To a birational self-map $f$ of an $n$-dimensional smooth projective variety $X$ one can associate an $(n-1)$-tuple of real numbers, called the dynamical degrees of $f$. They are invariants under birational conjugacy and related in several ways to both entropy-theoretic and spectral invariants of systems in topological or measure-theoretic dynamics. Their totality forms the dynamical spectrum of $X$, and we will discuss how these spectra might be used to distinguish rational from nearly rational varieties, i.e., in which ways they could be sensitive to subtle changes in birational type.
We will also present some computations of dynamical degrees of compositions of reflections in points on cubic fourfolds, for various configurations of points.