138 research outputs found
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
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
Self-Similar Corrections to the Ergodic Theorem for the Pascal-Adic Transformation
Let T be the Pascal-adic transformation. For any measurable function g, we
consider the corrections to the ergodic theorem sum_{k=0}^{j-1} g(T^k x) - j/l
sum_{k=0}^{l-1} g(T^k x). When seen as graphs of functions defined on
{0,...,l-1}, we show for a suitable class of functions g that these quantities,
once properly renormalized, converge to (part of) the graph of a self-affine
function. The latter only depends on the ergodic component of x, and is a
deformation of the so-called Blancmange function. We also briefly describe the
links with a series of works on Conway recursive recursive sequenc
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
Growth rate for the expected value of a generalized random Fibonacci sequence
A random Fibonacci sequence is defined by the relation g_n = | g_{n-1} +/-
g_{n-2} |, where the +/- sign is chosen by tossing a balanced coin for each n.
We generalize these sequences to the case when the coin is unbalanced (denoting
by p the probability of a +), and the recurrence relation is of the form g_n =
|\lambda g_{n-1} +/- g_{n-2} |. When \lambda >=2 and 0 < p <= 1, we prove that
the expected value of g_n grows exponentially fast. When \lambda = \lambda_k =
2 cos(\pi/k) for some fixed integer k>2, we show that the expected value of g_n
grows exponentially fast for p>(2-\lambda_k)/4 and give an algebraic expression
for the growth rate. The involved methods extend (and correct) those introduced
in a previous paper by the second author
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
Pinning by a sparse potential
We consider a directed polymer interacting with a diluted pinning potential
restricted to a line. We characterize explicitely the set of disorder
configurations that give rise to localization of the polymer. We study both
relevant cases of dimension 1+1 and 1+2. We also discuss the case of massless
effective interface models in dimension 2+1.Comment: to appear in Stochastic Processes and their Application
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