1,478 research outputs found
An extension which is relatively twofold mixing but not threefold mixing
We give an example of a dynamical system which is mixing relative to one of
its factors, but for which relative mixing of order three does not hold
Notes on Austin's multiple ergodic theorem
The purpose of this note is to present my understanding of Tim Austin's proof
of the multiple ergodic theorem for commuting transformations, emphasizing on
the use of joinings, extensions and factors. The existence of a sated
extension, which is a key argument in the proof, is presented in a general
context
2-fold and 3-fold mixing: why 3-dot-type counterexamples are impossible in one dimension
V.A. Rohlin asked in 1949 whether 2-fold mixing implies 3-fold mixing for a
stationary process indexed by Z, and the question remains open today. In 1978,
F. Ledrappier exhibited a counterexample to the 2-fold mixing implies 3-fold
mixing problem, the so-called "3-dot system", but in the context of stationary
random fields indexed by ZxZ. In this work, we first present an attempt to
adapt Ledrappier's construction to the one-dimensional case, which finally
leads to a stationary process which is 2-fold but not 3-fold mixing
conditionally to the sigma-algebra generated by some factor process. Then,
using arguments coming from the theory of joinings, we will give some strong
obstacles proving that Ledrappier's counterexample can not be fully adapted to
one-dimensional stationary processes
Zero Krengel Entropy does not kill Poisson Entropy
We prove that the notions of Krengel entropy and Poisson entropy for
infinite-measure-preserving transformations do not always coincide: We
construct a conservative infinite-measure-preserving transformation with zero
Krengel entropy (the induced transformation on a set of measure 1 is the Von
Neumann-Kakutani odometer), but whose associated Poisson suspension has
positive entropy
Averaging along Uniform Random Integers
Motivated by giving a meaning to "The probability that a random integer has
initial digit d", we define a URI-set as a random set E of natural integers
such that each n>0 belongs to E with probability 1/n, independently of other
integers. This enables us to introduce two notions of densities on natural
numbers: The URI-density, obtained by averaging along the elements of E, and
the local URI-density, which we get by considering the k-th element of E and
letting k go to infinity. We prove that the elements of E satisfy Benford's
law, both in the sense of URI-density and in the sense of local URI-density.
Moreover, if b_1 and b_2 are two multiplicatively independent integers, then
the mantissae of a natural number in base b_1 and in base b_2 are independent.
Connections of URI-density and local URI-density with other well-known notions
of densities are established: Both are stronger than the natural density, and
URI-density is equivalent to log-density. We also give a stochastic
interpretation, in terms of URI-set, of the H_\infty-density
Around King's Rank-One theorems: Flows and Z^n-actions
We study the generalizations of Jonathan King's rank-one theorems
(Weak-Closure Theorem and rigidity of factors) to the case of rank-one
R-actions (flows) and rank-one Z^n-actions. We prove that these results remain
valid in the case of rank-one flows. In the case of rank-one Z^n actions, where
counterexamples have already been given, we prove partial Weak-Closure Theorem
and partial rigidity of factors
Silicon-on-insulator photonic crystal miniature devices with slow light enhanced third-order nonlinearities
The effects of the slow-down factor on third-order nonlinear effects in silicon-on-insulator photonic crystal channel waveguides were investigated. In the slow light regime, with a group index equal to 99, these nonlinear effects are enhanced but the enhancement produced depends on the input peak power level. Simulations indicate the possibility of soliton-like propagation of 1 ps pulses at an input peak power level of 50 mW inside such a photonic crystal waveguide. The increase in the induced phase shift produced by lower group velocities can be used to decrease the size and power requirements needed to operate devices such as optical switches, logic gates, and wavelength translators
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