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We examine common supercyclic vectors for a path of operators. In particular, we show that the path consisting of convex combinations of two arbitrary unilateral weighted backward shifts has a dense G δ set of common supercyclic vectors. Moreover, we show there exists a path with a dense G δ set of common supercyclic vectors between a unilateral weighted backward shift which satisfies the Supercyclicity Criterion, and an operator which does not. Lastly, we provide an example of a path of unilateral weighted backward shifts that fails to have a common supercyclic vector.

Using the techniques of the hypercyclicity criterion, we prove that there is a meromorphic function f ( z ) on the complex plane whose translates f(z + n) for all n 1 1, are dense in the metric space of meromorphic functions on any region in the plane. In addition, we prove the analogue of the result for non-Euclidean translation on the unit disk.

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