In the paper we study what sets can be obtained as $\alpha$-limit of backward trajectories in graph maps. We show that case mixing maps, all those are $\omega$-limit and for but finitely many points $x$, obtain every $\omega$-limits set a trajectory starting $x$. For zero entropy is minimal set. maps with positive entropy, partial characterization which very close to complete picture possible s...

We consider the sequence of fractional parts {ξαn}, n = 1, 2, 3, . . . , where α > 1 is a Pisot number and ξ ∈ Q(α) is a positive number. We find the set of limit points of this sequence and describe all cases when it has a unique limit point. The case, where ξ = 1 and the unique limit point is zero, was earlier described by the author and Luca, independently.

Let N denote the set of positive integers. The asymptotic density of the set A ⊆ N is d(A) = limn→∞ |A ∩ [1, n]|/n, if this limit exists. Let AD denote the set of all sets of positive integers that have asymptotic density, and let SN denote the set of all permutations of the positive integers N. The group L consists of all permutations f ∈ SN such that A ∈ AD if and only if f(A) ∈ AD, and the g...

First a set of coherent statesá la Klauder is formally constructed for the Coulomb problem in a curved space of constant positive curvature. Then the flat-space limit is taken to reduce the set for the radial Coulomb problem to a set of hydrogen atom coherent states corresponding to both the discrete and the continuous portions of the spectrum for a fixed ℓ sector.

We study parabolic iterated function systems with overlaps on the real line. We show that if a d-parameter family of such systems satisfies a transversality condition, then for almost every parameter value the Hausdorff dimension of the limit set is the minimum of 1 and the least zero of the pressure function. Moreover, the local dimension of the exceptional set of parameters is estimated. If t...