Limit functions of discrete dynamical systems (Q2922921)

Let \((X,d)\) be a complete metric space and let \(\mathcal{K}(X)\) be the collection of nonempty and compact subsets of \(X\). For \(K \in \mathcal{K}(X)\), the set of continuous functions from \(K\) to \(X\) is denoted by \(C(K,X)\). We denote, as usual, by \(f^{\circ n}=f \circ \dots \circ f\) the \(n\)-th iterate of the continuous function \(f: X \to X\). If \(L\) is an arbitrary subset of \(X\), we write \(\Omega_p(L,f)\) for the collection of all functions \(g:L \to X\) that are pointwise limit of some subsequence of \((f^{\circ n})_n\), on \(L\). For \(K \in \mathcal{K}(X)\), \(\Omega_u(K,f)\) is the set of all functions that are limits of some subsequence of \((f^{\circ n})_n\) in \((C(K,X),d_{u,K})\), where \(d_{u,K}\) is the uniform metric given by NEWLINE\[NEWLINEd_{u,K}(f,g)=\sup_{x \in K}d(f(x),g(x)).NEWLINE\]NEWLINE Let \((Y,d_Y)\) be a complete metric space. A family \(\{f_\iota:X \to Y, \iota \in I\}\) is called topologically transitive, if for all \(U \subset X\) and \(V \subset Y\) open and nonempty, there is \(\iota \in I\) with \(f_\iota(U) \cap V \neq \emptyset.\) For \(M \in \mathbb{N}^*\), let \(f_\iota^{M}:X^M\to Y^M\) denote the \(M\)-fold product of \(f_\iota\). Then \(\{f_\iota^{\times M}:X^M \to Y^M, \iota \in I\}\) is called topological transitive, if for all open nonempty \(U_1 ,\dots,U_M\subset X\) and \(V_1,\dots,V_M \subset Y\), there is \(\iota \in I\) with \(f_\iota(U) \cap V \neq \emptyset.\)NEWLINENEWLINEIn this setting, the authors prove the following theorem.NEWLINENEWLINETheorem. Let \(X,Y_0\) be complete metric spaces so that for a countable set \(\{h_k: k \in \mathbb{N}\}\) of continuous functions \(h_k: X \to Y_0\), the restrictions \(h_k|_E\) form a dense set in \(C(E,Y_0 )\) for almost all \(E \in \mathcal{K}(X)\). Moreover, let \(Y \subset Y_0\) be closed and suppose that \(f_\iota: X \to Y~\) are continuous for every \(\iota \in I\) and that \(\{f_\iota^{\times N}, \iota \in I\}\) is topologically transitive for all \(N \in \mathbb{N}^*\). Then \(\{f_\iota|_E^{\times N}, \iota \in I\}\) is dense in \(C(E,Y)\) for almost all \(E \in \mathcal{K}(X).\)NEWLINENEWLINEApplying their result, the authors prove that for any rational function \(f\) with Julia set in \(\mathbb{C}\) and having a Siegel disk \(F\) with Jordan boundary \(\partial F\), the set of functions \(\Omega_u(E \cup \partial F,f)\) is equal to the set of functions \(C_H(E \cup \partial F)\) for almost all \(E \in \mathcal{K}(J)\), where \(J\) is the Julia set of \(f\), \(F\) is the Siegel disk and \(H\) is the set of functions \(h\) of the form \(h=\phi \circ (\beta. \mathrm{Id}) \circ \phi^{-1},\) \(\beta \in \mathbb{T}\) and \(\phi\) is a homeomorphism from \(\mathbb{T}\) to \(\partial F\) induced by the inverse Riemann mapping \(\phi : \mathbb{D} \to F\).