The Möbius Klein Transformation on Fractals: Twisted Sectors of Fractafolds

The Möbius Klein Transformation (MKT) splits the fields on a space with a twist into the untwisted and the twisted sector of a double cover. This article carries the split to fractafolds of finitely ramified fractals, with a sign or phase at a junction as the twist. For fully symmetric cells spectral decimation runs sector by sector for every phase; for Hanoi rings it follows from truncation. For every fully symmetric cell the sector difference at each level is the pullback of the one below. So each exceptional simple zero of the decimation function carries 1 extra state in a balanced sector. A twisted ring is the twisted sector of a ring twice as long; its lowest eigenvalue tends to a quarter of the lowest untwisted one. The symmetry group of a single cell splits the exceptional series by irreducible representation. The 6-series of the gasket carries the content of the whole coarser level, and the 2/3-series of the tetrahedral gasket and 2 Vicsek series inherit theirs from 1 level lower. The non-abelian sectors of the octahedron fractafold decimate. On fractafolds of 5 gasket families over any cell graph the difference of the 2 renormalized spectral sums is, at every level, a finite sum of preimage sums over the 2 level-0 spectra. It converges to an entire function. On large rings the twisted sector lies higher at every level, as in every model computed. On the diamond loop only the eigenvalues of a periodic and an antiperiodic cycle see the twist, and the difference is given by the Riemann zeta function. On rings of Sierpinski carpets the shared share falls with the level. A flux through a ring leaves the twisted sector stationary, and the predicted quarter rule tolerates disorder small against the gap. Deterministic scripts check every measured statement.