Some universality results for dynamical systems

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We prove some "universality" results for topological dynamical systems. In particular, we show that for any continuous self-map T of a perfect Polish space, one can find a dense, T-invariant set homeomorphic to the Baire space Nⁿ; that there exists a bounded linear operator U: ℓ₁ → ℓ₁ such that any linear operator T from a separable Banach space into itself with T≤ 1 is a linear factor of U; and that given any σ-compact family F of continuous self-maps of a compact metric space, there is a continuous self-map UF of Nⁿ such that each T∈ F is a factor of UF.