Structure of inflation/deflation on Penrose Tilings?

Question

Thinking about P2/P3 type penrose tilings (kites/darts or rhombs - for this question they should be equivalent?) we know we can "inflate"/"deflate" any tiling of the plane to get another. We also know that the number of Penrose tilings is uncountably infinite. Do we know what structure the "inflation/deflation" procedure bestows upon the set of penrose tilings?

To be more precise, for example we know that the 'sun' and the 'star' penrose tilings are dual under inflation/deflation. Are there any other 'cycles' of penrose tilings? In my mind, the inflation/deflation introduces a kind of 'local linear order' on the set of penrose tilings - is it known how to describe this structure more carefully?

Unsure of where to go for answers - I've been trying to find Penrose's paper “The Role of Aesthetics in Pure and Applied Mathematical Research.” in Bulletin of the Institute of Mathematics and Its Applications, 10, 1974. pp. 266–271. to no avail

Answer 1

The short answer is that there are infinitely many 'cycles' with arbitrarily long lengths. We would call them periodic orbits under the substitution action. (Not to be confused with periodic orbits under the translation action which obviously do not exist for aperiodic tiling spaces).

Periodic points for the substitution action on the tiling space are encoded in the dynamical zeta function which can be computed via the Lefschetz fixed point theorem and using inverse limit techniques from work of Anderson and Putnam.

https://www.cambridge.org/core/journals/ergodic-theory-and-dynamical-systems/article/topological-invariants-for-substitution-tilings-and-their-associated-castalgebras/5D7DB90543165287CA5C4CBF012323F5

There are many other ways of calculating the zeta function (And the associated cohomology groups), for example via Barge-Diamond complexes or the (dual) Walton homology groups of a tiling.

Answer 2

The minimal embedding of the rhombic version of the Penrose tiling, if to be constructed by cut and Project, is known to be the root lattice $A_4$.

All details of the according construction, lifting of tiling space paths to lattice paths (or re-projecting these to perp space), esp. the structure of the according acceptance domain, the "origins" of the local tiling configurations in the language of acceptance domain sub-regions, the rules of its in-/deflation symmetries, the consideration of isomorphism classes of tilings, etc. all were contained already within the very first according publication. Cf.:

M. Baake, P. Kramer, M. Schlottmann, D. Zeidler, "Planar Patterns with fivefold symmetry as sections of periodic structures in 4-space", Int. J. Mod. Phys. B4 (1990) 2217-2268.

--- rk