31 research outputs found
Invariant measures for Cartesian powers of Chacon infinite transformation
We describe all boundedly finite measures which are invariant by Cartesian
powers of an infinite measure preserving version of Chacon transformation. All
such ergodic measures are products of so-called diagonal measures, which are
measures generalizing in some way the measures supported on a graph. Unlike
what happens in the finite-measure case, this class of diagonal measures is not
reduced to measures supported on a graph arising from powers of the
transformation: it also contains some weird invariant measures, whose marginals
are singular with respect to the measure invariant by the transformation. We
derive from these results that the infinite Chacon transformation has trivial
centralizer, and has no nontrivial factor. At the end of the paper, we prove a
result of independent interest, providing sufficient conditions for an infinite
measure preserving dynamical system defined on a Cartesian product to decompose
into a direct product of two dynamical systems
Poisson suspensions and entropy for infinite transformations
The Poisson entropy of an infinite-measure-preserving transformation is
defined as the Kolmogorov entropy of its Poisson suspension. In this article,
we relate Poisson entropy with other definitions of entropy for infinite
transformations: For quasi-finite transformations we prove that Poisson entropy
coincides with Krengel's and Parry's entropy. In particular, this implies that
for null-recurrent Markov chains, the usual formula for the entropy holds in any of the definitions for entropy. Poisson
entropy dominates Parry's entropy in any conservative transformation. We also
prove that relative entropy (in the sense of Danilenko and Rudolph) coincides
with the relative Poisson entropy. Thus, for any factor of a conservative
transformation, difference of the Krengel's entropy is equal to the difference
of the Poisson entropies. In case there exists a factor with zero Poisson
entropy, we prove the existence of a maximum (Pinsker) factor with zero Poisson
entropy. Together with the preceding results, this answers affirmatively the
question raised in arXiv:0705.2148v3 about existence of a Pinsker factor in the
sense of Krengel for quasi-finite transformations.Comment: 25 pages, a final section with some more results and questions adde
La Llei de Benford
La llei de Benford reflecteix una irregularitat inesperada en la distribució dels
dígits de dades aleatòries. En aquest article es descriu el fenomen i se'n discuteixen
possibles explicacions.The Benford law reflects an unexpected irregularity in the distribution of digits
of random data. In this paper the law is described and some explanations are
discussed
Transposition game
We introduce a two-player game, in which each player extends a given sequence
by picking a free element in a domain D of the real line. The aim of the
players is to control the parity of the number of transpositions necessary to
put the final sequence in order. We will see that the winner can be the last
player, the second last player, the first player, the second player, the person
who wants the parity to end up even or the person who wants the parity to end
up odd. A special case of the game can be reduced to a game with nontrivial
winning strategy, but describable in so simple a way that children can
understand it and enjoy playing it
Almost-sure Growth Rate of Generalized Random Fibonacci sequences
We study the generalized random Fibonacci sequences defined by their first
nonnegative terms and for ,
(linear case) and (non-linear case), where each sign is independent and
either with probability or with probability (). Our
main result is that, when is of the form
for some integer , the exponential growth of for , and
of for , is almost surely positive and given
by where is an explicit
function of depending on the case we consider, taking values in ,
and is an explicit probability distribution on \RR_+ defined
inductively on generalized Stern-Brocot intervals. We also provide an integral
formula for in the easier case . Finally, we study the
variations of the exponent as a function of
Note about "First order correction for the hydrodynamic limit of symmetric simple exclusion processes with speed change in dimension d > 2".
International audienc
Bounds on Semigroups of Random Rotations on SO(n)
International audienceIn order to generate random orthogonal matrices, Hastings considered a Markov chain on the orthogonal group SO(n) generated by random rotations on randomly selected coordinate planes. We investigate different ways to measure the convergence to equilibrium of this walk. To this end, we prove, up to a multiplicative constant, that the spectral gap of this walk is bounded below by 1/n^2 and the entropy/entropy dissipation bound is bounded above by n^3