# Dense orbits and invertibility

Let g be the logistic map $g(x) = 4x(1-x)$ and define $\phi (x) = \sin^2(\frac{\pi}{2}x)$, for $x \in [0; 1]$. Show that $\phi$  is invertible and   $\phi \circ f = g \circ \phi$  , where $f$ is

$f(x) = \begin{cases} 2x &\mbox{if } x < 1/2 \\ 2(1-x) & \mbox{if } x \geq 1/2. \end{cases}$

Hence deduce that $g$ has a dense orbit.

How many orbits of least period 3 does it have?

Does any one have any suggestions or ways of answering this question?

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To find the inverse of $\phi$, probably the best way would be to just explicitly write one down and then prove that the inverse you've written down is well defined on the interval you're working in, $[0,1]$. The other way of doing this would be to show that $\phi$ is both injective and surjective which is also not too difficult (intermediately value theorem will be useful).
For the next part, recall the definition of a topologically conjugate dynamical system. Really, showing that $\phi\circ f=g\circ \phi$ is just a matter of showing that they are equal on all values $x\in [0,1]$. I would handle the cases $x\in[0,\frac{1}{2}]$ and $x\in[\frac{1}{2},1]$ separately.
To show that $g$ has dense orbits, recall that, by basic properties of topologically conjugate systems $f_1$ and $f_2$ with conjugation $\theta$ so that $f_1=\theta^{-1}\circ f_2\circ \theta$, we have $$f_1^n=\theta^{-1}\circ f_2^n\circ \theta$$ for all $n\geq 0$. And, as $\theta$ is a homeomorphism, if $x$ has dense orbit under $f_2$, then $y=\theta^{-1}(x)$ has dense orbit under $f_1$ (if you haven't been shown this yet, prove it). I'm assuming that you've proven the tent map $f$ has a dense orbit (this is normally done by showing that $f$ is conjugate itself to a system of symbolic dynamics) so this pretty much immediately shows that $g$ has a dense orbit.
Again, to show how many orbits of period $3$ that $g$ has, it's enough to use the above and work out how many orbits of period $3$ that $f$ has (also often shown using symbolic dynamics).