3,413 research outputs found
Poncelet pairs and the Twist Map associated to the Poncelet Billiard
We show that for a fixed curve and for a family of variables curves ,
the number of -Poncelet pairs is , where is the
number of natural numbers smaller than and which satisfies mcd . The curvee do not have to be part of the family.
In order to show this result we consider an associated billiard
transformation and a twist map which preserves area.
We use Aubry-Mather theory and the rotation number of invariant curves to
obtain our main result.
In the last section we estimate the derivative of the rotation number of a
general twist map using some properties of the continued fraction expansion
C*- Algebras and Thermodynamic Formalism
We present a detailed exposition (for a Dynamical System audience) of the
content of the paper: R. Exel and A. Lopes, Algebras, approximately
proper equivalence relations and Thermodynamic Formalism, {\it Erg. Theo. and
Dyn. Syst.}, Vol 24, pp 1051-1082 (2004). We show only the uniqueness of the
\beta-KMS (in a certain C*-Algebra obtained from the operators acting in
of a Gibbs invariant probability ) and its relation with the
eigen-probability for the dual of a certain Ruele operator. We
consider an example for a case of Hofbauer type where there exist a Phase
transition for the Gibbs state. There is no Phase transition for the KMS state.Comment: version update
Interactions, Specifications, DLR probabilities and the Ruelle Operator in the One-Dimensional Lattice
In this paper, we describe several different meanings for the concept of
Gibbs measure on the lattice in the context of finite alphabets
(or state space). We compare and analyze these "in principle" distinct notions:
DLR-Gibbs measures, Thermodynamic Limit and eigenprobabilities for the dual of
the Ruelle operator (also called conformal measures).
Among other things we extended the classical notion of a Gibbsian
specification on in such way that the similarity of many results
in Statistical Mechanics and Dynamical System becomes apparent. One of our main
result claims that the construction of the conformal Measures in Dynamical
Systems for Walters potentials, using the Ruelle operator, can be formulated in
terms of Specification. We also describe the Ising model, with
interaction energy, in the Thermodynamic Formalism
setting and prove that its associated potential is in Walters space - we
present an explicit expression. We also provide an alternative way for
obtaining the uniqueness of the DLR-Gibbs measures.Comment: to appear in Discrete and Continuous Dynamical Systems - Series
Functions for relative maximization
We introduce functions for relative maximization in a general context: the
beta and alpha applications. After a systematic study concerning regularities,
we investigate how to approximate certain values of these functions using
periodic orbits. We establish yet that the differential of an alpha application
dictates the asymptotic behavior of the optimal trajectories
Correlation Inequalities and Monotonicity Properties of the Ruelle Operator
Let be the symbolic space endowed with the product
order. A Borel probability measure over is said to satisfy the FKG
inequality if for any pair of continuous increasing functions and we
have .
In the first part of the paper we prove the validity of the FKG inequality on
Thermodynamic Formalism setting for a class of eigenmeasures of the dual of the
Ruelle operator, including several examples of interest in Statistical
Mechanics. In addition to deducing this inequality in cases not covered by
classical results about attractive specifications our proof has advantage of to
be easily adapted for suitable subshifts. We review (and provide proofs in our
setting) some classical results about the long-range Ising model on the lattice
and use them to deduce some monotonicity properties of the
associated Ruelle operator and their relations with phase transitions.
As is widely known, for some continuous potentials does not exists a positive
continuous eigenfunction associated to the spectral radius of the Ruelle
operator acting on . Here we employed some ideas related to the
involution kernel in order to solve the main eigenvalue problem in a suitable
sense - for a class of potentials having low regularity. From this we obtain an
explicit tight upper bound for the main eigenvalue (consequently for the
pressure) of the Ruelle operator associated to Ising models with
interaction energy. Extensions of the Ruelle operator to
suitable Hilbert Spaces are considered and a theorem solving to the main
eigenvalue problem (in a weak sense) is obtained by using the Lions-Lax-Milgram
theorem.Comment: 39 page
Selection of measure and a Large Deviation Principle for the general XY model
We consider a connected and compact manifold and we denote by the
Bernoulli space . The shift acting on is denoted by
.
We analyze the general XY model, as presented in a recent paper by A. T.
Baraviera, L. M. Cioletti, A. O. Lopes, J. Mohr and R. R. Souza. Denote the
Gibbs measure by , where is the eigenfunction,
and, is the eigenmeasure of the Ruelle operator associated to .
We are going to prove that any measure selected by , as , is a maximizing measure for . We also show, when the maximizing
probability measure is unique, that it is true a Large Deviation Principle,
with the deviation function , where , and, is any
calibrated subaction
On Bertelson-Gromov Dynamical Morse Entropy
In this mainly expository paper we present a detailed proof of several
results contained in a paper by M. Bertelson and M. Gromov on Dynamical Morse
Entropy. This is an introduction to the ideas presented in that work. Suppose
is compact oriented connected manifold of finite dimension.
Assume that is a surjective Morse function. For a given
natural number , consider the set and for , denote The
Dynamical Morse Entropy describes for a fixed interval the
asymptotic growth of the number of critical points of in , when . The part related to the Betti number entropy does not requires the
differentiable structure. One can describe generic properties of potentials
defined in the model of Statistical Mechanics with this machinery
Semiclassical limits, Lagrangian states and coboundary equations
Assume that is a continuous transformation . We consider
here the cases where is either the transformation or is a
smooth diffeomorphism of the circle . Consider a fixed continuous
potential , and (a quantum state). The transformation acting on
, , defined by
describes a discrete time dynamical evolution of
the quantum state . Given we define the
Lagrangian state In this case . Under
suitable conditions on the micro-support of , when , is . One of meanings of the semiclassical limit in Quantum
Mechanics is to take and . We address the
question of finding such that satisfies the property: , we have that has micro-support on the
graph of (which is the micro-support of ). In other
words: which is such that leaves the micro-support of
invariant? This is related to a coboundary equation for ,
twist conditions and the boundary of the fat attractor
Generic properties for random repeated quantum iterations
We denote by the set of by complex matrices. Given a fixed
density matrix and a fixed unitary
operator , the transformation describes the interaction of
with the external source . The result of this is . If
is a density operator then is also a density operator. The main
interest is to know what happen when we repeat several times the action of
in an initial fixed density operator . This procedure is known as
random repeated quantum iterations and is of course related to the existence of
one or more fixed points for . In \cite{NP}, among other things, the
authors show that for a fixed there exists a set of full probability
for the Haar measure such that the unitary operator satisfies the property
that for the associated there is a unique fixed point .
Moreover, there exists convergence of the iterates ,
when , for any given We show here that there is an open and
dense set of unitary operators such that the associated has a
unique fixed point. We will also consider a detailed analysis of the case when
. We will be able to show explicit results. We consider the topology
on the coefficients of . In this case we will exhibit the explicit
expression on the coefficients of which assures the existence of a unique
fixed point for . Moreover, we present the explicit expression of the
fixed point $Q_\Phi
Entropy and Variational principles for holonomic probabilities of IFS
Associated to a IFS one can consider a continuous map , defined by
were , is given
by and is the projection on the coordinate . A
-weighted system, , is a weighted system such that there exists a positive bounded function and probability on satisfying . A probability on is called holonomic for if . We denote the set of
holonomic probabilities by . Via disintegration, holonomic
probabilities on are naturally associated to a
-weighted system. More precisely, there exist a probability on
and on , such that is
. We consider holonomic ergodic probabilities. For a
holonomic probability we define entropy. Finally, we analyze the problem: given
, find the solution of the maximization pressure
problem p(\phi)=$
- β¦