Density of periodic points in $\mathbb{R}^d$.

Question

Let $f:\mathbb{R}^d\to\mathbb{R}^d$, $d\geq 2$ be a diffeomorphism. Suppose that $\bigcup_{n\geq 0}f^n(U)=\mathbb{R}^d$ for any open set $U$ of $\mathbb{R}^d$.

Questions:

1. Is it true then that the periodic points of $f$ are dense in $\mathbb{R}^d$? (Periodic points are points where $f\circ\dots\circ f=f^n(x)=x$ for some $n\geq 1$.)

2. What if $f$ is just differentiable?

3. If 1 is not true are there any other assumptions under which it is true?

Any references are welcome.