Sectors in Chaos: Covers, Twisted Ribbons and the Möbius–Klein Ladder of Phase Space

The Möbius Klein Transformation (MKT) splits the fields on a space with a twist into the untwisted and the twisted sectors of a double cover. This article applies it to chaos. A symmetric flow is a branched cover of its image, as Lorenz covers proto-Lorenz. Its transfer operator splits into 2 blocks, and the wing switching is the twisted sector, which decays asymptotically more slowly in an Ulam approximation. An orbit of a template carries 2 independent characters, orientation and cover. The spectrum of a symmetric horseshoe cover is the union of the spectra of the character-twisted image matrices. A period-doubling cascade is a tower of covers of the time circle, each adding a twisted sector. The new part is a ladder of orientation in phase space. By the Liouville pairing of Floquet multipliers, an invariant manifold of a periodic orbit can be a Möbius band from 3 dimensions on and a solid Klein bottle from 4 on. The hyperchaotic Rössler system has one. A symmetry with a half-period shift makes the quotient of the extended phase space non-orientable exactly when it reverses orientation. Under point, symplectic and antisymplectic symmetries, mechanical systems with 2 degrees of freedom keep that quotient orientable. A Klein-bottle hull commuting with the drive flow needs a periodic drive with a cyclic parameter. Under a quasiperiodic drive with a symmetry, odd observables carry only odd harmonics of the shifted frequency, and a symmetric torus doubling is not suppressed. For maps of the line the symmetry exchanges its 2 strands. For unstable orbits in 2 degrees of freedom, a negative multiplier, an eigenline with a Möbius twist, an odd zero count of the eigensolution and an odd Conley–Zehnder index are proved equivalent. Deterministic scripts check every measured statement.