to be proven, but nontrivial Hausdorff continuity properties have been obtained for Sturmian potentials whose rotation numbers are of bounded density [9] (see [13] for the original idea of the proof), and recently this property has been found to be stable under some perturbations when a singular continuous component is persistent [1]. Note that α-Hausdorff continuity implies α-packing continuity of a measure [11,16], but the reverse does not hold in general.…”