420 research outputs found
Integrability of dominated decompositions on three-dimensional manifolds
We investigate the integrability of two-dimensional invariant distributions (tangent sub-bundles) which arise naturally in the context of dynamical systems on 3-manifolds. In particular, we prove unique integrability of dynamically dominated and volume-dominated Lipschitz continuous invariant decompositions as well as distributions with some other regularity conditions
Integrability of continuous bundles
We give new sufficient conditions for the integrability and unique integrability of continuous tangent sub-bundles on manifolds of arbitrary dimension, generalizing Frobenius' classical Theorem for C^1 sub-bundles. Using these conditions we derive new criteria for uniqueness of solutions to ODE's and PDE's and for the integrability of invariant bundles in dynamical systems. In particular we give a novel proof of the Stable Manifold Theorem and prove some integrability results for dynamically defined dominated splittings
Decay of correlations in one-dimensional dynamics
We consider multimodal C3 interval maps f satisfying a summability condition on the derivatives Dn along the critical orbits which implies the existence of an absolutely continuous f-invariant probability measure mu. If f is non-renormalizable, mu is mixing and we show that the speed of mixing (decay of correlations) is strongly related to the rate of growth of the sequence (Dn) as n → infinity. We also give sufficient conditions for mu to satisfy the Central Limit Theorem. This applies for example to the quadratic Fibonacci map which is shown to have subexponential decay of correlations.</p
Statistical stability and limit laws for Rovella maps
We consider the family of one-dimensional maps arising from the contracting
Lorenz attractors studied by Rovella. Benedicks-Carleson techniques were used
by Rovella to prove that there is a one-parameter family of maps whose
derivatives along their critical orbits increase exponentially fast and the
critical orbits have slow recurrent to the critical point. Metzger proved that
these maps have a unique absolutely continuous ergodic invariant probability
measure (SRB measure).
Here we use the technique developed by Freitas and show that the tail set
(the set of points which at a given time have not achieved either the
exponential growth of derivative or the slow recurrence) decays exponentially
fast as time passes. As a consequence, we obtain the continuous variation of
the densities of the SRB measures and associated metric entropies with the
parameter. Our main result also implies some statistical properties for these
maps.Comment: 1 figur
Some remarks on the geometry of the Standard Map
We define and compute hyperbolic coordinates and associated foliations which
provide a new way to describe the geometry of the standard map. We also
identify a uniformly hyperbolic region and a complementary 'critical' region
containing a smooth curve of tangencies between certain canonical 'stable'
foliations.Comment: 25 pages, 11 figure
Statistical Consequences of Devroye Inequality for Processes. Applications to a Class of Non-Uniformly Hyperbolic Dynamical Systems
In this paper, we apply Devroye inequality to study various statistical
estimators and fluctuations of observables for processes. Most of these
observables are suggested by dynamical systems. These applications concern the
co-variance function, the integrated periodogram, the correlation dimension,
the kernel density estimator, the speed of convergence of empirical measure,
the shadowing property and the almost-sure central limit theorem. We proved in
\cite{CCS} that Devroye inequality holds for a class of non-uniformly
hyperbolic dynamical systems introduced in \cite{young}. In the second appendix
we prove that, if the decay of correlations holds with a common rate for all
pairs of functions, then it holds uniformly in the function spaces. In the last
appendix we prove that for the subclass of one-dimensional systems studied in
\cite{young} the density of the absolutely continuous invariant measure belongs
to a Besov space.Comment: 33 pages; companion of the paper math.DS/0412166; corrected version;
to appear in Nonlinearit
Urban society and the English Revolution : the archaeology of the new Jerusalem
The English Revolution has long been a defining subject of English historiography, with a large and varied literature that reflects continuing engagement with the central themes of civil conflict, and deep-rooted social, political and religious change. By contrast, this period has failed to catch the imagination of archaeologists. This research seeks to understand the world of the English Revolution through its material expression in English towns. Identifying the material expressions of the period is central to developing an archaeological understanding of the period. The clearest material expressions are found, in the fortifications that were built to protect towns, the destruction that was wrought on towns and in the reconstruction of the material world of English towns. Towns, like any other artefact, have their meanings. These meanings are multivalent and ever shifting, defined by the interaction of their material fabric and those who experience it. As these meanings change over time, they can be traced through the structures and artefacts of the town, and through the myths and legends that accrete on them. Understanding the interactions of material, myth and memory allows archaeologists to understand the true meaning of the urban built environment to generate a deeper and more nuanced understanding of the nature of the English urban culture of the period. Towns were fundamental to the English imagination as much as they were economically, politically or socially important. The English Revolution sits at the heart of the accepted conception of historical archaeology, but has been curiously neglected by historical archaeologists. The cultural conflict of this period embodies the themes that are central to historical archaeology, and nowhere is this more apparent than in urban culture.EThOS - Electronic Theses Online ServiceGBUnited Kingdo
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
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