Dense Orbit Research Articles

Large subspaces of compositionally universal functions with maximal cluster sets

Let ( φ n ) be a sequence of holomorphic self-maps of a Jordan domain G in the complex plane. Under appropriate conditions on ( φ n ) , we construct an H ( G ) -dense linear manifold–as well as a closed infinite-dimensional linear manifold–all of whose non-zero functions have H ( G ) -dense orbits under the action of a sequence of composition operators associated with ( φ n ) . Simultaneously, these functions also present maximal cluster sets along each member of a large class of curves in G tending to the boundary.

Density and equidistribution of half-horocycles on a geometrically finite hyperbolic surface

On geometrically finite negatively curved surfaces, we give necessary and sufficient conditions for a one-sided horocycle (hsu)s⩾0 to be dense in the nonwandering set of the horocyclic flow. We prove that all dense one-sided orbits (hsu)s⩾0 are equidistributed, extending results of Burger [‘Horocycle flows on geometrically finite surfaces’, Duke Math. J. 61 (1990) 779–803] and Schapira [‘Equidistribution of the horocycles of a geometrically finite surface’, Int. Math. Res. Not. 40 (2005) 2447–2471] where symmetric horocycles (hsu)−R⩽s⩽R were considered.

On weak mixing, minimality and weak disjointness of all iterates

AbstractThis article addresses some open questions about the relations between the topological weak mixing property and the transitivity of the map f×f2×⋯×fm, where f:X→X is a topological dynamical system on a compact metric space. The theorem stating that a weakly mixing and strongly transitive system is Δ-transitive is extended to a non-invertible case with a simple proof. Two examples are constructed, answering the questions posed by Moothathu [Diagonal points having dense orbit. Colloq. Math. 120(1) (2010), 127–138]. The first one is a multi-transitive non-weakly mixing system, and the second one is a weakly mixing non-multi-transitive system. The examples are special spacing shifts. The latter shows that the assumption of minimality in the multiple recurrence theorem cannot be replaced by weak mixing.

Partially hyperbolic sets with positive measure and $ACIP$ for partially hyperbolic systems

In [23] Xia introduced a simple dynamical density basis forpartially hyperbolic sets of volume preserving diffeomorphisms. Weapply the density basis to the study of the topological structure ofpartially hyperbolic sets. We show that if $\Lambda$ is a stronglypartially hyperbolic set with positive volume, then $\Lambda$contains the global stable manifolds over ${\alpha}(\Lambda^d)$ andthe global unstable manifolds over ${\omega}(\Lambda^d)$.  &nbspWe give several applications of the dynamical density to partiallyhyperbolic maps that preserve some $acip$. We show that if $f$ isessentially accessible and $\mu$ is an $acip$ of $f$, then$\text{supp}(\mu)=M$, the map $f$ is transitive, and $\mu$-a.e.$x\in M$ has a dense orbit in $M$. Moreover if $f$ is accessible andcenter bunched, then either $f$ preserves a smooth measure or thereis no $acip$ at all.

Semigroups of matrices with dense orbits

We give examples of n × n matrices A and B over the field 𝕂 = ℝ or ℂ such that for almost every column vector x ∈ 𝕂 n , the orbit of x under the action of the semigroup generated by A and B is dense in 𝕂 n .

On the complement of the Richardson orbit

We consider parabolic subgroups of a general linear group over an algebraically closed field k whose Levi part has exactly t factors. By a classical theorem of Richardson, the nilradical of a parabolic subgroup P has an open dense P-orbit. In the complement to this dense orbit, there are infinitely many orbits as soon as the number t of factors in the Levi part is ≥6. In this paper, we describe the irreducible components of the complement. In particular, we show that there are at most t − 1 irreducible components. We are also able to determine their codimensions.

On the transitivity of multifunctions and density of orbits in generalized topological spaces

The connection between transitivity and existence of a dense orbit for multifunctions φ : X X in generalized topological spaces is studied. Moreover strongly transitive multifunctions and functions in generalized topolog- ical spaces are investigated.

A Frobenius theorem for Cartan geometries, with applications

We prove analogues for Cartan geometries of Gromov's major theorems on automorphisms of rigid geometric structures. The starting point is a Frobenius theorem, which says that infinitesimal automorphisms of sufficiently high order integrate to local automorphisms. Consequences include a stratification theorem describing the configuration of orbits for local Killing fields in a compact real-analytic Cartan geometry, and an open-dense theorem in the smooth case, which says that if there is a dense orbit, then there is an open, dense, locally homogeneous subset. Combining the Frobenius theorem with the embedding theorem of Bader, Frances, and the author gives a representation theorem that relates the fundamental group of the manifold with the automorphism group.

On geometric properties of orbital varieties in type A

The intersection between a nilpotent orbit O ⊂ gl n ( C ) and the Lie algebra Lie ( B ) ⊂ gl n ( C ) of a Borel subgroup B ⊂ GL n ( C ) is an equidimensional, quasi-affine algebraic variety. Its irreducible components are called orbital varieties. In this Note, we provide criteria to guarantee that an orbital variety is smooth or has a dense orbit for the adjoint action of B. In addition, we point out a possible relation between these two properties.

Stable topological transitivity properties of ℝn-extensions of hyperbolic transformations

AbstractWe consider ℝn skew-products of a class of hyperbolic dynamical systems. It was proved by Niţică and Pollicott [Transitivity of Euclidean extensions of Anosov diffeomorphisms. Ergod. Th. & Dynam. Sys. 25 (2005), 257–269] that for an Anosov diffeomorphism ϕ of an infranilmanifold Λ there is (subject to avoiding natural obstructions) an open and dense set f:Λ→ℝN for which the skew-product ϕf(x,v)=(ϕ(x),v+f(x)) on Λ×ℝN has a dense orbit. We prove a similar result in the context of an Axiom A hyperbolic flow on an attractor.

Chaotic and topological properties of β-transformations

In this paper, we prove that the β-transformations are chaotic in the sense of both Li–Yorke and Devaney. The topological and metric properties of the sets of points with dense or non-dense orbits are investigated. We also prove the result that the set of points with non-dense orbits under the β-transformation is of full Hausdorff dimension for any β > 1 .

Rigid geometric structures, isometric actions, and algebraic quotients

By using a Borel density theorem for algebraic quotients, we prove a theorem concerning isometric actions of a Lie group G on a smooth or analytic manifold M with a rigid A-structure σ. It generalizes Gromov’s centralizer and representation theorems to the case where R(G) is split solvable and G/R(G) has no compact factors, strengthens a special case of Gromov’s open dense orbit theorem, and implies that for smooth M and simple G, if Gromov’s representation theorem does not hold, then the local Killing fields on $${\widetilde{M}}$$ are highly non-extendable. As applications of the generalized centralizer and representation theorems, we prove (1) a structural property of Iso(M) for simply connected compact analytic M with unimodular σ, (2) three results illustrating the phenomenon that if G is split solvable and large then π 1(M) is also large, and (3) two fixed point theorems for split solvable G and compact analytic M with non-unimodular σ.

On the problem of quantum control in infinite dimensions

In the framework of bilinear control of the Schrödinger equation, it has been proved that the reachable set has a dense complement in . Hence, in this setting, exact quantum control in infinite dimensions is not possible. On the other hand, it is known that there is a simple choice of operators which, when applied to an arbitrary state, generate dense orbits in Hilbert space. Compatibility of these two results is established in this paper and, in particular, it is proved that the closure of the reachable set of bilinear control is dense in . The requirements for controllability in infinite dimensions are also related to the properties of the infinite-dimensional unitary group.

Global Structure of Quaternion Polynomial Differential Equations

In this paper we mainly study the global structure of the quaternion Bernoulli equations $\dot q=aq+bq^n$ for $q\in \mathbb H$ the quaternion field and also some other form of cubic quaternion differential equations. By using the Liouvillian theorem of integrability and the topological characterization of $2$--dimensional torus: orientable compact connected surface of genus one, we prove that the quaternion Bernoulli equations may have invariant tori, which possesses a full Lebesgue measure subset of $\mathbb H$. Moreover, if $n=2$ all the invariant tori are full of periodic orbits; if $n=3$ there are nfiinitely many invariant tori fulfilling periodic orbits and also infinitely many invariant ones fulfilling dense orbits.

Minimal sets of non-resonant torus homeomorphisms

As was known to H. Poincare, an orientation preserving circle homeomorphism without periodic points is either minimal or has no dense orbits, and every orbit accumulates on the unique minimal set. In the first case the minimal set is the circle, in the latter case a Cantor set. In this paper we study a two-dimensional analogue of this classical result: we classify the minimal sets of non-resonant torus homeomorphisms; that is, torus homeomorphisms isotopic to the identity for which the rotation set is a point with rationally independent irrational coordinates.

On indecomposable trees in the boundary of outer space

Let T be an $${\mathbb{R}}$$ -tree, equipped with a very small action of the rank n free group F n , and let H ≤ F n be finitely generated. We consider the case where the action $${F_n \curvearrowright T}$$ is indecomposable–this is a strong mixing property introduced by Guirardel. In this case, we show that the action of H on its minimal invarinat subtree T H has dense orbits if and only if H is finite index in F n . There is an interesting application to dual algebraic laminations; we show that for T free and indecomposable and for H ≤ F n finitely generated, H carries a leaf of the dual lamination of T if and only if H is finite index in F n . This generalizes a result of Bestvina-Feighn-Handel regarding stable trees of fully irreducible automorphisms.

On the Lebesgue Measure of Li-Yorke Pairs for Interval Maps

We investigate the prevalence of Li-Yorke pairs for C 2 and C 3 multimodal maps f with non-flat critical points. We show that every measurable scrambled set has zero Lebesgue measure and that all strongly wandering sets have zero Lebesgue measure, as does the set of pairs of asymptotic (but not asymptotically periodic) points. If f is topologically mixing and has no Cantor attractor, then typical (w.r.t. two-dimensional Lebesgue measure) pairs are Li-Yorke; if additionally f admits an absolutely continuous invariant probability measure (acip), then typical pairs have a dense orbit for f × f. These results make use of so-called nice neighborhoods of the critical set of general multimodal maps, and hence uniformly expanding Markov induced maps, the existence of either is proved in this paper as well. For the setting where f has a Cantor attractor, we present a trichotomy explaining when the set of Li-Yorke pairs and distal pairs have positive two-dimensional Lebesgue measure.

Quiver representations with an irreducible null cone

Let d be a prehomogeneous dimension vector for a quiver Q. There is an action of the product Gl ( d ) of linear groups on the vector space rep ( Q , d ) of representations of Q with dimension vector d, and there is a representation T with a dense Gl ( d ) -orbit in rep ( Q , d ) . We give a construction for a dense subset F Q , d of the variety Z Q , d of common zeros of all semi-invariants in k [ rep ( Q , d ) ] of positive degree, and we show that this set is stable for big dimension vectors, i.e. F Q , N ⋅ d = { X ⊕ T N − 1 : X ∈ F Q , d } . Moreover, we show that the existence of a dense orbit in Z Q , d depends on a quiver Q ⊥ such that the category of representations of Q ⊥ is equivalent to the right perpendicular category T ⊥ .

Hypercyclictty and Countable Hypercyclicity for Adjoint of Operators

Let be an infinite dimensional separable complex Hilbert space and let , where is the Banach algebra of all bounded linear operators on . In this paper we prove the following results. If is a operator, then 1. is a hypercyclic operator if and only if D and for every hyperinvariant subspace of . 2. If is a pure, then is a countably hypercyclic operator if and only if and for every hyperinvariant subspace of . 3. has a bounded set with dense orbit if and only if for every hyperinvariant subspace of , .

POLYNOMIAL DYNAMIC AND LATTICE ORBITS IN S-ARITHMETIC HOMOGENEOUS SPACES

Consider a homogeneous space under a locally compact group G and a lattice Γ in G. Then the lattice naturally acts on the homogeneous space. Looking at a dense orbit, one may wonder how to describe its repartition. One then adopts a dynamical point of view and compare the asymptotic distribution of points in the orbits with the natural measure on the space. In the setting of Lie groups and their homogeneous spaces, several results show an equidistribution of points in the orbits. We address here this problem in the setting of p-adic and S-arithmetic groups.