Extension of semi-flows on a locally compact metric space

Question

Theorem 2.3, p. 26 from W Shen, Y Yi "Almost automorphic and almost periodic dynamics in skew-product semiflows" states that if there is a semi-flow $\varphi^{t}$, $t \geq 0$, on a locally compact metric space $\mathcal{S}$ such that every point $x \in \mathcal{S}$ has a unique backward orbit then the semi-flow $\varphi^{t}$ can be extended to a flow on $\mathcal{S}$. However, the proof given in the book is not satisfactory. In the most interesting part, where one has to show the continuity of $\varphi^{-t}(x) = (\varphi^{t})^{-1}(x)$ for $t > 0$, it refers to the "inverse function theorem" from a book on topology, in which I cannot find any such statement.

I know that the continuity of the inverse will follow if $S$ is compact (this is a well-known statement) or $S$ is a topological manifold (from the invariance of domain theorem). But I don't know any statement of automatic continuity for the inverse in the case of locally compact spaces.

Answer 1

This is not a complete answer, but an attempt. There exists a continuous bijective map $f \colon \mathcal{S} \to \mathcal{S}$ of a locally compact space to itself that is not a homeomorphism. This map can be contruscted from the standard conitunous bijective map of an half-open interval $[0,1)$ to the unit circle $\mathcal{S}^{1}$ that of course cannot be a homeomorphism. We just take a locally compact space $\mathcal{S} \subset \mathbb{R}^{2}$ as a disjoint union of inifite sequence of copies of $[0,1)$, say $\mathcal{I}_{k}$, and an infinite sequence of copies of $\mathcal{S}^{1}$, say $\mathcal{S}_{k}$, where $k=1,2,\ldots$. Let $f$ map $\mathcal{I}_{k+1}$ into $\mathcal{I}_{k}$, $\mathcal{I}_{1}$ into $\mathcal{S}_{1}$ and $\mathcal{S}_{k}$ into $\mathcal{S}_{k+1}$. Clearly, $f$ is a continuous bijection, but not a homeomorphism. This shows that the reasoning in the book is, indeed, at least unsatisfactory.

However, if we try to construct from this map a continuous analog with $\varphi^{1}=f$, we will have a problem with locally compactness (as for example, in the red point on the figure), which probably occurs since the dimension of the space is increased from one to two.