  This paper studies the relationship between the modified
Foata$\unicode{x2013}$Strehl action (a.k.a. valley-hopping)$\unicode{x2014}$a
group action on permutations used to demonstrate the $\gamma$-positivity of the
Eulerian polynomials$\unicode{x2014}$and the number of rixed points
$\operatorname{rix}$$\unicode{x2014}$a recursively-defined permutation
statistic introduced by Lin in the context of an equidistribution problem. We
give a linear-time iterative algorithm for computing the set of rixed points,
and prove that the $\operatorname{rix}$ statistic is homomesic under
valley-hopping. We also demonstrate that a bijection $\Phi$ introduced by Lin
and Zeng in the study of the $\operatorname{rix}$ statistic sends orbits of the
valley-hopping action to orbits of a cyclic version of valley-hopping, which
implies that the number of fixed points $\operatorname{fix}$ is homomesic under
cyclic valley-hopping.
