Ergodic Theory and Discrete One-dimensional Random Schr Odinger Operators: Uniform Existence of the Lyapunov Exponent
نویسنده
چکیده
We review recent results which relate spectral theory of discrete one-dimensional Schrödinger operators over strictly ergodic systems to uniform existence of the Lyapunov exponent. In combination with suitable ergodic theorems this allows one to establish Cantor spectrum of Lebesgue measure zero for a large class of quasicrystal Schrödinger operators. The results can also be used to study non-uniformity of cocycles. While most part of the paper discuss already known results, we also include new uniform ergodic theorems for Quasi-Sturmian systems.
منابع مشابه
Random Schrr Odinger Operators Arising from Lattice Gauge Elds I: Existence and Examples Mathematics Subject Classiication
We consider models of random Schrr odinger operators which exist thanks to a cohomological theorem in ergodic theory. Examples are ergodic Schrr odinger operators with random magnetic uxes on discrete two-dimensional lattices or non-periodic situations like Penrose lattices.
متن کاملSingular Spectrum of Lebesgue Measure Zero for One-dimensional Quasicrystals
The spectrum of one-dimensional discrete Schrödinger operators associated to strictly ergodic dynamical systems is shown to coincide with the set of zeros of the Lyapunov exponent if and only if the Lyapunov exponent exists uniformly. This is used to obtain Cantor spectrum of zero Lebesgue measure for all aperiodic subshifts with uniform positive weights. This covers, in particular, all aperiod...
متن کاملTheory of Hybrid Fractional Differential Equations with Complex Order
We develop the theory of hybrid fractional differential equations with the complex order $thetain mathbb{C}$, $theta=m+ialpha$, $0<mleq 1$, $alphain mathbb{R}$, in Caputo sense. Using Dhage's type fixed point theorem for the product of abstract nonlinear operators in Banach algebra; one of the operators is $mathfrak{D}$- Lipschitzian and the other one is completely continuous, we prove the exis...
متن کاملThe Lyapunov Exponents for Schr Odinger Operators with Slowly Oscillating Potentials
By studying the integrated density of states, we prove the existence of Lyapunov exponents and the Thouless formula for the Schrödinger operator −d2/dx2 + cos xν with 0 < ν < 1 on L2[0,∞). This yields an explicit formula for these Lyapunov exponents. By applying rank one perturbation theory, we also obtain some spectral consequences.
متن کاملUniform spectral properties of one-dimensional quasicrystals, II. The Lyapunov exponent
In this paper we introduce a method that allows one to prove uniform local results for one-dimensional discrete Schrödinger operators with Sturmian potentials. We apply this method to the transfer matrices in order to study the Lyapunov exponent and the growth rate of eigenfunctions. This gives uniform vanishing of the Lyapunov exponent on the spectrum for all irrational rotation numbers. For i...
متن کامل