  We present a new geometric approach to Floquet many-body systems described by
inhomogeneous conformal field theory in 1+1 dimensions. It is based on an exact
correspondence with dynamical systems on the circle that we establish and use
to prove existence of (non)heating phases characterized by the (absence)
presence of fixed or higher-periodic points of coordinate transformations
encoding the time evolution: Heating corresponds to energy and excitations
concentrating exponentially fast at unstable such points while nonheating to
pseudoperiodic motion. We show that the heating rate (serving as the order
parameter for transitions between these two) can have cusps, even within the
overall heating phase, and that there is a rich structure of phase diagrams
with different heating phases distinguishable through kinks in the entanglement
entropy, reminiscent of Lifshitz phase transitions. Our geometric approach
generalizes previous results for a subfamily of similar systems that used only
the $\mathfrak{sl}(2)$ algebra to general smooth deformations that require the
full infinite-dimensional Virasoro algebra, and we argue that it has wider
applicability, even beyond conformal field theory.
