225 research outputs found

### Asymptotic formula for the moments of Minkowski question mark function in the interval [0,1]

In this paper we prove the asymptotic formula for the moments of Minkowski
question mark function, which describes the distribution of rationals in the
Farey tree. The main idea is to demonstrate that certain a variation of a
Laplace method is applicable in this problem, hence the task reduces to a
number of technical calculations.Comment: 11 pages, 1 figure (final version). Lithuanian Math. J. (to appear

### Ducks on the torus: existence and uniqueness

We show that there exist generic slow-fast systems with only one
(time-scaling) parameter on the two-torus, which have canard cycles for
arbitrary small values of this parameter. This is in drastic contrast with the
planar case, where canards usually occur in two-parametric families. Here we
treat systems with a convex slow curve. In this case there is a set of
parameter values accumulating to zero for which the system has exactly one
attracting and one repelling canard cycle. The basin of the attracting cycle is
almost the whole torus.Comment: To appear in Journal of Dynamical and Control Systems, presumably
Vol. 16 (2010), No. 2; The final publication is available at
www.springerlink.co

### A toral diffeomorphism with a non-polygonal rotation set

We construct a diffeomorphism of the two-dimensional torus which is isotopic
to the identity and whose rotation set is not a polygon

### How do random Fibonacci sequences grow?

We study two kinds of random Fibonacci sequences defined by $F_1=F_2=1$ and
for $n\ge 1$, $F_{n+2} = F_{n+1} \pm F_{n}$ (linear case) or $F_{n+2} =
|F_{n+1} \pm F_{n}|$ (non-linear case), where each sign is independent and
either + with probability $p$ or - with probability $1-p$ ($0<p\le 1$). Our
main result is that the exponential growth of $F_n$ for $0<p\le 1$ (linear
case) or for $1/3\le p\le 1$ (non-linear case) is almost surely given by
$\int_0^\infty \log x d\nu_\alpha (x),$ where $\alpha$ is an explicit
function of $p$ depending on the case we consider, and $\nu_\alpha$ 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 $p$, since we prove that it is equal to zero for
$0<p\le1/3$. We also give some results about the variations of the largest
Lyapunov exponent, and provide a formula for its derivative

### A Phase Transition for Circle Maps and Cherry Flows

We study $C^{2}$ weakly order preserving circle maps with a flat interval.
The main result of the paper is about a sharp transition from degenerate
geometry to bounded geometry depending on the degree of the singularities at
the boundary of the flat interval. We prove that the non-wandering set has zero
Hausdorff dimension in the case of degenerate geometry and it has Hausdorff
dimension strictly greater than zero in the case of bounded geometry. Our
results about circle maps allow to establish a sharp phase transition in the
dynamics of Cherry flows

### Two ideals connected with strong right upper porosity at a point

Let $SP$ be the set of upper strongly porous at $0$ subsets of $\mathbb
R^{+}$ and let $\hat I(SP)$ be the intersection of maximal ideals $I \subseteq
SP$. Some characteristic properties of sets $E\in\hat I(SP)$ are obtained. It
is shown that the ideal generated by the so-called completely strongly porous
at $0$ subsets of $\mathbb R^{+}$ is a proper subideal of $\hat I(SP).$Comment: 18 page

### Renormalisation scheme for vector fields on T2 with a diophantine frequency

We construct a rigorous renormalisation scheme for analytic vector fields on
the 2-torus of Poincare type. We show that iterating this procedure there is
convergence to a limit set with a ``Gauss map'' dynamics on it, related to the
continued fraction expansion of the slope of the frequencies. This is valid for
diophantine frequency vectors.Comment: final versio

### Quasi-analyticity and determinacy of the full moment problem from finite to infinite dimensions

This paper is aimed to show the essential role played by the theory of
quasi-analytic functions in the study of the determinacy of the moment problem
on finite and infinite-dimensional spaces. In particular, the quasi-analytic
criterion of self-adjointness of operators and their commutativity are crucial
to establish whether or not a measure is uniquely determined by its moments.
Our main goal is to point out that this is a common feature of the determinacy
question in both the finite and the infinite-dimensional moment problem, by
reviewing some of the most known determinacy results from this perspective. We
also collect some properties of independent interest concerning the
characterization of quasi-analytic classes associated to log-convex sequences.Comment: 28 pages, Stochastic and Infinite Dimensional Analysis, Chapter 9,
Trends in Mathematics, Birkh\"auser Basel, 201

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