Linear (topological) dynamics studies the orbits of the action of

a semigroup of continuous linear operators on a topological vector space.

We are mostly interested in chaotic-type properties like the existence of a

dense orbit.

The aim of this talk is two-fold: to give an overview of the current state

of (chaotic) linear dynamics and to describe few recent results of the

author. One of the results, we are going to describe boils down to the

construction of a (non-unital) Banach algebra $A$ and an element $a$ of

$A$ such that both the set of scalar multiples of powers of $a$ and the set

$\{a(1+a)^n:n=1,2,...\}$ are dense in $A$. Apart from providing solutions

to a couple of open problems in linear dynamics, this example generates a

very interesting Banach algebra. In particular, $A$ is radical, generated

by one element and weakly amenable.

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