Hypercyclic, topologically mixing and chaotic semigroups on Banach spaces

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Hypercyclic, mixing, and chaotic C0-semigroups induced by semiflows (2007, unknown): “Bermúdez et al showed in [1] that the translation-semigroup is mixing on L p ρ (I ) or C0,ρ(I ), respectively, where I ∈ {[0,∞), R}, if and only if limt→±∞ ρ(t)= 0.”
Mixing operators and small subsets of the circle (2011, background): “(t) := sup x∈R+ ρ(x) ρ(x+t) < ∞ for any t ≥ 0 and C(t) is locally bounded on R+. Then the translation semigroup T = (Tt)t≥0 deﬁned by Ttf(x) = f(x+t) is a C0-semigroup on C0(R+,ρ). It is proved in [12] that T is strongly mixing in the topological sense if and only if ρ(x) −−−→x→∞ 0 as x → ∞. We n”