Self-Composition Deviation in Self-Affine Sets

• Self-Composition Deviation (SCD) is a phenomenon describing the asymptotic behavior of ergodic integrals in self-affine Delone sets, governed by a hierarchy of cohomological eigenexponents.
• The method uses expansive linear maps and pattern-equivariant cohomology to derive detailed expansions of ergodic averages with subleading corrections beyond classical boundary terms.
• Illustrative examples such as the Penrose, Chair, and Ammann–Beenker tilings demonstrate SCD's significance in linking dynamical fluctuations with topological and spectral invariants.

Self-Composition Deviation (SCD) refers to asymptotic phenomena in the growth of ergodic integrals for translation actions on pattern spaces arising from self-affine Delone sets. Under iteration by an expansive linear map, the rate of deviation of ergodic averages is governed by the spectrum of the induced action on the pattern-equivariant cohomology of the tiling/delone set. The deviation is characterized by a hierarchy of exponents determined by the eigenvalues of this cohomological action, leading to a detailed expansion of ergodic integrals with subleading cohomological corrections beyond classical boundary terms. This phenomenon was rigorously developed in the context of self-affine Delone set dynamics by Schmieding–Treviño (Schmieding et al., 2015).

1. Ergodic Integrals and Renormalization for Self-Affine Pattern Spaces

Let $\Lambda \subset \mathbb{R}^d$ be a repetitive, finite local complexity (FLC) Delone set, whose pattern space $\Omega = \Omega_\Lambda$ supports a unique translation-invariant probability measure $\mu$. For any bounded averaging set $B \subset \mathbb{R}^d$, and any $f \in C^0(\Omega)$, the ergodic integral is defined as

$$I(B, f; \Lambda_0) = \int_B f(\varphi_s(\Lambda_0))\,ds,$$

where $\varphi_s$ denotes translation by $s$ in $\mathbb{R}^d$. If $\Lambda$ is self-affine, there exists an expanding $A \in GL^+(d,\mathbb{R})$ and a measure-preserving homeomorphism $\Phi_A:\Omega \to \Omega$ such that $$\Phi_A \circ \varphi_t = \varphi_{A t} \circ \Phi_A$$.

A renormalized family of averaging sets is chosen as $B_T = g_{\sigma \log T}(B_0)$, with $g_t = \exp(t a)$, $\exp(a) = A$, and $\sigma = d/\log \det A$, ensuring $$\operatorname{Vol} B_T = \operatorname{Vol} B_0\,T^d$$. Under iteration, $A^n$ “magnifies” averaging, enabling comparison of ergodic integrals at scales $T$ and $T/\tau$ under the induced homeomorphism.

2. Deviation Hierarchy and Main Theorem

For a Delone set with “RFT” property (pattern-equivariant cohomology in top degree is finite-dimensional), the automorphism $\Phi_A$ induces a linear map $A^* = \Phi_A^*$ on $H^d(\Omega;\mathbb{C})$ with eigenvalues $|\nu_1| \geq |\nu_2| \geq \cdots \geq |\nu_r| > 0$ and $\nu_1 = \det A$. Writing any $f$ as a $d$-form using $f \star 1$, a Jordan basis decomposition yields

$$\star f = \sum_{i,j,k} \alpha_{i,j,k}(f)\,g_{i,j,k} + \star d\omega,$$

where $g_{i,j,k}$ are dual basis functions. The coefficients $\alpha_{i,j,k}(f)$ are determined by the cohomology class of $f \star 1$.

The rapidly-expanding subspace $E^+ \subset H^d(\Omega)$ is defined as the sum of generalized eigenspaces $E_i$ for those $\nu_i$ with

$$\frac{\log |\nu_i|}{\log \nu_1} \geq 1 - \frac{\log |\lambda_d|}{\log \nu_1},$$

where $\lambda_d$ is the smallest-modulus eigenvalue of $A$. The main SCD theorem states there are exactly $\dim E^+$ distributions $D_{i,j,k}$ such that:

• If $D_{i',j',k'}(f)=0$ for all $(i',j',k') < (i,j,k)$ but $D_{i,j,k}(f) \neq 0$, then for $T>3$

$$|I(B_T, f; \Lambda_0)| \leq C\,L(i,j,T)\,T^{d(\log |\nu_i|)/(\log \nu_1)}\, \|f\|_\infty$$

where $L(i,j,T) = (\log T)^{j-1}$ or $(\log T)^j$ depending on boundary-exponent equality, and $C$ depends only on $B_0$ and $A$.

• If all $D_{i,j,k}(f)=0$, the boundary term dominates:

$$|I(B_T,f)| = O\Big(T^{d(1 - (\log |\lambda_d|)/(\log \nu_1))}\Big).$$

In the pure-dilation case $A = \tau \operatorname{Id}$, the deviation reduces to classical estimates of the form

$I(B_R, f) = C(f)\,R^d + O(R^{d-1+\epsilon}),$

with $C(f)=D_{1,1,1}(f)$.

3. Cohomological Structure and Interpretation

The leading constant $C(f)=D_{1,1,1}(f)$ is interpreted topologically and dynamically using the asymptotic-cycle current $\mathcal{C}_\Lambda \in (H^d(\Omega;\mathbb{R}))'$, defined by

$$\mathcal{C}_\Lambda([\eta]) = \int_\Omega f_\eta\,d\mu, \quad \eta \in \Delta_\Lambda^d.$$

Via the Hodge star, $\mathcal{C}_\Lambda$ corresponds to a transverse invariant measure on the canonical transversal. The pure-point part of the diffraction spectrum is the evaluation of $\mathcal{C}_\Lambda$ on classes $[\lambda_x]$, $x \in \mathbb{R}^d$, supported on $\Lambda - \Lambda$, reproducing known spectral measures via Dworkin’s argument.

Diagonalizing $A^*$ on $H^d(\Omega)$, the eigenvector in the unstable line $E_{1,1,1}$ with eigenvalue $\nu_1$ is $[\star 1]$. The dual distribution $D_{1,1,1}$ recovers the density term:

$$D_{1,1,1}(f) = \mathcal{C}_\Lambda(f \star 1) = \int_\Omega f\,d\mu.$$

4. Explicit Examples

Several notable self-affine tilings and Delone sets exhibit the SCD phenomenon:

• Penrose tiling (inflation $\tau = (1+\sqrt{5})/2$): Substitution spectrum ${\tau^2, -\tau^{-1}, -\tau^{-1}, 0,\dots}$, with leading cohomological deviation $\tau^2$, boundary exponent $1/2$, and subleading terms at boundary order $O(R^1)$.
• Chair tiling ($A=2$Id): Top cohomology eigenvalues $\{4,0,0,...\}$; only the $4^n$ term is non-boundary.
• Ammann–Beenker tiling (octagonal): $A \in \mathrm{SL}(4, \mathbb{Z})$ with spectrum $\{1+\sqrt{2},1+\sqrt{2},1-\sqrt{2},1-\sqrt{2}\}$; two eigenvalues in $E^+$, the leading strictly and another at equality, yielding log corrections.
• Self-affine codim-1 cut-and-project sets: Top $H^d(\Omega)$ splits into a torus-factor and a singularity-part; products $\prod_i \lambda_i > \det A / |\lambda_d|$ lie strictly in $E^+$.

5. Structural Propositions and Asymptotic Expansions

Key results summarizing SCD include:

• Rapidly-expanding subspace:

$$E^+ = \bigoplus_{i\,:\,(\log |\nu_i|)/(\log \nu_1) \geq 1 - (\log |\lambda_d|)/(\log \nu_1)} E_i$$

• Boundary estimate: If $\psi \star 1 = d \omega$ is a coboundary,

$$\left|\int_{B_T} \psi \circ \varphi_s\,ds \right| \leq C\,T^{d(1 - (\log |\lambda_d|)/(\log \nu_1))}.$$

• Cohomological decomposition: For $\eta \in \Delta_\Lambda^d$,

$$\star\eta = \sum_{i,j,k} \alpha_{i,j,k}(\eta)\,g_{i,j,k} + \star d\omega.$$

• Induced-action estimate:

$$|\Upsilon^n(\eta_{i,j,k})| = \begin{cases} O(n^{j-1}|\nu_i|^n), & |\nu_i| > \nu_1/|\lambda_d|\ O(n^j|\nu_i|^n), & \text{if equality.} \end{cases}$$

Combining these elements, the full expansion of ergodic integrals is:

$$\int_{B_T} f \circ \varphi_s\,ds = \sum_{(i,j,k)} D_{i,j,k}(f)\,\Psi_{i,j,k}(T)\,L(i,j,T)\,T^{d(\log |\nu_i|)/(\log \nu_1)} + O\left(T^{d(1 - (\log |\lambda_d|)/(\log \nu_1))}\right).$$

In the dilation (self-similar) case, this specializes to

$I(R, f) = C(f)\,R^d + O(R^{d-1+\epsilon}),$

with $C(f)$ representing pairing with the unstable top class.

6. Broader Implications and Context

Self-Composition Deviation provides a generalization of classical deviation theorems and establishes a bridge between dynamical renormalization, cohomological invariants, and statistical properties of self-affine aperiodic structures. The SCD theorem extends beyond leading-order density terms by predicting intermediate orders of fluctuation (“Zorich–Forni” type exponents) directly from the induced cohomological dynamics and Jordan decomposition. It unifies understanding of deviation phenomena in substitution tilings, cut-and-project sets, and more general Delone sets, linking spectral, cohomological, and dynamical characteristics in a precise quantitative framework (Schmieding et al., 2015).