Hypercyclic orbits intersect subspaces in wild ways

Let T:X→XT:X→X be a hypercyclic operator of an infinite dimensional separable Banach space X. By modifying a construction of Grivaux, we will show that the T-orbit of a hypercyclic vector can intersect certain closed vector subspaces of X in many strange ways. Moreover, the set of visiting times of the orbit to the subspace can also be quite exotic, especially when the operator satisfies a stronger form of hypercyclicity. As a by-product, we will improve and/or supply new proofs to some of the recent results about subspace-hypercyclicity.