Size of the orbit of a dense set

Question

This question is a follow-up to: this post.

Let $X$ be a separable Banach space, $\phi\in C(X;X)$ be an injective continuous non-affine map, and $A$ be a dense $G_{\delta}$ subset of $X$. How big can $orb(\phi,A)$ be? Where: $$ \begin{aligned} &orb(\phi,A)\triangleq \bigcup_{a \in A} orb(\phi,a),\\ & orb(\phi,a) \triangleq \{\phi^n(a)\}_{n \in \mathbb{N}}, \end{aligned} $$

What can we say about the size of $orb(\phi,A)$? More specifically, does $X$ satisfy either of:

• If $\mu$ is a non-atomic and strictly positive Borel measure on $X$, then is $orb(\phi,A)$ of full $\mu$-measure,
• Is $orb(\phi,A)$ Haar-null,
• $orb(\phi,A)$ covers $X-E$, where $E$ is a finite-dimensional subspace of $X$?

Answer 1

The answer here is negative.

Given any infinite-dimensional Banach space $X$, fix any non-zero linear continuous functional $f:X\to\mathbb R$ and fix any vector $x_1\in f^{-1}(1)$. Take any dense $G_\delta$-set $G$ in the real line. Then its preimage $A=f^{-1}(G)$ is a dense $G_\delta$-set in $X$. Now take any continuous function $\psi:f^{-1}(0)\to f^{-1}(0)$ on the hyperplane $f^{-1}(0)$ and consider the continuous map $\phi:X\to X$, $\phi:x\mapsto f(x)x_1+\psi(x-f(x)x_1)$.

Observe that $\varphi(A)\subset A$ and hence $orb(\phi,A)=A$. Now observe that the complement $X\setminus A=f^{-1}(\mathbb R\setminus G)$ is infinite-dimensional and $\mu(A)=0$ for any measure $\mu$ supported by the set $X\setminus A$. Such measure $\mu$ can be non-atomic and strictly positive. If $G$ has Lebesgue measure zero, then the set $A$ is Haar-null. If $\mathbb R\setminus G$ has Lebesgue measure null, then $X\setminus A$ is Haar-null.