We revisit admissible deformations of self-similar tiling spaces, following the approach of Alex Clark and Lorenzo Sadun (2006), Johannes Kellendonk (2008), Antoine Julien and Lorenzo Sadun (2018), and especially Rodrigo Treviño (Quantitative weak mixing for random substitution tilings, preprint 2020), focusing on their spectral theory. The new feature is that we allow expanding similarities with irrational rotations, which necessitates dealing with fractal boundaries. This is handled by passing to mutually locally derivable pseudo-self-similar tilings having polygonal tiles, using the derived Voronoi construction of Natalie P. Frank (2000).

In this framework, we extend the construction of the spectral cocycle from a joint work with Alexander Bufetov (2014) and use pointwise Lyapunov exponents of this cocycle to bound local dimension of spectral measures for the deformed tilings.

This is a joint work with Rodrigo Treviño.