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Modifying Rudins original construction of the Rudin-Shapiro sequence, we derive a new substitution-based sequence with purely absolutely continuous diffraction spectrum.
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By generalising Rudins construction of an aperiodic sequence, we derive new substitution-based structures which have purely absolutely continuous diffraction and mixed dynamical spectrum, with absolutely continuous and pure point parts. We discuss several examples, including a construction based on Fourier matrices which yields constant-length substitutions for any length.
Given a self-adjoint operator H, a self-adjoint trace class operator V and a fixed Hilbert-Schmidt operator F with trivial kernel and co-kernel, using limiting absorption principle an explicit set of full Lebesgue measure is defined such that for all points of this set the wave and the scattering matrices can be defined and constructed unambiguously. Many well-known properties of the wave and scattering matrices and operators are proved, including the stationary formula for the scattering matrix. This new abstract scattering theory allows to prove that for any trace class perturbations of arbitrary self-adjoint operators the singular part of the spectral shift function is an almost everywhere integer-valued function.
We give sufficient conditions for the presence of the absolutely continuous spectrum of a Schrodinger operator on a regular rooted tree without loops (also called regular Bethe lattice or Cayley tree).
We continue to investigate absolutely continuous spectrum of generalized indefinite strings. By following an approach of Deift and Killip, we establish stability of the absolutely continuous spectra of two more model examples of generalized indefinite strings under rather wide perturbations. In particular, one of these results allows us to prove that the absolutely continuous spectrum of the isospectral problem associated with the two-component Camassa-Holm system in a certain dispersive regime is essentially supported on the set $(-infty,-1/2]cup [1/2,infty)$.
In this article, we study algebraic dynamical pairs $(f,a)$ parametrized by an irreducible quasi-projective curve $Lambda$ having an absolutely continuous bifurcation measure. We prove that, if $f$ is non-isotrivial and $(f,a)$ is unstable, this is equivalent to the fact that $f$ is a family of Latt`es maps. To do so, we prove the density of transversely prerepelling parameters in the bifucation locus of $(f,a)$ and a similarity property, at any transversely prerepelling parameter $lambda_0$, between the measure $mu_{f,a}$ and the maximal entropy measure of $f_{lambda_0}$. We also establish an equivalent result for dynamical pairs of $mathbb{P}^k$, under an additional assumption.
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