Submitted

with Paul Bourdon

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**.

A set F in a topological space is somewhere
dense if its closure has non-empty interior.

In this paper we prove the following theorem:

**Main Theorem:** If T is a bounded linear operator
on a Banach space and if there is a vector whose orbit is somewhere dense,
then that orbit is actually dense.

An operator is T is finitely-hypercyclic if there are a finite number of orbits whose union is dense. That is, if there are vectors {x1, x2,...,xn} such that the union of Orb(T,x1), Orb(T,x2),...,Orb(T,xn) is dense. D. Herrero conjectured in 1991 that a multi-hypercyclic operator was actually hypercyclic. In 1999 Alfredo Peris proved Herrero's conjecture and raised the question of whether or not an operator with a somewhere dense orbit must be hypercyclic. An answer to this question gives an immediate proof of Herrero's conjecture.

Thus our main Theorem immediately implies Herrero's conjecture
as well as a result due to S. Ansari [1993]
that says if T is hypercyclic then so is T^n. Furthermore T and T^n
have the same set of hypercyclic vectors.

**Corollary:** Suppose T is a bounded linear
operator on a Banach space.

(a) [Peris, 1999] If T is finitely-hypercyclic,
then T is hypercyclic.

(b) [Ansari, 1993] If T is hypercyclic, then T^n
is also hypercylic and T and T^n have the same set of hypercyclic vectors.