Weighted U-frequent hypercyclicity

Weighted

U

-frequent hypercyclicity

Céline ESSER [email protected]

University of Liège – Institute of Mathematics

Lens – October 24, 2018

Joint work with

Introduction and statement of the problem

LetT : X → X be a linear operator (X separable Banach space,dim X = ∞).

• TisU-frequently hypercyclic (ufhc)if∃x ∈ X such that∀U 6= ∅open

d {n ≥ 0 : Tnx ∈ U }
> 0 .

The set of such points is denotedUFHC(T )(Shkarin 2009).

• Tisreiteratively hypercyclic (rhc)if∃x ∈ Xsuch that∀U 6= ∅open

Bd {n ≥ 0 : Tnx ∈ U }
> 0 .

For_{E ⊆ N}, the densities used are

• the upper densityd(E) = lim sup n→+∞

# E ∩ [0, n]

n + 1 • the upper Banach densityBd(E) = lim

k→+∞lim supn→+∞

# E ∩ [n, n + k]

k + 1

One hasd(E) ≤ Bd(E)hence

T ufhc =⇒ T rhc

but one can prove thatTrhc _{; T}ufhc (Bès, Menet, Peris, Puig 2016)

Weighted upper densities

Leta = (an)_n∈Nbe a sequence of positive real numbers such thatP

n∈Nan= +∞.

Thea-upper densityof a set_{E ⊆ N}is defined by

da(E) = lim sup

n→+∞ n X k=0 ak₁E(k) n X k=0 ak

An operatorTisa-upper frequently hypercyclic (ufhca)if∃x ∈ X such that∀U 6= ∅

open,

da {n ≥ 0 : Tn_{x ∈ U }
> 0 .}

The set of such points is denotedUFHCa(T )and is a residual set

Remarks

• If an

bn & 0, thenda≤ db. Hence ufhca⇒ufhcb (Ernst, Mouze 2017)

• Ifan= 1for all_{n ∈ N}, thenda= d. Hence ufhca⇔ufhc

• Letνn= an

Pn−1

k=0ak

. If there isC > 0such thatνn ≥ Cfor all_{n ∈ N}, then

da(E) > 0as soon as#E = +∞

Consequences

• (H1) We consider sequencesa = (an)n∈Nsuch thatan % +∞ • (H2) We impose thatνn& 0

Example 1

For anyα > 0, the sequence(nα)_n∈Nbelongs toA:

• nα_{% +∞} • νn= Pn−1nα k=1kα ∼ nα nα+1 & 0

Example 2

For anyα > 1, the sequence(αn)_n∈Ndoes not belong toA:

νn= αn Pn−1 k=1αk = α n α−αn 1−α = (1 − α) 1 α1−n_{− 1} → α − 1

Comparison of the densities

Result

• For everya ∈ A, one hasd ≤ da.

• For everyα > 0, the sequencea = (nα₎

n∈N∈ Asatisfies 1

1 + αda≤ d ≤ da.

Consequence

For every operatorT, there isa ∈ Asuch thatUFHC(T ) = UFHCa(T ). In particular,

UFHC(T ) = \

a∈A

UFHCa(T )

and

Result

• For everya ∈ A, one hasda≤ Bd.

• Let(Ek)_k∈Nbe a sequence of subsets of_N. There existsa ∈ Asuch that

Bd(Ek) ≤ e da(Ek) ∀k ∈ N .

Consequence

For every operatorT,

RHC(T ) = [

a∈A

UFHCa(T )

and

IfT is rhc, thenRHC(T ) = HC(T )(Bès, Menet, Peris, Puig 2016)

Result

For everya ∈ A, one hasUFHCa(2B) 6= HC(2B).

However, ifT is rhc, then

HC(T ) = [

a∈A

UFHCa(T )

Chaos and ufhc

One has(Menet 2017) • chaos⇒rhc

• chaos;ufhc

Result

For everya ∈ A, there is a chaotic operator on`1_{which is not ufhc} a.

Product of operators

Question: Given an operatorTwhich is ufhc / ufhca/ rhc, what can we say about the

operatorT × · · · × T?

Result

IfT is ufhca, thenT × · · · × T is ufhca.

Keys:(using ideas from Bayart, Rusza 2015&Bès, Menet, Peris, Puig 2016) • ufhca⇒weakly-mixing

• Ifda(E) > 0, then there isδ > 0such that the set

k ≥ 0 : da E ∩ (E − k)) > δ

is syndetic

Consequence

Main references

F. Bayart and I. Z. Ruzsa

Difference sets and frequently hypercyclic weighted shifts

Ergod. Theory Dyn. Syst.,35, 691–709, 2015.

J. Bès, Q. Menet, A. Peris and Y. Puig

Recurrence properties of hypercyclic operators

Math. Ann.366, 545–572, 2016.

A. Bonilla and K-G. Grosse-Erdmann

Upper frequent hypercyclicity and related notions

Rev. Mat. Complut.,31, 673–711, 2018.

R. Ernst and A. Mouze

A quantitative interpretation of the frequent hypercyclicity criterion

Ergod. Theory Dyn. Syst., https://doi.org/10.1017/etds.2017.55, 2017.

Q. Menet

Linear chaos and frequent hypercyclicity

Trans. Amer. Math. Soc.369, 4977–4994, 2017.

S. Shkarin

On the spectrum of frequently hypercyclic operators