<p>An orbit γ of x ϵ x^r (M^n) , r≥1 , is said to be ω-recurrent if " γ ⊂ ω(γ)" . Let M be a compact manifold and “γ” an ω-recurrent orbit for x ϵ x^r (M^n) . If f=X<em>(t=1) and x ϵ "γ " show that x is ω-recurrent, that is x∈ ω</em>f “(x)” .
Remark: The Birkhoff Centre C(X) of x ϵ x^r (M^n) is defined as the closure of the set of those orbits that are both ω- and α-recurrent. The same definition works for f ∈ 〖Diff〗^r “(M)” . This exercise shows that C(X) =C (f) when f is the time 1 diffeo-morphism of X.</p>