  We study the appearance of powers of Hamilton cycles in pseudorandom graphs,
using the following comparatively weak pseudorandomness notion. A graph $G$ is
$(\varepsilon,p,k,\ell)$-pseudorandom if for all disjoint $X$ and $Y\subset
V(G)$ with $|X|\ge\varepsilon p^kn$ and $|Y|\ge\varepsilon p^\ell n$ we have
$e(X,Y)=(1\pm\varepsilon)p|X||Y|$. We prove that for all $\beta>0$ there is an
$\varepsilon>0$ such that an $(\varepsilon,p,1,2)$-pseudorandom graph on $n$
vertices with minimum degree at least $\beta pn$ contains the square of a
Hamilton cycle. In particular, this implies that $(n,d,\lambda)$-graphs with
$\lambda\ll d^{5/2 }n^{-3/2}$ contain the square of a Hamilton cycle, and thus
a triangle factor if $n$ is a multiple of $3$. This improves on a result of
Krivelevich, Sudakov and Szab\'o [Triangle factors in sparse pseudo-random
graphs, Combinatorica 24 (2004), no. 3, 403--426].
  We also extend our result to higher powers of Hamilton cycles and establish
corresponding counting versions.
