Here is a problem from Grosse-Erdmann and Peris' Linear Chaos book that I am trying to solve.
Exercise 1.3.4. Show that $T:X\rightarrow X$ is chaotic iff every finite family of nonempty open sets shares a periodic orbit, in the following sense: for each finite family $U_j\subset X, j=1,...,n,$ of nonempty open sets there is a periodic point $x\in U_1$ such that $T^{k_j}x\in U_j$ for some $k_j\geq 0, j=2,...,n$.
Here is my attempted solution, which I think is not correct.
The backwards implication is trivial. For the forward implication, suppose $T$ is chaotic (and that $X$ has no isolated points). Then there is a dense set of points in $X$ with dense orbit under $T$. Let $U_j, j=1,...n,$ be a finite family of nonempty open sets. Then there exists some $y\in U_1$ with dense orbit. Therefore, there exist nonnegative integers $k_2,...,k_n,$ such that $T^{k_j}y\in U_j$ for each $2\leq j\leq n$.
Let $\epsilon_j$ be such that $B_{\epsilon_j}(T^{k_j}y)\subset U_j$, and let $\epsilon=\min \{\epsilon_j\}$. Since $T^{k_j}$ is continuous at $y$, there exists $\delta_j>0$ such that $d(x,y)<\delta_j$ implies $d(T^{k_j}x,T^{k_j}y)<\epsilon$. Let $\delta=\min \{\delta_j\}$. Since $T$ is chaotic, there exists some periodic point $x\in U_1$ within $\delta$ of $y$.
Then $d(T^{k_j}x,T^{k_j}y)<\epsilon$, so $T^{k_j}x\in U_j$, and we are done.
Comments: I didn't use the hint in the book, which is states "use the continuity of $T$ and an induction process", which leads me to believe something is wrong. Also, my above proof asserts for any periodic $x\in U_1$ with $d(x,y)<\delta$, the claim holds. But if, say, $x$ is a fixed point, that would imply $U_i\cap U_j\neq \emptyset$. Thank you for any input or critiques you can provide.
P.s. Is there a link for how to format these posts to look better? Things like centering an equation on a new line or indenting a new paragraph. Thanks!