6,454 research outputs found
Quantitative recurrence statistics and convergence to an extreme value distribution for non-uniformly hyperbolic dynamical systems
For non-uniformly hyperbolic dynamical systems we consider the time series of
maxima along typical orbits. Using ideas based upon quantitative recurrence
time statistics we prove convergence of the maxima (under suitable
normalization) to an extreme value distribution, and obtain estimates on the
rate of convergence. We show that our results are applicable to a range of
examples, and include new results for Lorenz maps, certain partially hyperbolic
systems, and non-uniformly expanding systems with sub-exponential decay of
correlations. For applications where analytic results are not readily available
we show how to estimate the rate of convergence to an extreme value
distribution based upon numerical information of the quantitative recurrence
statistics. We envisage that such information will lead to more efficient
statistical parameter estimation schemes based upon the block-maxima method.Comment: This article is a revision of the previous article titled: "On the
convergence to an extreme value distribution for non-uniformly hyperbolic
dynamical systems." Relative to this older version, the revised article
includes new and up to date results and developments (based upon recent
advances in the field
Kick stability in groups and dynamical systems
We consider a general construction of ``kicked systems''. Let G be a group of
measure preserving transformations of a probability space. Given its
one-parameter/cyclic subgroup (the flow), and any sequence of elements (the
kicks) we define the kicked dynamics on the space by alternately flowing with
given period, then applying a kick. Our main finding is the following stability
phenomenon: the kicked system often inherits recurrence properties of the
original flow. We present three main examples. 1) G is the torus. We show that
for generic linear flows, and any sequence of kicks, the trajectories of the
kicked system are uniformly distributed for almost all periods. 2) G is a
discrete subgroup of PSL(2,R) acting on the unit tangent bundle of a Riemann
surface. The flow is generated by a single element of G, and we take any
bounded sequence of elements of G as our kicks. We prove that the kicked system
is mixing for all sufficiently large periods if and only if the generator is of
infinite order and is not conjugate to its inverse in G. 3) G is the group of
Hamiltonian diffeomorphisms of a closed symplectic manifold. We assume that the
flow is rapidly growing in the sense of Hofer's norm, and the kicks are
bounded. We prove that for a positive proportion of the periods the kicked
system inherits a kind of energy conservation law and is thus superrecurrent.
We use tools of geometric group theory and symplectic topology.Comment: Latex, 40 pages, revised versio
Algebraic properties of Gardner's deformations for integrable systems
An algebraic definition of Gardner's deformations for completely integrable
bi-Hamiltonian evolutionary systems is formulated. The proposed approach
extends the class of deformable equations and yields new integrable
evolutionary and hyperbolic Liouville-type systems. An exactly solvable
two-component extension of the Liouville equation is found.Comment: Proc. conf. "Nonlinear Physics: Theory and Experiment IV" (Gallipoli,
2006); Theor. Math. Phys. (2007) 151:3/152:1-2, 16p. (to appear
Recurrence statistics for the space of Interval Exchange maps and the Teichm\"uller flow on the space of translation surfaces
In this note we show that the transfer operator of a Rauzy-Veech-Zorich
renormalization map acting on a space of quasi-H\"older functions is
quasicompact and derive certain statistical recurrence properties for this map
and its associated Teichm\"uller flow. We establish Borel-Cantelli lemmas,
Extreme Value statistics and return time statistics for the map and flow.
Previous results have established quasicompactness in H\"older or analytic
function spaces, for example the work of M. Pollicott and T. Morita. The
quasi-H\"older function space is particularly useful for investigating return
time statistics. In particular we establish the shrinking target property for
nested balls in the setting of Teichm\"uller flow. Our point of view, approach
and terminology derives from the work of M. Pollicott augmented by that of M.
Viana
A trick around Fibonacci, Lucas and Chebyshev
In this article, we present a trick around Fibonacci numbers which can be
found in several magic books. It consists in computing quickly the sum of the
successive terms of a Fibonacci-like sequence. We give explanations and
extensions of this trick to more general sequences. This study leads us to
interesting connections between Fibonacci, Lucas sequences and Chebyshev
polynomials.Comment: 23 page
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