Hypercyclicity and subspace hypercyclicity in finite dimension

Abstract

We consider the remaining problem which claims that if A is a dense subset of a finite dimensional space X, then there is a nontrivial subspace M of X such that A ∩ M is dense in M . We show that the above problem has a negative answer when X = K n (K = R or C) for every n ≥ 2. However we answer positively the problem when the dense subset A is an orbit of an abelian semigroup of matrices on K n , K n = R 2 .