610 research outputs found
Cantor Spectrum for Schr\"odinger Operators with Potentials arising from Generalized Skew-shifts
We consider continuous -cocycles over a strictly ergodic
homeomorphism which fibers over an almost periodic dynamical system
(generalized skew-shifts). We prove that any cocycle which is not uniformly
hyperbolic can be approximated by one which is conjugate to an
-cocycle. Using this, we show that if a cocycle's homotopy
class does not display a certain obstruction to uniform hyperbolicity, then it
can be -perturbed to become uniformly hyperbolic. For cocycles arising
from Schr\"odinger operators, the obstruction vanishes and we conclude that
uniform hyperbolicity is dense, which implies that for a generic continuous
potential, the spectrum of the corresponding Schr\"odinger operator is a Cantor
set.Comment: Final version. To appear in Duke Mathematical Journa
Dominated Splitting and Pesin's Entropy Formula
Let be a compact manifold and be a diffeomorphism on
. If is an -invariant probability measure which is absolutely
continuous relative to Lebesgue measure and for
there is a dominated splitting on its orbit ,
then we give an estimation through Lyapunov characteristic exponents from below
in Pesin's entropy formula, i.e., the metric entropy satisfies
where
and
are the Lyapunov
exponents at with respect to Consequently, by using a dichotomy for
generic volume-preserving diffeomorphism we show that Pesin's entropy formula
holds for generic volume-preserving diffeomorphisms, which generalizes a result
of Tahzibi in dimension 2
Collision, explosion and collapse of homoclinic classes
Homoclinic classes of generic -diffeomorphisms are maximal transitive
sets and pairwise disjoint. We here present a model explaining how two
different homoclinic classes may intersect, failing to be disjoint. For that we
construct a one-parameter family of diffeomorphisms with
hyperbolic points and having nontrivial homoclinic classes, such that,
for , the classes of and are disjoint, for , they are equal,
and, for , their intersection is a saddle-node.Comment: This is the final version, accepted in 200
The Complexity of Reasoning for Fragments of Default Logic
Default logic was introduced by Reiter in 1980. In 1992, Gottlob classified
the complexity of the extension existence problem for propositional default
logic as \SigmaPtwo-complete, and the complexity of the credulous and
skeptical reasoning problem as SigmaP2-complete, resp. PiP2-complete.
Additionally, he investigated restrictions on the default rules, i.e.,
semi-normal default rules. Selman made in 1992 a similar approach with
disjunction-free and unary default rules. In this paper we systematically
restrict the set of allowed propositional connectives. We give a complete
complexity classification for all sets of Boolean functions in the meaning of
Post's lattice for all three common decision problems for propositional default
logic. We show that the complexity is a hexachotomy (SigmaP2-, DeltaP2-, NP-,
P-, NL-complete, trivial) for the extension existence problem, while for the
credulous and skeptical reasoning problem we obtain similar classifications
without trivial cases.Comment: Corrected versio
A striking correspondence between the dynamics generated by the vector fields and by the scalar parabolic equations
The purpose of this paper is to enhance a correspondence between the dynamics
of the differential equations on and those
of the parabolic equations on a bounded
domain . We give details on the similarities of these dynamics in the
cases , and and in the corresponding cases ,
and dim() respectively. In addition to
the beauty of such a correspondence, this could serve as a guideline for future
research on the dynamics of parabolic equations
Invariant manifolds and equilibrium states for non-uniformly hyperbolic horseshoes
In this paper we consider horseshoes containing an orbit of homoclinic
tangency accumulated by periodic points. We prove a version of the Invariant
Manifolds Theorem, construct finite Markov partitions and use them to prove the
existence and uniqueness of equilibrium states associated to H\"older
continuous potentials.Comment: 33 pages, 6 figure
Large deviations for non-uniformly expanding maps
We obtain large deviation results for non-uniformly expanding maps with
non-flat singularities or criticalities and for partially hyperbolic
non-uniformly expanding attracting sets. That is, given a continuous function
we consider its space average with respect to a physical measure and compare
this with the time averages along orbits of the map, showing that the Lebesgue
measure of the set of points whose time averages stay away from the space
average decays to zero exponentially fast with the number of iterates involved.
As easy by-products we deduce escape rates from subsets of the basins of
physical measures for these types of maps. The rates of decay are naturally
related to the metric entropy and pressure function of the system with respect
to a family of equilibrium states. The corrections added to the published
version of this text appear in bold; see last section for a list of changesComment: 36 pages, 1 figure. After many PhD students and colleagues having
pointed several errors in the statements and proofs, this is a correction to
published article answering those comments. List of main changes in a new
last sectio
Evaluation of the CPTEC/AGCM wind forecasts during the hurricane Catarina occurrence
International audienceIn March 2004 occurred the first hurricane registered at South Atlantic Ocean. The system named Catarina begun as an extratropical cyclone and remained quasi-stationary some days over the South Atlantic Ocean. The system displaced westward, acquiring characteristics of a hurricane and hit the Brazilian State of Santa Catarina (SC) between the 27 and the 28 March, causing destruction and deaths. The objective of this paper is to evaluate the Center for Weather Prediction and Climate Studies, Atmospheric Global Circulation Model (CPTEC/AGCM) forecast performance of some synoptic patterns associated with Catarina. The surface wind and reduced Sea Level Pressure (SLP) were examined. Moreover, the implementation of 10-m wind forecast (V10m) was evaluated. This variable was not available in the CPTEC/AGCM during the Catarina occurrence and in this study it was compared with the wind at first sigma-level of the AGCM. The CPTEC-Eta reanalyses were used to comparisons. According to reanalyses, more intense winds were observed in northeast, south and southwest edges of the cyclone. The system was not predicted by the CPTEC/AGCM forecasts longer than 24 h, then the analyses were carried out only for 24 h forecasts. In general, the first sigma-level wind forecasts underestimated the wind magnitude and the cyclone intensity. However, the Catarina formation and its displacement southeastward between the 20 and the 21 March were well represented by the model. The CPTEC/AGCM presents deficiencies to predict the system intensity, but in short-range forecasts it was possible to predict the system formation and its atypical trajectory. The wind results from the new implementation did not exhibit better performance compared with the wind at first sigma-level. These results will be better investigated in the future
Stochastic stability at the boundary of expanding maps
We consider endomorphisms of a compact manifold which are expanding except
for a finite number of points and prove the existence and uniqueness of a
physical measure and its stochastical stability. We also characterize the
zero-noise limit measures for a model of the intermittent map and obtain
stochastic stability for some values of the parameter. The physical measures
are obtained as zero-noise limits which are shown to satisfy Pesin?s Entropy
Formula
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