77 research outputs found

### A family of Schr\"odinger operators whose spectrum is an interval

By approximation, I show that the spectrum of the Schr\"odinger operator with
potential $V(n) = f(n\rho \pmod 1)$ for f continuous and $\rho > 0$, $\rho
\notin \N$ is an interval.Comment: Comm. Math. Phys. (to appear

### Generic Continuous Spectrum for Ergodic Schr"odinger Operators

We consider discrete Schr"odinger operators on the line with potentials
generated by a minimal homeomorphism on a compact metric space and a continuous
sampling function. We introduce the concepts of topological and metric
repetition property. Assuming that the underlying dynamical system satisfies
one of these repetition properties, we show using Gordon's Lemma that for a
generic continuous sampling function, the associated Schr"odinger operators
have no eigenvalues in a topological or metric sense, respectively. We present
a number of applications, particularly to shifts and skew-shifts on the torus.Comment: 14 page

### Borel-Cantelli sequences

A sequence $\{x_{n}\}_1^\infty$ in $[0,1)$ is called Borel-Cantelli (BC) if
for all non-increasing sequences of positive real numbers $\{a_n\}$ with
$\underset{i=1}{\overset{\infty}{\sum}}a_i=\infty$ the set
$\underset{k=1}{\overset{\infty}{\cap}} \underset{n=k}{\overset{\infty}{\cup}}
B(x_n, a_n))=\{x\in[0,1)\mid |x_n-x|<a_n \text{for} \infty
\text{many}n\geq1\}$ has full Lebesgue measure. (To put it informally, BC
sequences are sequences for which a natural converse to the Borel-Cantelli
Theorem holds).
The notion of BC sequences is motivated by the Monotone Shrinking Target
Property for dynamical systems, but our approach is from a geometric rather
than dynamical perspective. A sufficient condition, a necessary condition and a
necessary and sufficient condition for a sequence to be BC are established. A
number of examples of BC and not BC sequences are presented.
The property of a sequence to be BC is a delicate diophantine property. For
example, the orbits of a pseudo-Anosoff IET (interval exchange transformation)
are BC while the orbits of a "generic" IET are not.
The notion of BC sequences is extended to more general spaces.Comment: 20 pages. Some proofs clarifie

### An algorithm to identify automorphisms which arise from self-induced interval exchange transformations

We give an algorithm to determine if the dynamical system generated by a
positive automorphism of the free group can also be generated by a self-induced
interval exchange transformation. The algorithm effectively yields the interval
exchange transformation in case of success.Comment: 26 pages, 8 figures. v2: the article has been reorganized to make for
a more linear read. A few paragraphs have been added for clarit

### Geometric representation of interval exchange maps over algebraic number fields

We consider the restriction of interval exchange transformations to algebraic
number fields, which leads to maps on lattices. We characterize
renormalizability arithmetically, and study its relationships with a
geometrical quantity that we call the drift vector. We exhibit some examples of
renormalizable interval exchange maps with zero and non-zero drift vector, and
carry out some investigations of their properties. In particular, we look for
evidence of the finite decomposition property: each lattice is the union of
finitely many orbits.Comment: 34 pages, 8 postscript figure

### Irreversible Quantum Baker Map

We propose a generalization of the model of classical baker map on the torus,
in which the images of two parts of the phase space do overlap. This
transformation is irreversible and cannot be quantized by means of a unitary
Floquet operator. A corresponding quantum system is constructed as a completely
positive map acting in the space of density matrices. We investigate spectral
properties of this super-operator and their link with the increase of the
entropy of initially pure states.Comment: 4 pages, 3 figures include

### Escape orbits and Ergodicity in Infinite Step Billiards

In a previous paper we defined a class of non-compact polygonal billiards,
the infinite step billiards: to a given decreasing sequence of non-negative
numbers $\{p_{n}$, there corresponds a table \Bi := \bigcup_{n\in\N} [n,n+1]
\times [0,p_{n}].
In this article, first we generalize the main result of the previous paper to
a wider class of examples. That is, a.s. there is a unique escape orbit which
belongs to the alpha and omega-limit of every other trajectory. Then, following
a recent work of Troubetzkoy, we prove that generically these systems are
ergodic for almost all initial velocities, and the entropy with respect to a
wide class of ergodic measures is zero.Comment: 27 pages, 8 figure

### Invariant sets for discontinuous parabolic area-preserving torus maps

We analyze a class of piecewise linear parabolic maps on the torus, namely
those obtained by considering a linear map with double eigenvalue one and
taking modulo one in each component. We show that within this two parameter
family of maps, the set of noninvertible maps is open and dense. For cases
where the entries in the matrix are rational we show that the maximal invariant
set has positive Lebesgue measure and we give bounds on the measure. For
several examples we find expressions for the measure of the invariant set but
we leave open the question as to whether there are parameters for which this
measure is zero.Comment: 19 pages in Latex (with epsfig,amssymb,graphics) with 5 figures in
eps; revised version: section 2 rewritten, new example and picture adde

### Recurrence and algorithmic information

In this paper we initiate a somewhat detailed investigation of the
relationships between quantitative recurrence indicators and algorithmic
complexity of orbits in weakly chaotic dynamical systems. We mainly focus on
examples.Comment: 26 pages, no figure

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