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].     

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

     

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.     

Frequently hypercyclic operators

We investigate the subject of linear dynamics by studying the notion of frequent hypercyclicity for bounded operators on separable complex -spaces: is frequently hypercyclic if there exists a vector such that for every nonempty open subset of , the set of integers such that belongs to has positive lower density. We give several criteria for frequent hypercyclicity, and this leads us in particular to study linear transformations from the point of view of ergodic theory. Several other topics which are classical in hypercyclicity theory are also investigated in the frequent hypercyclicity setting.

     

Frequently hypercyclic subspaces

We study the existence of frequently hypercyclic subspaces for a given operator, that is, the existence of closed infinite-dimensional subspaces in which every non-zero vector is frequently hypercyclic. We attack the problem with any of the three methods that have been used for hypercyclic subspaces: a constructive approach, an approach via left-multiplication operators, and an approach via tensor products.     

Periodic orbits from nonperiodic orbits on an interval

This paper demonstrates that any continuous real-valued function which has an orbit with infinitely many limit points must necessarily have periodic cycles of arbitrarily large prime period. We present an example of a function with an orbit whose limit points are exactly Z⁺.     

On co-σ-porosity of the parameters with dense critical orbits for skew tent maps and matching on generalised β-transformations

We prove that the critical point and the point 1 have dense orbits for Lebesgue-a.e., parameter pairs in the two-parameter skew tent family and generalised $\beta$-transformations. As an application, we show that for the generalised $\beta$-transformation with the tribonacci number as slope, there is matching (i.e., $T^n\left(0\right)=T^n\left(1\right)$ for some $n\ge 1$) for Lebesgue-a.e. translation parameter.     

Quiver-graded Richardson orbits

In Lie theory, a dense orbit in the nilpotent radical of a parabolic group under the operation of the parabolic is called a Richardson orbit. We define a quiver-graded version of Richardson orbits generalizing the classical definition in the case of the general linear group. We define a quasi-hereditary algebra called the nilpotent quiver algebra whose isomorphism classes of Δ-filtered modules correspond to orbits in our generalized setting. We translate the existence of a Richardson orbit into the existence of a rigid Δ-filtered module of a given dimension vector. We study an idempotent recollement of this algebra whose associated intermediate extension functor can be used to produce Richardson orbits in some situations. This can be explicitly calculated in examples. We also give examples where no Richardson orbit exists.     

Maximal semigroup orbits

In this paper we show that certain actions are the a-homomorphic images of actions with disjoint maximal orbits. We also show that if T is a compact semigroup acting on a compact space X, then there is a compact right trivial semigroup Y and a congruenceN of T×Y such that (T×Y)/N homeomorphic to X and the action T×X→X is isomorphic the action of T on (T×Y)/N.     

Group actions and orbits

Let A be a group acting via automorphisms on a group G, and let Ω be the set of orbits in this action. Then C = C _G (A) acts on Ω in a natural manner, and using this action, we deduce some divisibility information about |Ω|.     

Periodic orbits and expansiveness

We prove that if a local diffeomorphism has expanding periodic points robustly then it is an expanding map. Using this, we reobtain a result due to Sakai: generic positively expansive maps are expanding. Our methods also show a global version of a result by Gan and Yang: generic expansive diffeomorphisms are Axiom A without cycles.     

Closure of conical orbits

Translated from Matematicheskie Zametki, Vol. 56, No. 4, pp. 153–155, October, 1994.