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Daniel Lenz

Publications and source records attributed to Daniel Lenz.

118 records · Page 7Linked to original sources
arXiv2002

Delone dynamical systems and associated random operators

We carry out a careful study of basic topological and ergodic features of Delone dynamical systems. We then investigate the associated topological groupoids and in particular their representations on certain direct integrals with non constant fibres. Via non-commutative-integration theory these representations give rise to von Neumann algebras of random operators. Features of these algebras and operators are discussed. Restricting our attention to a certain subalgebra of tight binding operators, we then discuss a Shubin trace formula.

math-ph↗
arXiv2001

Singular 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 aperiodic subshifts arising from primitive substitutions including new examples as e.g. the Rudin-Shapiro substitution. Our investigation is not based on trace maps. Instead it relies on an Oseledec type theorem due to A. Furman and a uniform ergodic theorem due to the author.

math-ph↗
arXiv2001

Uniform spectral properties of one-dimensional quasicrystals, IV. Quasi-Sturmian potentials

We consider discrete one-dimensional Schrödinger operators with quasi-Sturmian potentials. We present a new approach to the trace map dynamical system which is independent of the initial conditions and establish a characterization of the spectrum in terms of bounded trace map orbits. Using this, it is shown that the operators have purely singular continuous spectrum and their spectrum is a Cantor set of Lebesgue measure zero. We also exhibit a subclass having purely $α$-continuous spectrum. All these results hold uniformly on the hull generated by a given potential.

math-ph↗
arXiv2000

Hierarchical structures in Sturmian dynamical systems

Paper withdrawn; will be replaced by revised version containing application to lattice models as well. We study hierarchical properties of Sturmian words. These properties are similar to those of substitution dynamical systems. This approach allows one to carry over to Sturmian dynamical systems methods developed in the context of substitutions. For example, it allows for a very simple proof of a uniform ergodic type theorem for additive functions taking values in a Banach space. We then focus on establishing various versions of uniform subadditive ergodic type theorems. The main result states that linear repetitivity of a Sturmian system is equivalent to the validity of a uniform subadditive ergodic theorem which in turn is equivalent to uniform positivity of certain weights. Thus, the results of this paper completely cover the validity of uniform additive and subadditive ergodic theorems on Sturmian systems.

math.DS↗
arXiv2000

Uniform ergodic theorems on subshifts over a finite alphabet

We investigate uniform ergodic type theorems for additive and subadditive functions on a subshift over a finite alphabet. We show that every strictly ergodic subshift admits a uniform ergodic theorem for Banach-space-valued additive functions. We then give a necessary and sufficient condition on a minimal subshift to allow for a uniform subadditive ergodic theorem. This provides in particular a sufficient condition for unique ergodicity.

math.DS↗
arXiv2000

Linear repetitivity, I. Uniform subadditive ergodic theorems and applications

This paper is concerned with the concept of linear repetitivity in the theory of tilings. We prove a general uniform subadditive ergodic theorem for linearly repetitive tilings. This theorem unifies and extends various known (sub)additive ergodic theorems on tilings. The results of this paper can be applied in the study of both random operators and lattice gas models on tilings.

math.DS↗
arXiv1999

Half-line eigenfunction estimates and singular continuous spectrum of zero Lebesgue measure

We consider discrete one-dimensional Schrödinger operators with strictly ergodic, aperiodic potentials taking finitely many values. The well-known tendency of these operators to have purely singular continuous spectrum of zero Lebesgue measure is further elucidated. We provide a unified approach to both the study of the spectral type as well as the measure of the spectrum as a set. We apply this approach to Schrödinger operators with Sturmian potentials. Finally, in the appendix, we discuss the two different strictly ergodic dynamical systems associated to a circle map.

math.SP↗
arXiv1999

Uniform spectral properties of one-dimensional quasicrystals, III. $α$-continuity

We study the spectral properties of discrete one-dimensional Schrödinger operators with Sturmian potentials. It is shown that the point spectrum is always empty. Moreover, for rotation numbers with bounded density, we establish purely $α$-continuous spectrum, uniformly for all phases. The proofs rely on the unique decomposition property of Sturmian potentials, a mass-reproduction technique based upon a Gordon-type argument, and on the Jitomirskaya-Last extension of the Gilbert-Pearson theory of subordinacy.

math-ph↗
arXiv1999

Uniform 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 irrational rotation numbers with bounded continued fraction expansion, it gives uniform existence of the Lyapunov exponent on the whole complex plane. Moreover, it yields uniform polynomial upper bounds on the growth rate of transfer matrices for irrational rotation numbers with bounded density. In particular, all our results apply to the Fibonacci case.

math-ph↗