1,525 research outputs found
Asymptotically exact spectral estimates for left triangular matrices
For a family of left triangular matrices with binary entries we derive
asymptotically exact (as ) representation for the complete
eigenvalues-eigenvectors problem. In particular we show that the dependence of
all eigenvalues on is asymptotically linear for large . A similar result
is obtained for more general (with specially scaled entries) left triangular
matrices as well. As an application we study ergodic properties of a family of
chaotic maps.Comment: 7 pages, LaTe
Perron-Frobenius spectrum for random maps and its approximation
To study the convergence to equilibrium in random maps we developed the
spectral theory of the corresponding transfer (Perron-Frobenius) operators
acting in a certain Banach space of generalized functions. The random maps
under study in a sense fill the gap between expanding and hyperbolic systems
since among their (deterministic) components there are both expanding and
contracting ones. We prove stochastic stability of the Perron-Frobenius
spectrum and developed its finite rank operator approximations by means of a
``stochastically smoothed'' Ulam approximation scheme. A counterexample to the
original Ulam conjecture about the approximation of the SBR measure and the
discussion of the instability of spectral approximations by means of the
original Ulam scheme are presented as well.Comment: 24 pages, LaTe
Ergodic properties of a simple deterministic traffic flow model re(al)visited
We study statistical properties of a family of maps acting in the space of
integer valued sequences, which model dynamics of simple deterministic traffic
flows. We obtain asymptotic (as time goes to infinity) properties of
trajectories of those maps corresponding to arbitrary initial configurations in
terms of statistics of densities of various patterns and describe weak
attractors of these systems and the rate of convergence to them. Previously
only the so called regular initial configurations (having a density with only
finite fluctuations of partial sums around it) in the case of a slow particles
model (with the maximal velocity 1) have been studied rigorously. Applying
ideas borrowed from substitution dynamics we are able to reduce the analysis of
the traffic flow models corresponding to the multi-lane traffic and to the flow
with fast particles (with velocities greater than 1) to the simplest case of
the flow with the one-lane traffic and slow particles, where the crucial
technical step is the derivation of the exact life-time for a given cluster of
particles. Applications to the optimal redirection of the multi-lane traffic
flow are discussed as well.Comment: 20 pages, LaTe
Ergodicity of a collective random walk on a circle
We discuss conditions for unique ergodicity of a collective random walk on a
continuous circle. Individual particles in this collective motion perform
independent (and different in general) random walks conditioned by the
assumption that the particles cannot overrun each other. Additionally to
sufficient conditions for the unique ergodicity we discover a new and
unexpected way for its violation due to excessively large local jumps.
Necessary and sufficient conditions for the unique ergodicity of the
deterministic version of this system are obtained as well. Technically our
approach is based on the interlacing property of the spin function which
describes states of pairs of particles in coupled processes under study.Comment: 20 pages, 6 figure
Generalized phase transitions in finite coupled map lattices
We investigate generalized phase transitions of type localization -
delocalization from one to several Sinai-Bowen-Ruelle invariant measures in
finite networks of chaotic elements (coupled map lattices) with general graphs
of connections in the limit of weak coupling.Comment: 21 pages, LaTe
On exclusion type inhomogeneous interacting particle systems
For a large class of inhomogeneous interacting particle systems (IPS) on a
lattice we develop a rigorous method for mapping them onto homogeneous IPS. Our
novel approach provides a direct way of obtaining the statistical properties of
such inhomogeneous systems by studying the far simpler homogeneous systems. In
the cases when the latter can be solved exactly our method yields an exact
solution for the statistical properties of an inhomogeneous IPS. This approach
is illustrated by studies of three of IPS, namely those with particles of
different sizes, or with varying (between particles) maximal velocities, or
accelerations.Comment: 12 pages, 4 figure
Condensation versus independence in weakly interacting CMLs
We propose a simple model unifying two major approaches to the analysis of
large multicomponent systems: interacting particle systems (IPS) and couple map
lattices (CML) and show that in the weak interaction limit depending on fine
properties of the interaction potential this model may demonstrate both
condensation/synchronization and independent motions. Note that one of the main
paradigms of the CML theory is that the latter behavior is supposed to be
generic. The model under consideration is related to dynamical networks and
sheds a new light to the problem of synchronization under weak interactions.Comment: 18 page
Topological and metric recurrence for general Markov chains
Using ideas borrowed from topological dynamics and ergodic theory we
introduce topological and metric versions of the recurrence property for
general Markov chains. The main question of interest here is how large is the
set of recurrent points. We show that under some mild technical assumptions the
set of non recurrent points is of zero reference measure. Necessary and
sufficient conditions for a reference measure (which needs not to be
dynamically invariant) to satisfy this property are obtained. These results are
new even in the purely deterministic setting.Comment: 17 pages, 2 figures, to appear in MM
Stochastic stability of traffic maps
We study ergodic properties of a family of traffic maps acting in the space
of bi-infinite sequences of real numbers. The corresponding dynamics mimics the
motion of vehicles in a simple traffic flow, which explains the name. Using
connections to topological Markov chains we obtain nontrivial invariant
measures, prove their stochastic stability, and calculate the topological
entropy. Technically these results in the deterministic setting are related to
the construction of measures of maximal entropy via measures uniformly
distributed on periodic points of a given period, while in the random setting
we directly construct (spatially) Markov invariant measures. In distinction to
conventional results the limiting measures in non-lattice case are non-ergodic.
Average velocity of individual ``vehicles'' as a function of their density and
its stochastic stability is studied as well.Comment: 23 pages, accepted by "Nonlinearity
Finite rank approximations of expanding maps with neutral singularities
For a class of expanding maps with neutral singularities we prove the
validity of a finite rank approximation scheme for the analysis of
Sinai-Ruelle-Bowen measures. Earlier results of this sort were known only in
the case of hyperbolic systems.Comment: 13 pages, 3 figures, to appear in Discrete and Continuous Dynamical
Systems
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