  In view of the complexity of the dynamics of learning in games, we seek to
decompose a game into simpler components where the dynamics' long-run behavior
is well understood. A natural starting point for this is Helmholtz's theorem,
which decomposes a vector field into a potential and an incompressible
component. However, the geometry of game dynamics - and, in particular, the
dynamics of exponential / multiplicative weights (EW) schemes - is not
compatible with the Euclidean underpinnings of Helmholtz's theorem. This leads
us to consider a specific Riemannian framework based on the so-called
Shahshahani metric, and introduce the class of incompressible games, for which
we establish the following results: First, in addition to being
volume-preserving, the continuous-time EW dynamics in incompressible games
admit a constant of motion and are Poincar\'e recurrent - i.e., almost every
trajectory of play comes arbitrarily close to its starting point infinitely
often. Second, we establish a deep connection with a well-known decomposition
of games into a potential and harmonic component (where the players' objectives
are aligned and anti-aligned respectively): a game is incompressible if and
only if it is harmonic, implying in turn that the EW dynamics lead to
Poincar\'e recurrence in harmonic games.
