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Perturbations of Hypercyclic Vectors and Lattice-Like Orbits

J. Math Analysis & Appl. 273 No. 1(2002), 67-74

A bounded linear operator T is hypercyclic if there is a vector with dense orbit; so if T acts on a Banach space X, then T is hypercyclic if there is a vector x in X such that Orb(T,x) = {x, Tx, T^2x, ...} is dense in X.  This paper addresses questions of the following type:  if x has a dense orbit and the orbit of y is "small", then does (x+y) have dense orbit?

In this paper we prove the following theorems:

Theorem 1:  Suppose that T is a bounded linear operator on a Banach space X.  If there is a d > 0 and a vector whose orbit comes within a distance d of every point in X, then T is hypercyclic.

Theorem 1 allows us to give a more general form of the well known Hypercyclicity Criterion.  While the more general form is equivalent to the Hypercyclicity Criterion, the more general form may be easier to apply in certain cases.

Theorem 2:  Given an $\epsilon > 0$, there exists a vector x whose orbit under twice the backward shift comes within a distance $\epsilon$ of every point, yet the orbit of x is not dense.

Corollary:  For each $\epsilon > 0$, there exists vectors x and y such that x has a dense orbit, and $\|T^nx - T^ny\| < \epsilon$ for all n, yet the orbit of y is not dense.

Theorem 3: If T is a bounded linear operator on a Banach space X and x is a vector with dense orbit under T and y is a vector satisfying that the closure of Orb(T,y) is countable and compact, then (x+y) also has dense orbit under T.