arXiv subjects
Publications and source records attributed to Kristian Bjerklöv.
We show that for a large class of potential functions and big coupling constant $λ$ the Schrödinger cocycle over the expanding map $x\mapsto bx ~( \text{mod} 1)$ on $\mathbb{T}$ has a Lyapunov exponent $>(\logλ)/4$ for all energies, provided that the integer $b\geq λ^3$.
We introduce a class of real analytic "peaky" potentials for which the corresponding quasi-periodic 1D Schrödinger operators exhibit, for quasiperiodic frequencies in a set of positive Lebesgue measure, both absolutely continuous and pure point spectrum.
In this paper we investigate maps of the two-torus $\mathbb{T}^2$ of the form $T(x,y)=(x+ω,g(x)+f(y))$ for Diophantine $ω\in\mathbb{T}$ and for a class of maps $f,g:\mathbb{T}\to\mathbb{T}$, where each $g$ is strictly monotone and of degree 2, and each $f$ is an orientation preserving circle homeomorphism. For our class of $f$ and $g$ we show that $T$ is minimal and has exactly two invariant and ergodic Borel probability measures. Moreover, these measures are supported on two $T$-invariant graphs. One of the graphs is a Strange Nonchaotic Attractor whose basin of attraction consists of (Lebesgue) almost all points in $\mathbb{T}^2$. Only a low regularity assumption (Lipschitz) is needed on the maps $f$ and $g$, and the results are robust with respect to Lipschitz-small perturbations of $f$ and $g$.
Let $f:\mathbb{T}\to\mathbb{R}$ be a Morse function of class $C^2$ with exactly two critical points, let $ω\in\mathbb{T}$ be Diophantine, and let $λ>0$ be sufficiently large (depending on $f$ and $ω$). For any value of the parameter $E\in \mathbb{R}$ we make a careful analysis of the dynamics of the skew-product map $$Φ_E(θ,r)=\left(θ+ω,λf(θ)-E-1/r\right),$$ acting on the "torus" $\mathbb{T}\times\widehat{\mathbb{R}}$. The map $Φ_E$ is intimately related to the quasi-periodic Schrödinger cocycle $(ω,A_E): \mathbb{T}\times \mathbb{R}^2 \to \mathbb{T}\times \mathbb{R}^2$, $(θ,x)\mapsto (θ+ω, A_E(θ)\cdot x)$, where $A_E:\mathbb{T}\to \text{SL}(2,\mathbb{R})$ is given by $A_{E}(θ)=\left(\begin{matrix}0 & 1 \\ -1 & λf(θ)-E \end{matrix} \right), \quad E\in \mathbb{R}. $ More precisely, $(ω,A_E)$ naturally acts on the space $\mathbb{T}\times\widehat{\mathbb{R}}$, and $Φ_E$ is the map thus obtained. The analysis of $Φ_E$ allows us to derive three main results: (1) The (maximal) Lyapunov exponent of the Schrödinger cocycle $(ω,A_E)$ is $\gtrsim \log λ$, uniformly in $E\in \mathbb{R}$. This implies that the map $Φ_E$ has exactly two ergodic probability measures for all $E\in \mathbb{R}$; (2) If $E$ is on the edge of an open gap in the spectrum $σ(H)$ of the associated Schrödinger operator $H_θ$, then there exist a phase $θ\in\mathbb{T}$ and a vector $u\in l^2(\mathbb{Z})$, exponentially decaying at $\pm\infty$, such that $H_θu=Eu$; (3) The map $Φ_E$ is minimal iff $E\in σ(H)\setminus\{\text{edges of open gaps}\}$. In particular, $Φ_E$ is minimal for all $E$ for which the fibered rotation number $α(E)$ associated to $(ω,A_E)$ is irrational with respect to $ω$.