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 $\mathbb P=\mathbb R\setminus \mathbb Q$.



Transitive homeomorphisms of Erdős spaces
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,](http://matwbn.icm.edu.pl/ksiazki/fm/fm150/fm15021.pdf) 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](http://www.few.vu.nl/~dijkstra/research/papers/2004ERA.pdf). 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](https://www.springer.com/us/book/9780387943749)), 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](https://www.ams.org/journals/proc/1986-096-04/S0002-9939-1986-0826501-2/S0002-9939-1986-0826501-2.pdf), 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 $E$.


