  We study the discrete dynamics of standard (or left) polynomials $f(x)$ over
division rings $D$. We define their fixed points to be the points $\lambda \in
D$ for which $f^{\circ n}(\lambda)=\lambda$ for any $n \in \mathbb{N}$, where
$f^{\circ n}(x)$ is defined recursively by $f^{\circ n}(x)=f(f^{\circ
(n-1)}(x))$ and $f^{\circ 1}(x)=f(x)$. Periodic points are similarly defined.
We prove that $\lambda$ is a fixed point of $f(x)$ if and only if
$f(\lambda)=\lambda$, which enables the use of known results from the theory of
polynomial equations, to conclude that any polynomial of degree $m \geq 2$ has
at most $m$ conjugacy classes of fixed points. We also consider arbitrary
periodic points, and show that in general, they do not behave as in the
commutative case. We provide a sufficient condition for periodic points to
behave as expected.
