Self Affine Delone Sets and Deviation Phenomena

Abstract

Abstract. We study the growth of norms of ergodic integrals for the translation action on spaces coming from expansive, self-affine Delone sets. The linear map giving the self-affinity induces a renormalization map on the pattern space and we show that the rate of growth of ergodic integrals is controlled by the induced action of the renormalizing map on the cohomology of the pattern space up to boundary errors. We explore the consequences for the diffraction of such Delone sets, and explore in detail what the picture is for substitution tilings as well as for cut and project sets which are self-affine. We also explicitly compute some examples.

Traces of Random Operators Associated with Self-Affine Delone Sets and Shubin’s Formula

“…We briefly summarize the setup from [ST17] relevant in order to state the main results of this paper (see §5 for details). Given any RFT Delone set Λ, there exist d Λ ≥ 1 numbers ν 1 > |ν 2 | ≥ · · · ≥ |ν d Λ | > 1 which are the eigenvalues of an induced map on the cohomology space H d (Ω Λ ; R) (this is defined in §3).…”

“…It was showed in [ST17] that elements in the dual space C + Λ = (E + Λ ) to the rapidly expanding subspace E + Λ are represented by R d -invariant Λ-equivariant currents. It admits a decomposition into eigenvectors of the induced action by Φ Λ :…”

“…We now recall the relevant ergodic theoretic results from [ST17]. Let Λ be an RFT Delone set and recall the definition of the rapidly expanding subspace E + Λ ⊂ H d (Ω Λ ; R d ).…”

“…where L(i, j, T ) is a non-negative power of log T (see (15) and (16) below), for all T > 0. The main result of [ST17] states that there exist |I + Λ | closed, R d -invariant, Λ-equivariant currents {C i,j,k } (i,j,k)∈I + Λ which control the growth of ergodic integrals (12).…”

Equilibrium Configurations for Generalized Frenkel–Kontorova Models on Quasicrystals

“…For a nite dimensional vector space V and Delone set Λ, the set C r Λ (R d ; V ) denotes the space of C r Λ-equivariant functions with values in V and will denote by C r Λ the space of Λ-equivariant functions f ∈ C r (R d ; R). For a self-a ne Delone set Λ with self-a nity matrix A, if f is a continuous Λ-equivariant function, so is A * f (see [ST15,§4]). For f ∈ C 3 Λ its Hessian matrix at the point z is denoted by H(f )(z).…”