Frequently dense orbits

We study the notion of frequent hypercyclicity that was recently introduced by Bayart and Grivaux. We show that frequently hypercyclic operators satisfy the Hypercyclicity Criterion, answering a question of Bayart and Grivaux [Trans. Amer. Math. Soc., in press]. We also disprove a conjecture therein concerning frequently hypercyclic weighted shifts, and we prove that vectors which have a somewhere frequently dense orbit are frequently hypercyclic. To cite this article: K.-G. Grosse-Erdmann, A. Peris, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

Linear semigroups with coarsely dense orbits

Let S be a finitely generated abelian semigroup of invertible linear operators on a finite-dimensional real or complex vector space V. We show that every coarsely dense orbit of S is actually dense in V. More generally, if the orbit contains a coarsely dense subset of some open cone C in V, then the closure of the orbit contains the closure of C. In the complex case the orbit is then actually dense in V. For the real case we give precise information about the possible cases for the closure of the orbit.     

Area preserving analytic flows with dense orbits

The aim of this paper is to give sufficient conditions on area-preserving flows that guarantee the existence of dense orbits. We also answer a question by M.D. Hirsch [M.D. Hirsch, Dense recurrence in area-preserving flows on surfaces, Nonlinearity 12 (1999) 1545-1553]. The results of this work are a generalization of the ones in [M.D. Hirsch, Dense recurrence in area-preserving flows on surfaces, Nonlinearity 12 (1999) 1545-1553] and [H. Marzougui, Area preserving flows with a dense orbit, Nonlinearity 15 (2002) 1379-1384].     

Ordered differential fields

We introduce the abstract notion of an ordered differential field and show that some of the basic asymptotic growth properties of differentiable functions can be discussed within this setting. However, in order to ensure that log x → ∞ as x → ∞, we have to assume that the field of constants is archimedean.     

Ordered differential fields

The paper is devoted to the study of a new algebraic structure - an oredred differential field. The structure is a differential field and an ordered field, the order being connected with a derivation. This paper is the first one in a series devoted to new algebraic methods in analysis. The essence of the methods is to exploit the connection between a linear order and a derivation for the study of function sets and operators on these sets.     

On operators with orbits dense relative to nontrivial subspaces

In the present paper we consider bounded linear operators which have orbits dense relative to nontrivial subspaces. We give nontrivial examples of such operators and establish many of their basic properties. An example of an operator which has an orbit dense relative to a certain subspace but is not subspace-hypercyclic for this subspace is given. This, in turn, provides a new answer to a question posed in [18]. Other hypercyclic-like properties of such operators are also considered.     

Hypercyclic tuples of operators and somewhere dense orbits

In this paper we prove that there are hypercyclic (n+1)-tuples of diagonal matrices on Cn and that there are no hypercyclic n-tuples of diagonalizable matrices on Cn. We use the last result to show that there are no hypercyclic subnormal tuples in infinite dimensions. We then show that on real Hilbert spaces there are tuples with somewhere dense orbits that are not dense, but we also give sufficient conditions on a tuple to insure that a somewhere dense orbit, on a real or complex space, must be dense.     

Periodic billiard orbits are dense in rational polygons

We show that periodic orbits are dense in the phase space for billiards in polygons for which the angle between each pair of sides is a rational multiple of

     

Foliated vector fields without periodic orbits

We investigate the behavior of the divisor function in both short intervals and in arithmetic progressions. The latter problem was recently studied by É. Fouvry, S. Ganguly, E. Kowalski and Ph. Michel. We prove a complementary result to their main theorem. We also show that in short intervals of certain lengths the divisor function has a Gaussian limiting distribution. The analogous problems for Hecke eigenvalues are also considered.     

Ordered tournaments and ordered triplewhist tournaments with the three person property

It is well known that an ordered tournament OWh(v) exists if and only if v ≡ 1 (mod 4), v ≥ 5. An ordered triplewhist tournament on v players is said to have the three person property if no two games in the tournament have three common players. We briefly denote such a design as a 3POTWh(v). In this article, we show that a 3POTWh(v) exists whenever v>17 and v ≡ 1 (mod 4) with few possible exceptions. We also show that an ordered whist tournament on v players with the three person property, denoted 3POWh(v), exists if and only if v ≡ 1 (mod 4), v ≥ 9. © 2008 Wiley Periodicals, Inc. J Combin Designs 17: 39–52, 2009     

A minimizing property of hyperbolic Keplerian orbits

In this short note, we characterize hyperbolic Keplerian orbits as minimizing paths of the Keplerian action functional in the space of curves from a ray emanating from the attractive focus to a point in space. Variants of this result have been previously proved by different methods. Our proof based on hyperbolic anomaly is simple and informative.     

Rationality of singularities and the Gorenstein property for nilpotent orbits

Ordzhonikidze Aviation Institute, Moscow. Translated from Funktsional'yi Analiz i Ego Prilozheniya, Vol. 25, No. 3, pp. 76–78, July–September, 1991.     

Nilpotent orbits over ground fields of good characteristic

Let X be an F-rational nilpotent element in the Lie algebra of a connected and reductive group G defined over the ground field F. Suppose that the Lie algebra has a non-degenerate invariant bilinear form. We show that the unipotent radical of the centralizer of X is F-split. This property has several consequences. When F is complete with respect to a discrete valuation with either finite or algebraically closed residue field, we deduce a uniform proof that G(F) has finitely many nilpotent orbits in (F). When the residue field is finite, we obtain a proof that nilpotent orbital integrals converge. Under some further (fairly mild) assumptions on G, we prove convergence for arbitrary orbital integrals on the Lie algebra and on the group. The convergence of orbital integrals in the case where F has characteristic 0 was obtained by Deligne and Ranga Rao (1972).