Generating and zeta functions, structure, spectral and analytic properties of the moments of Minkowski question mark function

In this paper we are interested in moments of Minkowski question mark function ?(x). It appears that, to certain extent, the results are analogous to the results obtained for objects associated with Maass wave forms: period functions, L-series, distributions, spectral properties. These objects can be naturally defined for ?(x) as well. Despite the fact that there are various nice results about the nature of ?(x), these investigations are mainly motivated from the perspective of metric number theory, Hausdorff dimension, singularity and generalizations. In this work it is shown that analytic and spectral properties of various integral transforms of ?(x) do reveal significant information about the question mark function. We prove asymptotic and structural results about the moments, calculate certain integrals involving ?(x), define an associated zeta function, generating functions, Fourier series, and establish intrinsic relations among these objects. At the end of the paper it is shown that certain object associated with ?(x) establish a bridge between realms of imaginary and real quadratic irrationals.Comment: 34 pages, 4 figures (submitted 01/2008). Minor revisions and typos. A graph of dyadic zeta function on the critical line was added. Theorem 3 was strengthene

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

Dynamical consequences of a free interval: minimality, transitivity, mixing and topological entropy

We study dynamics of continuous maps on compact metrizable spaces containing a free interval (i.e., an open subset homeomorphic to an open interval). A special attention is paid to relationships between topological transitivity, weak and strong topological mixing, dense periodicity and topological entropy as well as to the topological structure of minimal sets. In particular, a trichotomy for minimal sets and a dichotomy for transitive maps are proved.Comment: 21 page

Fractal analysis for sets of non-differentiability of Minkowski's question mark function

In this paper we study various fractal geometric aspects of the Minkowski question mark function $Q.$ We show that the unit interval can be written as the union of the three sets $\Lambda_{0}:=\{x:Q'(x)=0\}$, $\Lambda_{\infty}:=\{x:Q'(x)=\infty\}$, and $\Lambda_{\sim}:=\{x:Q'(x)$ does not exist and $Q'(x)\not=\infty\}.$ The main result is that the Hausdorff dimensions of these sets are related in the following way. $\dim_{H}(\nu_{F})<\dim_{H}(\Lambda_{\sim})= \dim_{H} (\Lambda_{\infty}) = \dim_{H} (\mathcal{L}(h_{\mathrm{top}}))<\dim_{H}(\Lambda_{0})=1.$ Here, $\mathcal{L}(h_{\mathrm{top}})$ refers to the level set of the Stern-Brocot multifractal decomposition at the topological entropy $h_{\mathrm{top}}=\log2$ of the Farey map $F,$ and $\dim_{H}(\nu_{F})$ denotes the Hausdorff dimension of the measure of maximal entropy of the dynamical system associated with $F.$ The proofs rely partially on the multifractal formalism for Stern-Brocot intervals and give non-trivial applications of this formalism.Comment: 22 pages, 2 figure

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

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

Sur les produits canoniques d'ordre infini

Mode of access: Internet