---
title: Sierpinski-Type Self-Affine Measures
url: https://www.emergentmind.com/topics/sierpinski-type-self-affine-measures
type: topic
---

# Sierpinski-Type Self-Affine Measures

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Sierpinski-type self-affine measures are Borel probability measures supported on attractors of affine iterated function systems whose digit structure models the planar Sierpinski gasket and a range of non-conformal, random, or level-dependent analogues. In the planar three-digit setting, for an expanding integer matrix \(M\in M_2(\mathbb Z)\) and
\[
D=\Bigl\{(0,0)^T,(\alpha_1,\alpha_2)^T,(\beta_1,\beta_2)^T\Bigr\},
\qquad \alpha_1\beta_2-\alpha_2\beta_1\neq 0,
\]
the associated measure \(\mu_{M,D}\) is defined by a self-affine refinement equation and is supported on a compact attractor \(T(M,D)\). The subject combines affine IFS theory, Fourier analysis, orthogonal exponential systems, exact-dimensionality, pressure methods, and random fractal geometry. A central milestone is the complete spectral classification of generalized planar Sierpinski-type self-affine measures in the remaining arithmetic case \(\det(M)\in 3\mathbb Z\) and \(\alpha_1\beta_2-\alpha_2\beta_1\in 3\mathbb Z\) [2010.14724].

## 1. Canonical constructions and model classes

For the generalized planar model, the affine maps are
\[
\tau_d(x)=M^{-1}(x+d),\qquad d\in D,
\]
and the compact attractor is the unique nonempty set satisfying
\[
T(M,D)=\bigcup_{d\in D}\tau_d\bigl(T(M,D)\bigr).
\]
The corresponding probability measure is the unique Borel measure satisfying
\[
\mu_{M,D}(E)=\frac13\sum_{d\in D}\mu_{M,D}\bigl(M^{-1}(E-d)\bigr)
\]
for all Borel \(E\subset \mathbb R^2\). This is the basic generalized Sierpinski-type self-affine measure studied in the spectral classification problem [2010.14724].

A broader planar class replaces the three-digit, integer-matrix setting by
\[
\mu_{M,D}(\cdot)=p^{-1}\sum_{d\in D}\mu_{M,D}(M(\cdot)-d),
\]
where \(p\) is prime, \(M=\begin{pmatrix}\rho_1^{-1}&c\\0&\rho_2^{-1}\end{pmatrix}\) is a real upper triangular expanding matrix, and \(D=\{d_0,\dots,d_{p-1}\}\subset\mathbb Z^2\) satisfies a prescribed zero-set condition for \(\widehat{\delta}_D\). In that setting the attractor is
\[
T=\sum_{k=1}^\infty M^{-k}D.
\]
The paper introducing this class refers to \(\mu_{M,D}\) as a planar Sierpinski-type self-affine measure [2508.20809].

Random and level-dependent analogues also appear in the literature. In random statistically self-affine Sierpinski sponges \(K\subset[0,1]^k\), a Galton–Watson percolation on an \(m_1\times\cdots\times m_k\) grid yields a random limit set and a Mandelbrot measure \(\mu\) on cylinders via multiplicative weights \(Q(u)\) and a martingale limit \(Y(u)\), with \(\mu([u])=Q(u)Y(u)\) [2002.00642]. In self-affine Moran constructions in \(\mathbb R^2\), the affine maps are level-dependent,
\[
\Phi_w(x)=T_k(x+w),\qquad T_k=\operatorname{diag}(n_k^{-1},m_k^{-1}),
\]
and a Bernoulli measure on the code space pushes forward to a self-affine Moran measure \(\mu_p\) on the attractor \(E\) [2309.08145].

These constructions share the same structural feature: the measure is determined recursively by affine contractions and digit data. What changes from one model class to another is the arithmetic of the digits, the geometry of the linear part, and the degree of randomness or level dependence.

## 2. Fourier analysis, orthogonal exponentials, and Hadamard triples

For the planar three-digit model, the Fourier transform has the infinite-product form
\[
\widehat\mu_{M,D}(\xi)
=\int_{\mathbb R^2}e^{-2\pi i\langle x,\xi\rangle}\,d\mu_{M,D}(x)
=\prod_{j=1}^\infty m_D\bigl((M^*)^{-j}\xi\bigr),
\]
where \(M^*\) is the transpose of \(M\) and
\[
m_D(x)=\frac13\sum_{d\in D}e^{-2\pi i\langle d,x\rangle}
\]
is the mask polynomial. The zero set \(Z(\widehat\mu_{M,D})\) controls the orthogonality of exponential functions in \(L^2(\mu_{M,D})\) [2010.14724].

A measure \(\mu_{M,D}\) is spectral if there exists a countable set \(\Lambda\subset\mathbb R^2\) such that
\[
\bigl\{e^{2\pi i\langle \lambda,x\rangle}:\lambda\in\Lambda\bigr\}
\]
is an orthonormal basis of \(L^2(\mu_{M,D})\). Orthogonality is equivalent to the bi-zero condition
\[
(\Lambda-\Lambda)\setminus\{0\}\subset Z\bigl(\widehat\mu_{M,D}\bigr).
\]
This reduces the existence of orthogonal exponential systems to a problem about the arithmetic structure of the Fourier zero set [2010.14724].

The central algebraic object in the classification is the Hadamard triple. For a pair \((M,D)\), this means there exists a set \(S\subset\mathbb Z^2\) with \(\#S=3\) such that
\[
\Bigl[e^{-2\pi i\langle M^{-1}d,s\rangle}\Bigr]_{d\in D,\;s\in S}
\]
is unitary. The 2020 classification shows that spectrality is equivalent to existence of a Hadamard triple after an integral similarity transformation. This makes Hadamard admissibility, up to conjugation by \(Q\in GL_2(\mathbb Z)\), the decisive criterion in the unresolved arithmetic regime [2010.14724].

Two auxiliary facts are especially important. First, similarity invariance: if \(M_1=A M A^{-1}\) and \(D_1=A D\), then \(\mu_{M_1,D_1}\) is spectral if and only if \(\mu_{M,D}\) is spectral. Second, a finiteness test: if \(Z(m_D)\cap[0,1)^2\) is finite but nonempty, then \(\mu_{M,D}\) admits only finitely many orthogonal exponentials. Together these results permit reduction to normalized digit configurations and residue-class analysis modulo \(3\) [2010.14724].

## 3. Complete spectral classification for generalized planar three-digit measures

Earlier work had already treated the cases \(\det(M)\notin 3\mathbb Z\) or \(\alpha_1\beta_2-\alpha_2\beta_1\notin 3\mathbb Z\). The remaining case is
\[
\det(M)\in 3\mathbb Z,\qquad
\Delta:=\alpha_1\beta_2-\alpha_2\beta_1\in 3\mathbb Z.
\]
In this regime, the complete criterion is
\[
\mu_{M,D}\text{ is spectral}
\quad\Longleftrightarrow\quad
\exists\,Q\in M_2(\mathbb Z)\text{ invertible over }\mathbb Z
\text{ such that }(QMQ^{-1},QD)\text{ is a Hadamard triple}.
\]
This settles the spectrality of the generalized Sierpinski-type self-affine measure \(\mu_{M,D}\) completely [2010.14724].

A particularly transparent subcase occurs when
\[
\beta_1-\alpha_1\in 3\mathbb Z,\qquad
\beta_2-\alpha_2\in 3\mathbb Z.
\]
Then \(\mu_{M,D}\) is spectral if and only if \(L^2(\mu_{M,D})\) admits an infinite orthogonal set of exponentials. In this form, the distinction between finite orthogonality and full spectrality collapses [2010.14724].

The proof splits according to the residues of
\[
\delta_1=\beta_1-\alpha_1,\qquad \delta_2=\beta_2-\alpha_2
\]
modulo \(3\). In Case I, where \(\delta_1,\delta_2\in 3\mathbb Z\), a unimodular similarity reduces the digit set to a normalized form \(\widetilde D=\{(0,0),(o,0),(0,w)\}\). The zero set of the mask polynomial is then analyzed, showing \(Z(m_{\widetilde D})\subset \frac13\mathbb Z^2\) and in fact containing a full lattice coset. Writing \(\widetilde M\) modulo \(3\) as one of ten residue types, a combinatorial analysis proves that eight types admit only finitely many orthogonal exponentials and hence are non-spectral, while the remaining two are converted to a standard Hadamard-triple form by a second similarity \(Q_n\) [2010.14724].

In Case II, where \(\delta_1\) or \(\delta_2\notin 3\mathbb Z\), the analysis again proceeds through residue classes mod \(3\), zero-set geometry, and conjugacy. Most residue classes are non-spectral; only when the “3-adic height” \(s\) of \(M\) exceeds a certain threshold \(n\) can one construct a Hadamard triple after suitable conjugation. The resulting equivalence is summarized in the formulation “spectral \(\Longleftrightarrow\) admissible up to conjugation” [2010.14724].

The classical Sierpinski gasket appears as
\[
M=3I,\qquad D=\{(0,0),(1,0),(0,1)\},
\]
for which \(\mu_{M,D}\) is spectral with spectrum \(\mathbb Z^2\). In the framework of the classification theorem, this is the basic spectral example among planar three-digit attractors [2010.14724].

## 4. Prime-digit triangular extensions and refined orthogonality criteria

A later extension studies
\[
M=\begin{pmatrix}\rho_1^{-1}&c\\0&\rho_2^{-1}\end{pmatrix},
\qquad
D=\{d_0,\dots,d_{p-1}\}\subset\mathbb Z^2,
\]
with \(p\) prime and with the rank-one zero-condition
\[
\mathcal Z\bigl(\widehat{\delta}_D\bigr)
=
\bigcup_{j=1}^{p-1}\Bigl(\tfrac jp\,\mathbf a+\mathbb Z^2\Bigr)
\]
for some nonzero \(\mathbf a\in\mathbb Z^2\) whose coordinates lie in \(\{1,\dots,p-1\}\). In this setting
\[
\mathcal Z(\widehat\mu_{M,D})
=
\bigcup_{k\ge 1}(M^*)^k\,\mathcal Z(\widehat{\delta}_D),
\]
so the same bi-zero philosophy remains in force, but the arithmetic now involves \(p\)-divisibility rather than only congruence modulo \(3\) [2508.20809].

In the diagonal case \(\rho_1=\rho_2=\rho\), infinite orthogonal systems exist if and only if
\[
\rho^{-1}=\Bigl(\frac ts\Bigr)^{1/r},
\qquad s,t,r\in\mathbb N,\quad \gcd(s,t)=1,\quad t\in p\mathbb Z,
\]
and
\[
c=\kappa\,\rho^{-1},\qquad \kappa\in\mathbb Q.
\]
The same paper gives sharp finite-size statements when these conditions fail partially. If \(c\notin\mathbb Q\,\rho^{-1}\), there can be at most \(p\) mutually orthogonal exponentials, and that bound is attained. If \(c=\kappa\rho^{-1}\) with \(\kappa\in\mathbb Q\setminus\{t/p\text{-integer}\}\), the maximal orthogonal size is exactly \(p\) when \(s\notin p\mathbb Z\), whereas \(s\in p\mathbb Z\) allows arbitrarily large orthogonal families, still without a basis in general [2508.20809].

The full diagonal spectral criterion is sharper:
\[
\mu_{M,D}\text{ is spectral}
\quad\Longleftrightarrow\quad
\rho^{-1}\in p\mathbb Z
\quad\text{and}\quad
c\in\Bigl\{\tfrac ts:t\in p\mathbb Z,\ \gcd(s,t)=1\Bigr\}\cup\{0\}.
\]
Thus infinite orthogonality, arbitrarily large orthogonal families, and complete spectrality become distinct arithmetic regimes [2508.20809].

When \(\rho_1\neq\rho_2\), a shear conjugation
\[
R=\begin{pmatrix}1&c''\\0&1\end{pmatrix},
\qquad
c''=\frac{c}{d_{1,1}(\rho_1^{-1}-\rho_2^{-1})},
\]
reduces the matrix to diagonal form and transfers the arithmetic to the transformed digit set. Spectrality forces \(c''\in\mathbb Q\) and \(\rho_1^{-1}\in p\mathbb Z\). There is then a dichotomy depending on whether \(c''\) lies in the exceptional set
\[
E_\mathbf a
=
\Bigl\{\tfrac hl:\;l\,a_2-h\,a_1\in\mathbb Z\setminus p\mathbb Z,\ \gcd(h,l)=1\Bigr\}\cup\{0\}.
\]
If \(c''\in\mathbb Q\setminus E_\mathbf a\), the only necessary-and-sufficient condition is \(\rho_1^{-1}\in p\mathbb Z\). If \(c''=s/t\in E_\mathbf a\) and \(t\) is divisible by exactly \(p^\ell\) but not by \(p^{\ell+1}\), then spectrality is equivalent to
\[
\rho_1^{-1},\rho_2^{-1}\in p\mathbb Z
\quad\text{and}\quad
p^{\ell+1}\mid (\rho_1^{-1}-\rho_2^{-1}).
\]
The paper packages these cases into a master criterion and also gives Example 6.1 as a counterexample to Conjecture 4.2 in [CWZ] [2508.20809].

These results suggest a broader pattern: for planar Sierpinski-type self-affine measures, spectrality is governed by a combination of zero-set geometry, conjugacy, and explicit \(p\)-adic or mod-\(3\) divisibility conditions.

## 5. Dimension theory, pressure, and measures of full dimension

Fourier spectrality is only one axis of the theory. A second axis concerns local dimensions and pressure. In a generic setting for self-affine and almost self-affine measures, the projected measure \(\nu\) is exact-dimensional, and for \(\nu\)-almost every \(x\),
\[
\dim_{\mathrm{loc}}\nu(x)=D_1(\nu)=\min\{d_1,n\},
\]
where \(d_1\) is the unique zero of the pressure function \(P(s,1)\). The pressure is defined from the singular-value function \(\phi^s(A)\) and the Bernoulli weights \(p_i\), and its zero furnishes the information dimension in this generic regime [1105.2411].

For random statistically self-affine Sierpinski sponges, the dimensional theory has a Ledrappier–Young form. Writing \(T(q)=-\log \mathbb E(\sum_a W_a^q)\), and denoting by \(\nu_i\) the averaged projected Bernoulli measure on the \(i\)-th factor, the Mandelbrot measure \(\mu\) is exact-dimensional and satisfies
\[
\dim_H\mu
=
\frac{T'(1^-)}{\ln m_1}
+
\sum_{i=2}^k
\Bigl(\frac{1}{\ln m_i}-\frac{1}{\ln m_{i-1}}\Bigr)
\min\bigl(T'(1^-),h_{\nu_i}(\pi_i)\bigr).
\]
The same work establishes the variational principle
\[
\dim_H K=\sup\{\dim_H\mu:\mu\text{ is Mandelbrot on }K\},
\]
and shows that the supremum is attained by a unique Mandelbrot measure \(\mu_*\) [2002.00642].

Level-dependent planar models admit analogous formulas. For self-affine Moran measures \(\mu_p\), the entropy dimensions are expressed through the approximate-square entropy \(H_k(p)\):
\[
\overline{\dim}_e\mu_p
=
\limsup_{k\to\infty}\frac{H_k(p)}{\sum_{i=1}^k\log m_i},
\qquad
\underline{\dim}_e\mu_p
=
\liminf_{k\to\infty}\frac{H_k(p)}{\sum_{i=1}^k\log m_i}.
\]
Under either frequency-separation (FSC) or measure-separation (MSC), one obtains for \(\mu_p\)-almost every \(x\) the corresponding local-dimension limits, hence
\[
\dim_H\mu_p=\underline{\dim}_e\mu_p,
\qquad
\dim_P\mu_p=\overline{\dim}_e\mu_p.
\]
A corollary identifies settings in which one can choose \(p\) so that \(\dim_H\mu_p=\dim_P\mu_p=\dim_H E=\dim_P E\) [2309.08145].

A related existence result comes from the thermodynamic formalism of subadditive cylinder functions. There exists an ergodic invariant equilibrium measure, and for typical self-affine sets there exists an ergodic invariant measure having the same Hausdorff dimension as the set. In the affine setting the natural cylinder function is the singular-value function, and the full-dimension measure arises at the zero of the corresponding pressure [1701.08578].

## 6. Random models, spectral asymptotics, and density dualities

In the literature on V-variable affine nested Sierpinski gaskets, the word “spectral” refers to eigenvalue asymptotics of Dirichlet forms rather than to orthonormal Fourier bases. For a product-type measure \(\mu\) on a V-variable fractal \(K(T)\), each cell \(\Delta_i\) carries a resistance scale \(r_i=\rho_i^{-1}\) and mass \(\mu_i\), and the crossing time is
\[
t_i:=\mu_i r_i.
\]
The spectral exponent \(\beta\) is the unique solution of
\[
\gamma(\beta):=\mathbb E_V\log\sum_{|i|=n(1)} t_i^{\beta/2}=0,
\]
and
\[
\frac{\beta}{2}
=
\lim_{\lambda\to\infty}\frac{\log N_D(\lambda)}{\log\lambda}
\qquad\text{a.s.}
\]
For the flat measure in the resistance metric,
\[
\beta=\frac{2\,d_f^r}{d_f^r+1},
\]
which is called the spectral dimension \(d_s\). The same work proves two-sided neck-cut heat-kernel bounds and almost-sure global small-\(t\) fluctuations of the form
\[
c_1 t^{-d_s/2}\phi(1/t)^{-c_2}
\le p_t(x,x)\le
c_3 t^{-d_s/2}\phi(1/t)^{c_4},
\qquad
\phi(u)=\exp(\sqrt{u\log\log u}),
\]
for the flat measure \(\nu\). This is a different notion of spectrality from the Fourier-analytic one used for \(\mu_{M,D}\) [1502.00711].

Another line of work connects discrete address distributions to geometric measure. For a self-affine set \(K(B,\mathcal D)\) and the associated address measure
\[
\mu=\lim_{s\to\infty}\sum_{\ell_0,\dots,\ell_{s-1}\in\mathcal D}
\delta_{\ell_0+B\ell_1+\cdots+B^{s-1}\ell_{s-1}},
\]
the upper Beurling density is
\[
D^+(\mu)
=
\limsup_{R\to\infty}\sup_{x\in\mathbb R^n}
\frac{\mu(x+[-R/2,R/2]^n)}{R^n}.
\]
When \(|\mathcal D|=|\det B|\), positivity of the Lebesgue measure of the self-affine tile is equivalent to the OSC, and
\[
|K|=\frac{1}{D^+(\mu)}.
\]
When \(|\mathcal D|<|\det B|\) and \(B\) is a similarity, the Hausdorff measure obeys
\[
\mathcal H^s(K)=\frac{1}{D_s^+(\mu)}
\]
under the OSC. The paper gives Sierpinski-type examples including the planar Sierpinski gasket and the Sierpinski carpet [1306.0079].

Taken together, these directions show that Sierpinski-type self-affine measures sit at the intersection of several theories. In one branch, the decisive question is whether \(L^2(\mu)\) admits an orthonormal basis of exponentials; in another, the focus is exact-dimensionality and pressure; in a third, the object of study is eigenvalue asymptotics and heat kernels on random fractals. The common thread is that affine recursion and digit arithmetic produce highly rigid measure-theoretic phenomena whose classification is often expressible in explicit algebraic, pressure-theoretic, or density-theoretic terms.

Source: https://www.emergentmind.com/topics/sierpinski-type-self-affine-measures