Singular continuous spectrum in ergodic theory
نویسنده
چکیده
We prove that in the weak topology of measure preserving transformations , a dense G has purely singular continuous spectrum in the orthocomplement of the constant functions. In the uniform topology, a dense G of aperiodic transformations has singular continuous spectrum. We show that a dense G of shift-invariant measures has purely singular continuous spectrum. These results stay true for Z d actions of measure preserving transformations. There exist smooth unitary cocy-cles over an irrational rotation which have purely singular continuous spectrum.
منابع مشابه
Lyapunov Exponents and Spectral Analysis of Ergodic Schrödinger Operators: a Survey of Kotani Theory and Its Applications
The absolutely continuous spectrum of an ergodic family of onedimensional Schrödinger operators is completely determined by the Lyapunov exponent as shown by Ishii, Kotani and Pastur. Moreover, the part of the theory developed by Kotani gives powerful tools for proving the absence of absolutely continuous spectrum, the presence of absolutely continuous spectrum, and even the presence of purely ...
متن کاملGeneric Singular Spectrum for Ergodic Schrödinger Operators
We consider Schrödinger operators with ergodic potential Vω(n) = f (T (ω)), n ∈ Z, ω ∈ , where T : → is a nonperiodic homeomorphism. We show that for generic f ∈ C( ), the spectrum has no absolutely continuous component. The proof is based on approximation by discontinuous potentials which can be treated via Kotani theory.
متن کاملStrictly Ergodic Subshifts and Associated Operators
We consider ergodic families of Schrödinger operators over base dynamics given by strictly ergodic subshifts on finite alphabets. It is expected that the majority of these operators have purely singular continuous spectrum supported on a Cantor set of zero Lebesgue measure. These properties have indeed been established for large classes of operators of this type over the course of the last twen...
متن کاملSingular Continuous Spectrum for Palindromic Schrödinger Operators
We give new examples of discrete Schrödinger operators with potentials taking finitely many values that have purely singular continuous spectrum. If the hull X of the potential is strictly ergodic, then the existence of just one potential x in X for which the operator has no eigenvalues implies that there is a generic set in X for which the operator has purely singular continuous spectrum. A su...
متن کاملSingular Continuous Spectrum for Palindromic Schrr Odinger Operators
We give new examples of discrete Schrr odinger operators with potentials taking nitely many values that have purely singular continuous spectrum. If the hull X of the potential is strictly ergodic, then the existence of just one potential x in X for which the operator has no eigenvalues implies that there is a generic set in X for which the operator has purely singular continuous spectrum. A su...
متن کامل