Fixed points and zero holonomy in a cyclic skew product
Abstract
Fixed points of a finite cyclic skew product are exactly zero-holonomy base returns, with one factor for every fiber phase.
Theorem 1.1 (Fixed-point counting through the holonomy kernel).
Lean statement: D5/S1/Dynamics/PeriodicOrbits/SkewZModFixedCount.fixed_count_skew_zmod
Proof. Machine-checked in Lean as D5/S1/Dynamics/PeriodicOrbits/SkewZModFixedCount.fixed_count_skew_zmod (✓ std3). ∎
Source. Repository-derived.
Commentary.
For an additive skew map with fiber ZMod q, an n-step fixed point must first have an n-step fixed base point. Its fiber coordinate then returns exactly when the n-step Birkhoff sum is zero in ZMod q.
When the holonomy vanishes, every one of the q fiber phases returns; when it does not, no phase returns. The fixed-point count is therefore q times the cardinality of the zero-holonomy return set. This is the counting bridge from the skew group action to the Möbius/Dold layer.