Topological Spaces with Low Reflexivity Indices

We determine the reflexivity indices of certain topological spaces. The reflexivity index is a measure of certain type of dynamical complexity of the space. The reflexivity index is 1 exactly when there is a continuous selfmap on the space so that the orbit at each point is dense in the space. Spaces with index 1 include the rational numbers, the irrational numbers, an n-torus with 1≤n≤2ℵ0, and X∖0 where X is any infinite-dimensional separable Banach space. The index of 0,1 and Rn (n∈N) is 2.