I am trying to solve exercise 1.2.7 from Grosse-Erdmann and Peris' book Linear Chaos. It is stated as follows:
Let $T:X\rightarrow X$ be continuous on a separable and complete metric space $X$ without isolated points. A backward orbit of a vector $x$ is a sequence $(x_n)$ in $X$ (if it exists) such that $x_0=x$ and $Tx_n=x_{n-1}$. Show that if $T$ is topologically transitive then there is a dense set of points with dense backward orbits.
There is a hint that says "see previous exercise." The previous exercise involved proving constructively (not using Baire's theorem) that such a $T$ as above has a dense set of points with dense orbit in the following way (this is the hint for solving the previous problem):
Let $(y_n)$ be dense in $X$ and let $x_0\in X$. Find $x_1$ close to $x_0$ and a positive integer $m_1$ so that $T^{m_1}x_1$ is close to $y_1$. Then find $x_2$ close to $x_1$ and a positive integer $m_2$ so that $T^{m_1}x_2$ is close to $T^{m_1}x_1$ and $T^{m_1+m_2}x_2$ is close to $y_2$. Continue.
I was able to solve the previous problem, but I needed to use the continuity of $T^{m_1}$ to find a suitable $m_2$ and $x_2$. If I am to repeat the spirit of this problem instead with preimages like $T^{-m_1}$ (since backwards orbits are of the form $\{x, T^{-1}x, T^{-2}x, ...\}$), then I no longer necessarily have continuity of the $T^{-n}$.
Anyway, this is what I have been thinking about for this problem. Any hints or suggestions would be very much appreciated. Thank you!
Be careful that $T$ is not assumed to be bijective, so things like $T^{-1}x$ do not make sense.
Maybe the following hint (in the spirit of the one you indicated for forward orbits) will be enough for you.
As above, let $(y_n)_{n\geq 1}$ be dense in $X$, and let $x_0\in X$. Find $x_1$ close to $y_1$ and an integer $n_1$ such that $T^{n_1}x_1$ is close to $x_0$. Then find $x_2$ close to $y_2$ and an integer $n_2$ such that $T^{n_2}x_2$ is close to $x_1$ and $T^{n_1+n_2}x_2$ is close to $T^{n_1}x_1$. Continue.