arXiv subjects

Yan-Hui Qu

Publications and source records attributed to Yan-Hui Qu.

4 recordsLinked to original sources

arXiv2015

Spectral Properties of Schrödinger Operators With Pattern Sturmian Potentials

We consider discrete Schrödinger operators with pattern Sturmian potentials. This class of potentials strictly contains the class of Sturmian potentials, for which the spectral properties of the associated Schrödinger operators are well understood. In particular, it is known that for every Sturmian potential, the associated Schrödinger operator has zero-measure spectrum and purely singular continuous spectral measures. We conjecture that the same statements hold in the more general class of pattern Sturmian potentials. We prove partial results in support of this conjecture. In particular, we confirm the conjecture for all pattern Sturmian potentials that belong to the family of Toeplitz sequences.

math.SP↗

arXiv2014

Transport Exponents of Sturmian Hamiltonians

We consider discrete Schrödinger operators with Sturmian potentials and study the transport exponents associated with them. Under suitable assumptions on the frequency, we establish upper and lower bounds for the upper transport exponents. As an application of these bounds, we identify the large coupling asymptotics of the upper transport exponents for frequencies of constant type. We also bound the large coupling asymptotics uniformly from above for Lebesgue-typical frequency. A particular consequence of these results is that for most frequencies of constant type, transport is faster than for Lebesgue almost every frequency. We also show quasi-ballistic transport for all coupling constants, generic frequencies, and suitable phases.

math.SP↗

arXiv2011

Localized asymptotic behavior for almost additive potentials

We conduct the multifractal analysis of the level sets of the asymptotic behavior of almost additive continuous potentials $(ϕ_n)_{n=1}^\infty$ on a topologically mixing subshift of finite type $X$ endowed itself with a metric associated with such a potential. We work without additional regularity assumption other than continuity. Our approach differs from those used previously to deal with this question under stronger assumptions on the potentials. As a consequence, it provides a new description of the structure of the spectrum in terms of {\it weak} concavity. Also, the lower bound for the spectrum is obtained as a consequence of the study sets of points at which the asymptotic behavior of $ϕ_n(x)$ is localized, i.e. depends on the point $x$ rather than being equal to a constant. Specifically, we compute the Hausdorff dimension of sets of the form $\{x\in X: \lim_{n\to\infty} ϕ_n(x)/n=ξ(x)\}$, where $ξ$ is a given continuous function. This has interesting geometric applications to fixed points in the asymptotic average for dynamical systems in $\R^d$, as well as the fine local behavior of the harmonic measure on conformal planar Cantor sets.

math.DS↗

arXiv2010

Multifractal analysis and localized asymptotic behavior for almost additive potentials

We conduct the multifractal analysis of the level sets of the asymptotic behavior of almost-additive continuous potentials $(ϕ_n)_{n=1}^\infty$ on a topologically mixing subshift of finite type $X$ endowed itself with a metric associated with such a potential. We work without bounded distorsion property assumption. We express the whole Hausdorff spectrum in terms of a conditional variational principle, as well as a new large deviations principle. Our approach provides a new description of the structure of the spectrum in terms of {\it weak} concavity. Another new point is that we consider sets of points at which the asymptotic behavior of $ϕ_n(x)$ is localized, i.e. depends on the point $x$ rather than being equal to a constant. Specifically, we compute the Hausdorff dimension of sets of the form $\{x\in X: \lim_{n\to\infty} ϕ_n(x)/n=ξ(x)\}$, where $ξ$ is a given continuous function. This is naturally related to Birkhoff's ergodic theorem and has interesting geometric applications to fixed points in the asymptotic average for dynamical systems in $\R^d$, as well as the fine local behavior of the harmonic measure on conformal planar Cantor sets.

math.DS↗