601 research outputs found
DE MINIMIS NON CURAT LEX
An age-old maxim often applied but infrequently rationalized is that of de minimus non curat lex. In the recent case of Steve Anderson v. Mt. Clemens Pottery Company, the United States Supreme Court focused attention upon the doctrine by ruling that it should be applied in determining whether walking time and other preliminary activities constitute work for which employees are entitled to compensation under the Fair Labor Standards Act of 1938. The so-called portal-to-portal problems which have arisen as a result of the last mentioned ruling make timely a discussion of the origin, meaning, function and application of the maxim
Connecting geodesics and security of configurations in compact locally symmetric spaces
A pair of points in a riemannian manifold makes a secure configuration if the
totality of geodesics connecting them can be blocked by a finite set. The
manifold is secure if every configuration is secure. We investigate the
security of compact, locally symmetric spaces.Comment: 27 pages, 2 figure
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
Invariant curves and explosion of periodic Islands in systems of piecewise rotations
Copyright © 2005 Society for Industrial and Applied MathematicsInvertible piecewise isometric maps (PWIs) of the plane, in spite of their apparent simplicity, can show a remarkable number of dynamical features analogous to those found in nonlinear smooth area preserving maps. There is a natural partition of the phase space into an exceptional set, ⋶, consisting of the closure of the set of points whose orbits accumulate on discontinuities of the map, and its complement. In this paper we examine a family of noninvertible PWIs on the plane that consist of rotations on each of four atoms, each of which is a quadrant. We show that this family gives examples of global attractors with a variety of geometric structures. On some of these attractors, there appear to be nonsmooth invariant curves within ⋶ that form barriers to ergodicity of any invariant measure supported on ⋶. These invariant curves are observed to appear on perturbations of an “integrable” case where the exceptional set is a union of annuli and it decomposes into a one-dimensional family of interval exchange maps that may be minimal but nonergodic. We have no adequate theoretical explanation for the curves in the nonsmooth case, but they appear to come into existence at the same times as an explosion of periodic islands near where the interval exchanges used to be located. We exhibit another example—a piecewise rotation on the plane with two atoms that also appears to have nonsmooth invariant curves
Integrability of one degree of freedom symplectic maps with polar singularities
In this paper, we treat symplectic difference equations with one degree of
freedom. For such cases, we resolve the relation between that the dynamics on
the two dimensional phase space is reduced to on one dimensional level sets by
a conserved quantity and that the dynamics is integrable, under some
assumptions. The process which we introduce is related to interval exchange
transformations.Comment: 10 pages, 2 figure
A series of coverings of the regular n-gon
We define an infinite series of translation coverings of Veech's double-n-gon
for odd n greater or equal to 5 which share the same Veech group. Additionally
we give an infinite series of translation coverings with constant Veech group
of a regular n-gon for even n greater or equal to 8. These families give rise
to explicit examples of infinite translation surfaces with lattice Veech group.Comment: A missing case in step 1 in the proof of Thm. 1 b was added. (To
appear in Geometriae Dedicata.
The content of pentose-cycle intermediates in liver in starved, fed ad libitum and meal-fed rats
The increased rate of ethanol removal from blood of clofibrate-treated rats (Short Communication)
A Classification of Minimal Sets of Torus Homeomorphisms
We provide a classification of minimal sets of homeomorphisms of the
two-torus, in terms of the structure of their complement. We show that this
structure is exactly one of the following types: (1) a disjoint union of
topological disks, or (2) a disjoint union of essential annuli and topological
disks, or (3) a disjoint union of one doubly essential component and bounded
topological disks. Periodic bounded disks can only occur in type 3. This result
provides a framework for more detailed investigations, and additional
information on the torus homeomorphism allows to draw further conclusions. In
the non-wandering case, the classification can be significantly strengthened
and we obtain that a minimal set other than the whole torus is either a
periodic orbit, or the orbit of a periodic circloid, or the extension of a
Cantor set. Further special cases are given by torus homeomorphisms homotopic
to an Anosov, in which types 1 and 2 cannot occur, and the same holds for
homeomorphisms homotopic to the identity with a rotation set which has
non-empty interior. If a non-wandering torus homeomorphism has a unique and
totally irrational rotation vector, then any minimal set other than the whole
torus has to be the extension of a Cantor set.Comment: Published in Mathematische Zeitschrift, June 2013, Volume 274, Issue
1-2, pp 405-42
Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmuller geodesic flow
We compute the sum of the positive Lyapunov exponents of the Hodge bundle
with respect to the Teichmuller geodesic flow. The computation is based on the
analytic Riemann-Roch Theorem and uses a comparison of determinants of flat and
hyperbolic Laplacians when the underlying Riemann surface degenerates.Comment: Minor corrections. To appear in Publications mathematiques de l'IHE
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