Consider the tent map $f:[0,1] \to [0,1]$, defined by $f(x)=2x$ if $x<0.5$, or $x =0.5$ and $f(x)=2-2x$ if $x>0.5$. Describe a point in $[0,1]$ whose orbit under $f$ is dense in $[0,1]$.
This is a question which is part of a course in fractal geometry and chaotic dynamics. I realise that the functions in this equation are the inverse of the equations in an iterated function system which maps $[0,1]$ onto itself (i.e., $[0,1]$ is the attractor). However, I'm not quite sure how to find the points with periodic orbits, and I'm not sure if it is best to examine the original equation or the iterated functions system to find the points.