Transitive homeomorphisms of Erdős spaces

Question

A surjective homeomorphism $h:X\to X$ is minimal if $$\overline{\{h^n(x):n\in \mathbb N\}}=X$$ for every $x\in X$. In other words, the orbit of each point is dense.

Does either of the Erdös spaces $\mathfrak E$ or $\mathfrak E_c$ have a minimal homeomorphism?

The Erdös spaces are defined as:

$\mathfrak E=\{x\in \ell^2:x_n\in \mathbb Q\text{ for all $n<\omega$}\}$; and

$\mathfrak E_c=\{x\in \ell^2:x_n\in \mathbb P\text{ for all $n<\omega$}\},$

$\ell^2$ is the Hilbert space, $\mathbb Q$ is the set of rational numbers, and $\mathbb P=\mathbb R\setminus \mathbb Q$.

Answer 1

The answer to both questions is affirmative.

Theorem 1. The complete Erdos space $\mathfrak E_c$ has a self-homeomorphism whose every orbit is dense in $\mathfrak E_c$.

Proof. We use a known result of Kawamura, Oversteegen and Tymchantyn, that the complete Erdos space $\mathfrak E_c$ is homeomorphic to the set $E$ of enpoints of the Lelek fan. So, it suffices to construct a self-homeomorphism $h$ of the Lelek fan $L$ such that the orbit of each point $x\in E$ is dense in $E$. To construct such a homeomorphism, represent the Lelek fan as the inverse limit of a sequence $(L_n)$ of finite geometric graphs such that the preimage $p_n^{-1}(x)$ of any end-point $x$ of the graph $L_n$ under the projection $p_n:L_{n+1}\to L_n$ has some fixed odd number of points, depending only on $n$, but not on $x$ (the odd number will be used in the proof of Theorem 2). In this case we can construct a sequence of homeomorphisms $(h_n:L_n\to L_n)_{n=1}^\infty$ such that for every $n\in\mathbb N$ we have $p_n\circ h_{n+1}=h_n\circ p_n$ and the $h_n$-orbit of any end-point $x$ of $L_n$ coincides with the (finite) set of end-points of the graph $L_n$. Then the limit of the sequence $(h_n)$ is a required homeomorphism $h$ of the Lelek fan such that the $h$-orbit for any end-point $x$ of $L$ is dense in the set of end-points $E$ of $L$.

Theorem 2. The rational Erdos space $\mathfrak E$ has a self-homeomorphism whose every orbit is dense in $\mathfrak E$.

Proof. We shall use Corollary 5.4 of this paper of Dijkstra and van Mill. This Corollary states that the rational Erdos space $\mathfrak E$ is homeomorphic to the product $\mathfrak E_c\times\mathbb Q^\omega$. Using the argument of the proof of Theorem 1, we can construct a self-homeomorphism $h_1$ of $\mathfrak E_c$ and a self-homeomorphism $h_2$ of the Cantor cube $2^{\omega}$ such that each orbit of the homeomorphism $h:\mathfrak E_c\times 2^\omega\to\mathfrak E_c\times 2^\omega$, $h:(x,y)\mapsto (h_1(x),h_2(y))$, is dense in $\mathfrak E_c\times 2^\omega$.

Using Mycielski-Kuratowski Theorem (19.1 in this book of Kechris), we can find a topological copy $C\subset 2^\omega$ of the Cantor set such that the sets $h^n(C)$, $n\in\mathbb Z$, are pairwise disjoint. Then take a topological copy $D\subset C$ of the space $\mathbb Q^\omega$ in $C$ and observe that the space $M=\bigcup_{n\in\mathbb Z}h^n(D)$ is meager and each non-empty closed-and-open set in $M$ is of type $F_{\sigma\delta}$, but not $G_{\delta\sigma}$. By a theorem of van Engelen, up to a homeomorphism, $\mathbb Q^\omega$ is a unique meager zero-dimensional metrizable space whose every closed-and-open set is of type $F_{\sigma\delta}$ but not $G_{\delta\sigma}$. This characterization of van Engelen implies that the space $M$ is homeomorphic to $\mathbb Q^\omega$. Then the product $E:=\mathfrak E_c\times M$ is homeomorphic to the rational Erdos space $\mathfrak E$ and the restriction of the homeomorphism $h$ to $E$ has the required property: the $h$-orbit of any point of $E$ is dense in $E$.