645 research outputs found
Dynamics of meromorphic maps with small topological degree III: geometric currents and ergodic theory
We continue our study of the dynamics of mappings with small topological
degree on (projective) complex surfaces. Previously, under mild hypotheses, we
have constructed an ergodic ``equilibrium'' measure for each such mapping. Here
we study the dynamical properties of this measure in detail: we give optimal
bounds for its Lyapunov exponents, prove that it has maximal entropy, and show
that it has product structure in the natural extension. Under a natural further
assumption, we show that saddle points are equidistributed towards this
measure. This generalize results that were known in the invertible case and is,
to our knowledge, one among not very many instances in which a natural
invariant measure for a non-invertible dynamical system is well-understood.Comment: v3. Exposition improved. Final version, to appear in Ann. Scient. de
l'EN
Solving Dynamic Discrete Choice Models Using Smoothing and Sieve Methods
We propose to combine smoothing, simulations and sieve approximations to
solve for either the integrated or expected value function in a general class
of dynamic discrete choice (DDC) models. We use importance sampling to
approximate the Bellman operators defining the two functions. The random
Bellman operators, and therefore also the corresponding solutions, are
generally non-smooth which is undesirable. To circumvent this issue, we
introduce a smoothed version of the random Bellman operator and solve for the
corresponding smoothed value function using sieve methods. We show that one can
avoid using sieves by generalizing and adapting the `self-approximating' method
of Rust (1997) to our setting. We provide an asymptotic theory for the
approximate solutions and show that they converge with root-N-rate, where
is number of Monte Carlo draws, towards Gaussian processes. We examine their
performance in practice through a set of numerical experiments and find that
both methods perform well with the sieve method being particularly attractive
in terms of computational speed and accuracy
Stability and bifurcations for dissipative polynomial automorphisms of C^2
We study stability and bifurcations in holomorphic families of polynomial
automorphisms of C^2. We say that such a family is weakly stable over some
parameter domain if periodic orbits do not bifurcate there. We first show that
this defines a meaningful notion of stability, which parallels in many ways the
classical notion of J-stability in one-dimensional dynamics. In the second part
of the paper, we prove that under an assumption of moderate dissipativity, the
parameters displaying homoclinic tangencies are dense in the bifurcation locus.
This confirms one of Palis' Conjectures in the complex setting. The proof
relies on the formalism of semi-parabolic bifurcation and the construction of
"critical points" in semi-parabolic basins (which makes use of the classical
Denjoy-Carleman-Ahlfors and Wiman Theorems).Comment: Revised version. Part 1 on holomorphic motions and stability was
reorganize
A note on optimal algorithms for fixed points
technical reportWe present a constructive lemma that we believe will make possible the design of nearly optimal 0(dlog | ) cost algorithms for computing eresidual approximations to the fixed points of d-dimensional nonexpansive mappings with respect to the infinity norm. This lemma is a generalization of a two-dimensional result that we proved in [lj
Invariant template matching in systems with spatiotemporal coding: a vote for instability
We consider the design of a pattern recognition that matches templates to
images, both of which are spatially sampled and encoded as temporal sequences.
The image is subject to a combination of various perturbations. These include
ones that can be modeled as parameterized uncertainties such as image blur,
luminance, translation, and rotation as well as unmodeled ones. Biological and
neural systems require that these perturbations be processed through a minimal
number of channels by simple adaptation mechanisms. We found that the most
suitable mathematical framework to meet this requirement is that of weakly
attracting sets. This framework provides us with a normative and unifying
solution to the pattern recognition problem. We analyze the consequences of its
explicit implementation in neural systems. Several properties inherent to the
systems designed in accordance with our normative mathematical argument
coincide with known empirical facts. This is illustrated in mental rotation,
visual search and blur/intensity adaptation. We demonstrate how our results can
be applied to a range of practical problems in template matching and pattern
recognition.Comment: 52 pages, 12 figure
Statistics of torus piecewise isometries
By now, we have learned quite well how to study hyperbolic (locally
expanding/contracting or both) chaotic dynamical systems, thanks to a large
extent to the development of the so called operator approach. Contrary to this
almost nothing is known about piecewise isometries, except for a special case
of one-dimensional interval exchange mappings. The last case is fundamentally
different from the general situation in the presence of an invariant measure
(Lebesgue measure), which helps a lot in the analysis. We start by showing that
already the restriction of the rotation of the plane to a torus demonstrates a
number of rather unexpected properties. Our main results describe sufficient
conditions for the existence/absence of invariant measures of torus piecewise
isometries. Technically these results are based on the approximation of the
maps under study by weakly periodic ones.Comment: 17 page
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