30 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
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
A central limit theorem for the variation of the sum of digits
We prove a Central Limit Theorem for probability measures defined via the
variation of the sum-of-digits function, in base . For and , we consider as the density of integers for which the sum of digits increases by when we add to
. We give a probabilistic interpretation of on the probability
space given by the group of -adic integers equipped with the normalized Haar
measure. We split the base- expansion of the integer into so-called
"blocks", and we consider the asymptotic behaviour of as the number
of blocks goes to infinity. We show that, up to renormalization,
converges to the standard normal law as the number of blocks of grows to
infinity. We provide an estimate of the speed of convergence. The proof relies,
in particular, on a -mixing process defined on the -adic integers
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
How do random Fibonacci sequences grow?
We study two kinds of random Fibonacci sequences defined by and
for , (linear case) or (non-linear case), where each sign is independent and
either + with probability or - with probability (). Our
main result is that the exponential growth of for (linear
case) or for (non-linear case) is almost surely given by
where is an explicit
function of depending on the case we consider, and is an
explicit probability distribution on \RR_+ defined inductively on
Stern-Brocot intervals. In the non-linear case, the largest Lyapunov exponent
is not an analytic function of , since we prove that it is equal to zero for
. We also give some results about the variations of the largest
Lyapunov exponent, and provide a formula for its derivative