Let $f:H\rightarrow H$ be a countinuous map from the separable hilbert space into itself, for every $x\in H$ define the discrete dynamical system $$ \xi_x^{n+1}\triangleq f(\xi^n_x);\qquad \xi^0_x\triangleq x. $$ Define the map $\Phi_f$ from $H$ to itself by $$ \Phi(x)\triangleq \lim\limits_{n \mapsto \infty} \xi_x^n. $$
What properties does $f$ need to satisfy so that $\Phi_f$ has dense orbit in $H$?