$\newcommand{\O}{\mathcal{O}}\newcommand{\G}{\mathcal{G}}\newcommand{\T}{\mathcal{T}}$TLDR; skip to the end of the preamble - we know that the tent map system is topologically transitive. However, do we know explicit descriptions of points which are transitive? We know they exist, but can we find the needle in the haystack?
This question is about topological dynamical systems, in particular about the transitivity of the tent map. Some definitions:
A topological dynamical system is a pair $(K;\varphi)$ where $K$ is a nonempty compact Hausdorff space and $\varphi:K\to K$ is continuous.
The system is (topologically) transitive if there exists at least one $x_0\in K$ such that the orbit of $x_0$ is dense: $$\overline{\{\varphi^n(x_0):n\in\Bbb N_0\}}=K$$
It is a theorem that if the following holds:
($\ast$) Let $(K;\varphi)$ be a topological dynamical system. For all nonempty open $U,V$ in $K$, there exists an $n\in\Bbb N$ such that $\varphi^n(U)\cap V\neq\emptyset$.
And if the system is metrisable, then it the system is also topologically transitive. Proof:
Let ($\ast$) hold in a topological dynamical system $(K;\varphi)$. $(\ast)$ is equivalent to the assertion that $\varphi^{-n}(U)\cap V\neq\emptyset$ for some $n$ as well, which we use below. If $K$ is metrisable, its compactness implies it is also second countable and complete. Let $\{\O_m\}_{m\in\Bbb N}$ be a countable base for the topology on $K$. By $(\ast)$, each of the following sets is dense in $K$: $$\G_n:=\bigcup_{m\in\Bbb N}\varphi^{-m}(\O_n)$$And by continuity of $\varphi$ each $\G_n$ is also open. By the Baire Category Theorem, using metrisability and completeness, we have that: $$\T:=\bigcap_{n\in\Bbb N}\G_n$$Is dense in $K$. Fixing any $x_0\in\T$, if $\mathcal{O}\subseteq K$ is an arbitrary nonempty set we have that there exists at least one $n\in\Bbb N$, $\O_n\subseteq\O$, by basis, and by definition of $\T$ there must exist $k_n\in\Bbb N$, $\varphi^{k_n}(x_0)\in\O_n\subseteq\O$, so the orbit of $x_0$ is dense for each $x_0\in\T$.
Consider now the "tent map":
The topological dynamical system $([0,1];T)$ is the Euclidean subspace $[0,1]$ and the map $T:[0,1]\to[0,1]$ which maps $x\mapsto1-|2x-1|$.
I recently showed that the tent map system is topologically transitive by showing it satisfies $(\ast)$ (the OP uses $(\ast)$ as their base definition, I however find it a less interesting definition). So, we know that there are infinitely many, dense, transitive points. How can we pin down the mystery set $\T$? I could not find any research online pertaining to this, so I am asking here as I am not equipped with the tools to tackle this myself.
Intuitively, any point $x_0$ for which $T^n(x_0)$ is eventually in any open set must be irrational. It certainly cannot be a rational with a power-of-two denominator. Calculating the preimages in the $\G_n$ is nightmarish, and computing their intersection would be even worse. As far as I know, that approach is intractable. Does anyone know any technique for identifying the elements of $\T$? Many thanks. I think it would be really interesting if we could find examples, e.g. it would be nice if, say, $1/e$ were a transitive point.
