Recurrent and almost automorphic selections of multivalued mappings

Let $(U,\rho )$ be a complete metric space and $({\mathrm {cl}}_{\, b}\, U,{\mathrm {dist}})$ be the metric space of nonempty closed bounded subsets of the space $U$ with the Hausdorff metric ${\mathrm {dist}}$. On the set $M({\mathbb R},U)$ of strongly measurable functions $f\colon{\mathbb R}\to U$ we introduce the metric $d^{(\rho )}$ such that the convergence in this metric is equivalent to the convergence in Lebesgue measure on every closed interval $[-l,l]$, $l>0$. The metric $d^{({\mathrm {dist}})}$ on the set $M({\mathbb R},{\mathrm {cl}}_{\, b}\, U)$ of strongly measurable multivalued mappings $f\colon{\mathbb R}\to {\mathrm {cl}}_{\, b}\, U$ (which are considered as functions with the range in ${\mathrm {cl}}_{\, b}\, U$) is defined by analogy with the metric $d^{(\rho )}.$ The spaces $M({\mathbb R},U)$ and $M({\mathbb R},{\mathrm {cl}}_{\, b}\, U)$ are the phase spaces of the dynamical systems of translations. For a multivalued Stepanov-like recurrent mapping $F\in {\mathcal R}({\mathbb R},{\mathrm {cl}}_{\, b}\, U)\subseteq M({\mathbb R},{\mathrm {cl}}_{\, b}\, U)$ and for any $x_0\in U$ and any nondecreasing function $\eta \colon[0,+\infty )\to [0,+\infty )$ for which $\eta (0)=0$ and $\eta (\xi )>0$ for $\xi >0$, it is proved that there exists a homomorphism of dynamical systems ${\mathcal F}:\overline {{\mathrm {orb}}\, F}=\overline {\{ F(\cdot +t):t\in {\mathbb R}\} }\to M({\mathbb R},U)$ such that $({\mathcal F}F^{\, \prime })(t)\in F^{\, \prime }(t)$ and $\rho (({\mathcal F}F^{\, \prime })(t),x_0)\leqslant \rho (x_0,F^{\, \prime }(t))+\eta \bigl( \rho (x_0,F^{\, \prime }(t))\bigr) $ for all $F^{\, \prime }\in \overline {{\mathrm {orb}}\, F}$ and a.e. $t\in {\mathbb R}$. Furthermore, the functions ${\mathcal F}F^{\, \prime }$ are Stepanov-like recurrent. If the multivalued mapping $F$ is Stepanov-like almost automorphic, then the function ${\mathcal F}F$ is Stepanov-like almost automorphic as well.