I'm studying some topics related to the invariant subspace problem, and consequently I find myself dealing with hypercyclic operators.
(An operator $T:X\rightarrow X$ is hypercyclic if there is some vector $x\in X$ such that the set $\left\{ x,Tx,T^{2}x,...\right\}$ is dense in $X$, $x$ is then called a hypercyclic vector)
So I'm trying to find references on spectral properties of these kind of operators, and how it depends on the iterates of the hypercyclic vectors.
My first intuition is that the spectrum of a hypercyclic operator is always connected, so if anyone can tell me something about this suspicion or any other related fact I will be very grateful.
Thank you !