The Katok-Spatzier conjecture, generalized symmetries, and equilibrium-free flows (Q1948379)

The author studies and classifies equilibrium-free flows on compact manifolds without boundary under nontrival generalized symmetries. It is shown that these flows are rare, i.e., such flows not possessing a generalized symmetry form a residual set. Furthermore, on a \(2\)-torus these equilibrium-free flows are topologically conjugate to a minimal flow. In case its Lyapunov exponent in the flow direction does not vanish, a generalized symmetry is shown to be nontrivial. One finds additional conditions under which the multiplier of a nontrivial generalized symmetry is an algebraic number of norm \(\pm 1\). Finally, conditions are given (including the Katok-Spatzier conjecture) such that an equilibrium-free flow on an \(n\)-torus with generalized symmetries is projectively conjugated to an irrational flow of Koch type.