Orbits of homogeneous polynomials on Banach spaces

Abstract

We study the dynamics induced by homogeneous polynomials on Banach spaces. It is known that no homogeneous polynomial defined on a Banach space can have a dense orbit. We show, a simple and natural example of a homogeneous polynomial with an orbit that is at the same time d-dense (the orbit meets every ball of radius d), weakly dense and such that Γ · OrbP (x) is dense for every Γ ⊂ C that is either unbounded or that has 0 as an accumulation point. Moreover we generalize the construction to arbitrary infinite dimensional separable Fréchet spaces. To prove this we study Julia sets of homogeneous polynomials on Banach spaces.However, the behavior of the orbits induced by a homogeneous polynomial can be highly nontrivial and it is far from being understood. For example, in [4] Bernardes showed that the orbits may oscillate between infinity and the boundary of the limit ball. He also proved that every infinite dimensional and separable Banach space supports supercyclic homogeneous polynomials. More recently Peris, Kim and Song [16,17] proved that every separable Banach space of dimension greater than one supports numerically hypercyclic homogeneous polynomials. This means that there are vectors x ∈ S X , x * ∈ S X * for which its numerical orbit, N orb P (x, x * ) := {x * (P n (x)) : n ∈ N 0 } is dense in C.The following result shows that the Julia set of a homogeneous polynomial share some of the properties satisfied by the Julia set of a holomorphic function on the complex plane.Proposition 2.12 (Properties of J P ). Let P be a non linear homogeneous polynomial. The Julia set J P satisfies the following properties:Proof. i) This is clear since J P is the boundary of an open set.ii) First note that y ∈ A P if and only if P (y) ∈ A P . Thus, if x ∈ J P then P (x) is not in A P .On the other hand, there exists (a n ) n ⊆ A P such that a n → x. Since P (a n ) belongs also to A P and P (a n ) → P (x), we have that P (x) ∈ J P .iii) Just note that if x ∈ J P then λx ∈ J P for every |λ| = 1.iv) It suffices to show that A P n = A P . Clearly if P k (x) → 0 then (P n ) k (x) → 0 and therefore A P ⊆ A P n . The converse follows thanks to the existence of the limit ball (Proposition 2.1). If P nk (x) → 0 then Orb P n (x) meets eventually the limit ball of P and therefore P n (x) must tend to zero.In contrast to the one dimensional case, the Julia set J P may be empty. Indeed, if P ∈ P( 2 ℓ 2 ; ℓ 2 ) is defined as P (x) = (x 2 2 , x 2 3 , x 2 4 , . . .), then P n (x) → 0 for every x ∈ ℓ 2 . Thus A P = ℓ 2 and hence J P = R P = ∅.Recall that a set A is completely invariant under P if P (A) ⊆ A and P −1 (A) ⊆ A. The Julia set of a homogeneous polynomials need not to be completely invariant, as we will show in Example 3.19. On the other hand, under certain conditions on P , J P results completely invariant.have that P n (x) → 0 and hence P (x) ∈ A P . If R P = ∅ this implies that x ∈ J P . If R P = ∅ and x ∈ R P , there is some ǫ > 0 so that P n (y) → ∞ for every y ∈ B ǫ (x).Since P is open, P (B ǫ (x)) is an open neighborhood of...

Hypercyclic bilinear operators on Banach spaces

“…It remains to see that Φ(z) is well defined and that z is universal. The well definition of Φ(z) is deduced from (7).…”

“…This lemma was a key result to prove the existence of hypercyclic operators on arbitrary separable infinite dimensional Fréchet spaces. The version we will apply was proved in [7,Lemma 4.3], where it was needed to show the existence of homogeneous polynomials with special dynamics. The main difference with the original version is that we provide a control of the asymptotic behavior of the sequence α(n) = x * n (x n ).…”

“…We will show that if P is the homogeneous polynomial induced by M , P (x) = M (x, x) = e ′ 1 (x)B ω (x), then there is some vector x for which {M (P n (x), P m (x)) : n, m ∈ N} = ℓ 1 . This polynomial P was studied in [7]. We will need the following definitions and results, which were posed in [7].…”

“…This polynomial P was studied in [7]. We will need the following definitions and results, which were posed in [7]. Given a homogeneous polynomial Q acting on a Banach space it is worth considering its Julia set J Q = ∂{x : lim Q n (x) = 0}.…”

“…Notice that if X is a Banach space, then no m-linear operator M can be strongly transitive. Indeed, in the same way as in the case of homogeneous polynomials [3,7] and of bihypercyclic operators [10], it is possible to define a notion of limit ball: if…”