In this manuscript, we construct and analyze dynamical systems on the Klein bottle using anti-periodic cocycles that respect the non-orientable structure of the surface. We establish basic topological properties, including minimality and zero entropy for irrational parameters, and identify two ergodic measures. Using Whitney immersion and frame-based observables, we compute spectral measures for the Koopman operator. The analysis reveals a mixed spectrum consisting of discrete eigenvalues, a flip symmetry eigenvalue, and singular continuous components.
The spectrum features eigenvalues obtained from the Jacobi–Anger expansion, a flip eigenvalue of –1 arising from the non-orientable geometry, and singular continuous parts associated with cocycles that are not cohomologous to constants. We demonstrate spectral non-rigidity: in the family of systems with nonzero cocycle parameter, many examples share the same spectral type while differing topologically, distinguished by homological Birkhoff variances. These results combine harmonic analysis and topological invariants to study the dynamics of the Klein bottle.