Exact-Dimensional Property of Density of States Measure of Sturm Hamiltonian

Abstract

Abstract. For frequency α of bounded type and coupling λ > 20, we show that the density of states measure N α,λ of the related Sturm Hamiltonian is exact upper and lower dimensional, however, in general it is not exact-dimensional.

Multifractal spectra of Moran measures without local dimension

“…Then, using a similar method in the proof of inequality (12), but replace (10), (11) and ( 9) by ( 17), ( 18) and ( 16),we can conclude and then dim loc (µ , x) s 2 α − β 2 (s 2 ) from lemma 3.4. • For α ∈ (β 1 (1), β 1 (−∞)] and α ∈ [h(α), β 2 (1)).…”

“…Then, using a similar method in the proof of inequality (12), but replace (10), (11) and (9) by (17), 18and (16),we can conclude d(µ , x, n) ≤ s 2 (α + ) − β 2 (s 2 ) + s 2 (1 + ) + (s 1 α 3 − β 1 (s 1 ))…”

“…In [12], the author studies the density of states measure of certain Sturm Hamiltonians. He shows that such class of measures is always exact upper and lower dimensional.…”

“…Then, using a similar method in the proof of inequality (12), but replace (10), (11) and ( 9) by ( 17 • For α ∈ (β 1 (1), β 1 (−∞)] and α ∈ [h(α), β 2 (1)). For the lower bound, use the same method as the previous item, we can also define the point (α 0 , s 1 α 0 − β 1 (s 1 )) and (α 2 , s 2 α 2 − β 2 (s 2 )).…”