Question: Prove that if $f$ and $g$ are topologically conjugate functions and $f$ has chaos, then $g$ has chaos.
I can see a bit of the reason behind of the claim but I can't prove it.
To prove that $g:X \rightarrow X$ has chaos, I have to prove if $(i)$ The set of periodic points of $g$ is dense in $Y$, and $(ii)$ For every $U$, $V$ open in $Y$, there exists $x \in U$ and $n \in \mathbb{N}$ such that $g^n(x)\in V$. By the way, since the functions $f:X \rightarrow X$ and $g:Y \rightarrow Y$ are topologically conjugate, there exists a homeomorphism $h:Y \rightarrow X$ such that $f o h = h o g$, which implies that for each $x \in X$ and $n \in \mathbb{N}$, we have $h(g^n(x)) = f^n(h(x))$ (by induction method).
If the set of periodic points of $f$ is dense in $X$, then for some $x \in X$ and $\epsilon > 0$, then there is a periodic point $p$ such that $|x - p| < \epsilon$. Since $f(x)\in X$ and $h(x)\in X$ so it holds that $|f(h(x)) - p_1| < \epsilon_1$ so for any $x\in X$ $h(g(x))$ in dense in $X$; since $X$ is homomorphic to $Y$ so arbitrary close in $X$ implies arbitrary close in $Y$ so $g$ is dense in $Y$. (?)
For every $U$, $V$ open in $X$, $f^n(h(x))\in V$ since $h$ is homomorphism $g^n(x)\in h^{-1}(V)$ which is open in $Y$ (?)
Could someone please help me how to prove the theorem mentioned in the first line?
Thank you.
This is proved as Proposition 7.2.1 on page 147 of Lecture Notes on Dynamical Systems, Chaos and fractal geometry by Geoffrey R.Goodson of the Towson University Mathematics Department.