939 research outputs found

### Dephasing representation: Employing the shadowing theorem to calculate quantum correlation functions

Due to the Heisenberg uncertainty principle, various classical systems
differing only on the scale smaller than Planck's cell correspond to the same
quantum system. This fact is used to find a unique semiclassical representation
without the Van Vleck determinant, applicable to a large class of correlation
functions expressible as quantum fidelity. As in the Feynman path integral
formulation of quantum mechanics, all contributing trajectories have the same
amplitude: that is why it is denoted the ``dephasing representation.'' By
relating the present approach to the problem of existence of true trajectories
near numerically-computed chaotic trajectories, the approximation is made
rigorous for any system in which the shadowing theorem holds. Numerical
implementation only requires computing actions along the unperturbed
trajectories and not finding the shadowing trajectories. While semiclassical
linear-response theory was used before in quasi-integrable and chaotic systems,
here its validity is justified in the most generic, mixed systems. Dephasing
representation appears to be a rare practical method to calculate quantum
correlation functions in nonuniversal regimes in many-dimensional systems where
exact quantum calculations are impossible.Comment: 5 pages, 1 figure, to appear in Phys. Rev. E (R

### Holder Shadowing on Finite Intervals

For any $\theta, \omega > 1/2$ we prove that, if any $d$-pseudotrajectory of
length $\sim 1/d^{\omega}$ of a diffeomorphism $f\in C^2$ can be
$d^{\theta}$-shadowed by an exact trajectory, then $f$ is structurally stable.
Previously it was conjectured by Hammel-Grebogi-Yorke that for $\theta = \omega
= 1/2$ this property holds for a wide class of non-uniformly hyperbolic
diffeomorphisms. In the proof we introduce the notion of sublinear growth
property for inhomogenious linear equations and prove that it implies
exponential dichotomy.Comment: 19 pages. Minor change

### Hopf type rigidity for thermostats

We show a Hopf type rigidity for thermostats without conjugate points on a
2-torusComment: 9 pages; minor revisions to reflect published versio

### Absolute Continuity Theorem for Random Dynamical Systems on $R^d$

In this article we provide a proof of the so called absolute continuity
theorem for random dynamical systems on $R^d$ which have an invariant
probability measure. First we present the construction of local stable
manifolds in this case. Then the absolute continuity theorem basically states
that for any two transversal manifolds to the family of local stable manifolds
the induced Lebesgue measures on these transversal manifolds are absolutely
continuous under the map that transports every point on the first manifold
along the local stable manifold to the second manifold, the so-called
Poincar\'e map or holonomy map. In contrast to known results, we have to deal
with the non-compactness of the state space and the randomness of the random
dynamical system.Comment: 46 page

### Pseudo-Anosov flows in toroidal manifolds

We first prove rigidity results for pseudo-Anosov flows in prototypes of
toroidal 3-manifolds: we show that a pseudo-Anosov flow in a Seifert fibered
manifold is up to finite covers topologically equivalent to a geodesic flow and
we show that a pseudo-Anosov flow in a solv manifold is topologically
equivalent to a suspension Anosov flow. Then we study the interaction of a
general pseudo-Anosov flow with possible Seifert fibered pieces in the torus
decomposition: if the fiber is associated with a periodic orbit of the flow, we
show that there is a standard and very simple form for the flow in the piece
using Birkhoff annuli. This form is strongly connected with the topology of the
Seifert piece. We also construct a large new class of examples in many graph
manifolds, which is extremely general and flexible. We construct other new
classes of examples, some of which are generalized pseudo-Anosov flows which
have one prong singularities and which show that the above results in Seifert
fibered and solvable manifolds do not apply to one prong pseudo-Anosov flows.
Finally we also analyse immersed and embedded incompressible tori in optimal
position with respect to a pseudo-Anosov flow.Comment: 44 pages, 4 figures. Version 2. New section 9: questions and
comments. Overall revision, some simplified proofs, more explanation

### Stable Flags and the Riemann-Hilbert Problem

We tackle the Riemann-Hilbert problem on the Riemann sphere as stalk-wise
logarithmic modifications of the classical R\"ohrl-Deligne vector bundle. We
show that the solutions of the Riemann-Hilbert problem are in bijection with
some families of local filtrations which are stable under the prescribed
monodromy maps. We introduce the notion of Birkhoff-Grothendieck
trivialisation, and show that its computation corresponds to geodesic paths in
some local affine Bruhat-Tits building. We use this to compute how the type of
a bundle changes under stalk modifications, and give several corresponding
algorithmic procedures.Comment: 39 page

### Analyticity of the SRB measure for a class of simple Anosov flows

We consider perturbations of the Hamiltonian flow associated with the
geodesic flow on a surface of constant negative curvature. We prove that, under
a small perturbation, not necessarely of Hamiltonian character, the SRB measure
associated to the flow exists and is analytic in the strength of the
perturbation. An explicit example of "thermostatted" dissipative dynamics is
constructed.Comment: 23 pages, corrected typo

### Black Hole Thermodynamics in Carath\'eodory's Approach

We show that, in the framework of Carath\'eodory's approach to
thermodynamics, one can implement black hole thermodynamics by realizing that
there exixts a quasi-homogeneity symmetry of the Pfaffian form \deq
representing the infinitesimal heat exchanged reversibly by a Kerr-Newman black
hole; this allow us to calculate readily an integrating factor, and, as a
consequence, a foliation of the thermodynamic manifold can be recovered.Comment: 10 pages; elsart styl

### Hyperbolic Chaos of Turing Patterns

We consider time evolution of Turing patterns in an extended system governed
by an equation of the Swift-Hohenberg type, where due to an external periodic
parameter modulation long-wave and short-wave patterns with length scales
related as 1:3 emerge in succession. We show theoretically and demonstrate
numerically that the spatial phases of the patterns, being observed
stroboscopically, are governed by an expanding circle map, so that the
corresponding chaos of Turing patterns is hyperbolic, associated with a strange
attractor of the Smale-Williams solenoid type. This chaos is shown to be robust
with respect to variations of parameters and boundary conditions.Comment: 4 pages, 4 figure

### On a problem of A. Weil

A topological invariant of the geodesic laminations on a modular surface is
constructed. The invariant has a continuous part (the tail of a continued
fraction) and a combinatorial part (the singularity data). It is shown, that
the invariant is complete, i.e. the geodesic lamination can be recovered from
the invariant. The continuous part of the invariant has geometric meaning of a
slope of lamination on the surface.Comment: to appear Beitr\"age zur Algebra und Geometri

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