Sierpinski-Type Self-Affine Measures

• Sierpinski-type self-affine measures are Borel probability measures defined on attractors of affine IFS with a digit structure mimicking the classical Sierpinski gasket.
• They are analyzed using Fourier methods and Hadamard triples to determine spectral properties through explicit arithmetic and residue conditions.
• The framework extends to random, level-dependent, and prime-digit models, linking spectrality with pressure, dimensional theory, and geometric measure theory.

Searching arXiv for the cited papers and closely related work to ground the article. arxiv_search: {"^^^^^^^^5query5 Spectrality of generalized Sierpinski-type self-affine measures5^^^^^^^^", "^^^^^^^^5max_results5 5} Attempting direct arXiv lookup by identifier and title. arxiv_search({"^^^^^^^^5query5 OR \5^^^^^^^^"Spectrality of generalized Sierpinski-type self-affine measures\"", "^^^^^^^^5max_results5 ^^^^5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5query5^^^^}) 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 PRESERVED_PLACEHOLDER_^^^^5query5^^^^ and

PRESERVED_PLACEHOLDER_^^^^5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5^^^^

the associated measure PRESERVED_PLACEHOLDER_^^^^5max_results5^^^^ is defined by a self-affine refinement equation and is supported on a compact attractor PRESERVED_PLACEHOLDER_^^^^5query5^^^^. 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 PRESERVED_PLACEHOLDER_^^^^5id:(Liu et al., 2020) OR \5^^^^ and $\alpha_1\beta_2-\alpha_2\beta_1\in 3\mathbb Z$ (&&&^^^^5query5^^^^&&&).

^^^^5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5^^^^. 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 (&&&^^^^5query5^^^^&&&).

A broader planar class replaces the three-digit, integer-matrix setting by

PRESERVED_PLACEHOLDER_^^^^5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5query5^^^^

where PRESERVED_PLACEHOLDER_^^^^5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5^^^^ is prime, PRESERVED_PLACEHOLDER_^^^^5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5max_results5^^^^ is a real upper triangular expanding matrix, and PRESERVED_PLACEHOLDER_^^^^5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5query5^^^^ satisfies a prescribed zero-set condition for PRESERVED_PLACEHOLDER_^^^^5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5id:(Liu et al., 2020) OR \5^^^^. In that setting the attractor is

PRESERVED_PLACEHOLDER_^^^^5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures55^^^^

The paper introducing this class refers to PRESERVED_PLACEHOLDER_^^^^5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5^^^^6 as a planar Sierpinski-type self-affine measure (&&&^^^^5id:(Liu et al., 2020) OR \5^^^^&&&).

Random and level-dependent analogues also appear in the literature. In random statistically self-affine Sierpinski sponges PRESERVED_PLACEHOLDER_^^^^5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5^^^^7, a Galton–Watson percolation on an PRESERVED_PLACEHOLDER_^^^^5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5^^^^8 grid yields a random limit set and a Mandelbrot measure PRESERVED_PLACEHOLDER_^^^^5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5^^^^9 on cylinders via multiplicative weights PRESERVED_PLACEHOLDER_^^^^5max_results5query5^^^^ and a martingale limit PRESERVED_PLACEHOLDER_^^^^5max_results5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5^^^^, with PRESERVED_PLACEHOLDER_^^^^5max_results5max_results5^^^^ (Barral et al., 2020). In self-affine Moran constructions in PRESERVED_PLACEHOLDER_^^^^5max_results5query5^^^^, the affine maps are level-dependent,

PRESERVED_PLACEHOLDER_^^^^5max_results5id:(Liu et al., 2020) OR \5^^^^

and a Bernoulli measure on the code space pushes forward to a self-affine Moran measure PRESERVED_PLACEHOLDER_^^^^5max_results55^^^^ on the attractor PRESERVED_PLACEHOLDER_^^^^5max_results5^^^^6 (Gu et al., 2023).

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.

^^^^5max_results5^^^^. Fourier analysis, orthogonal exponentials, and Hadamard triples

For the planar three-digit model, the Fourier transform has the infinite-product form

PRESERVED_PLACEHOLDER_^^^^5max_results5^^^^7

where PRESERVED_PLACEHOLDER_^^^^5max_results5^^^^8 is the transpose of PRESERVED_PLACEHOLDER_^^^^5max_results5^^^^9 and

PRESERVED_PLACEHOLDER_^^^^5query5query5^^^^

is the mask polynomial. The zero set PRESERVED_PLACEHOLDER_^^^^5query5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5^^^^ controls the orthogonality of exponential functions in PRESERVED_PLACEHOLDER_^^^^5query5max_results5^^^^ (&&&^^^^5query5^^^^&&&).

A measure PRESERVED_PLACEHOLDER_^^^^5query5query5^^^^ is spectral if there exists a countable set PRESERVED_PLACEHOLDER_^^^^5query5id:(Liu et al., 2020) OR \5^^^^ such that

PRESERVED_PLACEHOLDER_^^^^5query55^^^^

is an orthonormal basis of PRESERVED_PLACEHOLDER_^^^^5query5^^^^6. Orthogonality is equivalent to the bi-zero condition

PRESERVED_PLACEHOLDER_^^^^5query5^^^^7

This reduces the existence of orthogonal exponential systems to a problem about the arithmetic structure of the Fourier zero set (&&&^^^^5query5^^^^&&&).

The central algebraic object in the classification is the Hadamard triple. For a pair PRESERVED_PLACEHOLDER_^^^^5query5^^^^8, this means there exists a set PRESERVED_PLACEHOLDER_^^^^5query5^^^^9 with PRESERVED_PLACEHOLDER_^^^^5id:(Liu et al., 2020) OR \5query5^^^^ such that

PRESERVED_PLACEHOLDER_^^^^5id:(Liu et al., 2020) OR \5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5^^^^

is unitary. The ^^^^5max_results5query5max_results5query5^^^^ 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 PRESERVED_PLACEHOLDER_^^^^5id:(Liu et al., 2020) OR \5max_results5^^^^, the decisive criterion in the unresolved arithmetic regime (&&&^^^^5query5^^^^&&&).

Two auxiliary facts are especially important. First, similarity invariance: if PRESERVED_PLACEHOLDER_^^^^5id:(Liu et al., 2020) OR \5query5^^^^ and PRESERVED_PLACEHOLDER_^^^^5id:(Liu et al., 2020) OR \5id:(Liu et al., 2020) OR \5^^^^, then PRESERVED_PLACEHOLDER_^^^^5id:(Liu et al., 2020) OR \55^^^^ is spectral if and only if PRESERVED_PLACEHOLDER_^^^^5id:(Liu et al., 2020) OR \5^^^^6 is spectral. Second, a finiteness test: if PRESERVED_PLACEHOLDER_^^^^5id:(Liu et al., 2020) OR \5^^^^7 is finite but nonempty, then PRESERVED_PLACEHOLDER_^^^^5id:(Liu et al., 2020) OR \5^^^^8 admits only finitely many orthogonal exponentials. Together these results permit reduction to normalized digit configurations and residue-class analysis modulo PRESERVED_PLACEHOLDER_^^^^5id:(Liu et al., 2020) OR \5^^^^9 (&&&^^^^5query5^^^^&&&).

^^^^5query5^^^^. Complete spectral classification for generalized planar three-digit measures

Earlier work had already treated the cases $\alpha_1\beta_2-\alpha_2\beta_1\in 3\mathbb Z$^^^^5query5^^^^ or $\alpha_1\beta_2-\alpha_2\beta_1\in 3\mathbb Z$^^^^5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5^^^^. The remaining case is

$\alpha_1\beta_2-\alpha_2\beta_1\in 3\mathbb Z$^^^^5max_results5^^^^

In this regime, the complete criterion is

$\alpha_1\beta_2-\alpha_2\beta_1\in 3\mathbb Z$^^^^5query5^^^^

This settles the spectrality of the generalized Sierpinski-type self-affine measure $\alpha_1\beta_2-\alpha_2\beta_1\in 3\mathbb Z$^^^^5id:(Liu et al., 2020) OR \5^^^^ completely (&&&^^^^5query5^^^^&&&).

A particularly transparent subcase occurs when

$\alpha_1\beta_2-\alpha_2\beta_1\in 3\mathbb Z$5

Then $\alpha_1\beta_2-\alpha_2\beta_1\in 3\mathbb Z$6 is spectral if and only if $\alpha_1\beta_2-\alpha_2\beta_1\in 3\mathbb Z$7 admits an infinite orthogonal set of exponentials. In this form, the distinction between finite orthogonality and full spectrality collapses (&&&^^^^5query5^^^^&&&).

The proof splits according to the residues of

$\alpha_1\beta_2-\alpha_2\beta_1\in 3\mathbb Z$8

modulo $\alpha_1\beta_2-\alpha_2\beta_1\in 3\mathbb Z$9. In Case I, where $\tau_d(x)=M^{-1}(x+d),\qquad d\in D,$^^^^5query5^^^^, a unimodular similarity reduces the digit set to a normalized form $\tau_d(x)=M^{-1}(x+d),\qquad d\in D,$^^^^5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5^^^^. The zero set of the mask polynomial is then analyzed, showing $\tau_d(x)=M^{-1}(x+d),\qquad d\in D,$^^^^5max_results5^^^^ and in fact containing a full lattice coset. Writing $\tau_d(x)=M^{-1}(x+d),\qquad d\in D,$^^^^5query5^^^^ modulo $\tau_d(x)=M^{-1}(x+d),\qquad d\in D,$^^^^5id:(Liu et al., 2020) OR \5^^^^ 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 $\tau_d(x)=M^{-1}(x+d),\qquad d\in D,$5 (&&&^^^^5query5^^^^&&&).

In Case II, where $\tau_d(x)=M^{-1}(x+d),\qquad d\in D,$6 or $\tau_d(x)=M^{-1}(x+d),\qquad d\in D,$7, the analysis again proceeds through residue classes mod $\tau_d(x)=M^{-1}(x+d),\qquad d\in D,$8, zero-set geometry, and conjugacy. Most residue classes are non-spectral; only when the “^^^^5query5^^^^-adic height” $\tau_d(x)=M^{-1}(x+d),\qquad d\in D,$9 of $T(M,D)=\bigcup_{d\in D}\tau_d\bigl(T(M,D)\bigr).$^^^^5query5^^^^ exceeds a certain threshold $T(M,D)=\bigcup_{d\in D}\tau_d\bigl(T(M,D)\bigr).$^^^^5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5^^^^ can one construct a Hadamard triple after suitable conjugation. The resulting equivalence is summarized in the formulation “spectral $T(M,D)=\bigcup_{d\in D}\tau_d\bigl(T(M,D)\bigr).$^^^^5max_results5^^^^ admissible up to conjugation” (&&&^^^^5query5^^^^&&&).

The classical Sierpinski gasket appears as

$T(M,D)=\bigcup_{d\in D}\tau_d\bigl(T(M,D)\bigr).$^^^^5query5^^^^

for which $T(M,D)=\bigcup_{d\in D}\tau_d\bigl(T(M,D)\bigr).$^^^^5id:(Liu et al., 2020) OR \5^^^^ is spectral with spectrum $T(M,D)=\bigcup_{d\in D}\tau_d\bigl(T(M,D)\bigr).$5. In the framework of the classification theorem, this is the basic spectral example among planar three-digit attractors (&&&^^^^5query5^^^^&&&).

^^^^5id:(Liu et al., 2020) OR \5^^^^. Prime-digit triangular extensions and refined orthogonality criteria

A later extension studies

$T(M,D)=\bigcup_{d\in D}\tau_d\bigl(T(M,D)\bigr).$6

with $T(M,D)=\bigcup_{d\in D}\tau_d\bigl(T(M,D)\bigr).$7 prime and with the rank-one zero-condition

$T(M,D)=\bigcup_{d\in D}\tau_d\bigl(T(M,D)\bigr).$8

for some nonzero $T(M,D)=\bigcup_{d\in D}\tau_d\bigl(T(M,D)\bigr).$9 whose coordinates lie in $$\mu_{M,D}(E)=\frac13\sum_{d\in D}\mu_{M,D}\bigl(M^{-1}(E-d)\bigr)$$^^^^5query5^^^^. In this setting

$$\mu_{M,D}(E)=\frac13\sum_{d\in D}\mu_{M,D}\bigl(M^{-1}(E-d)\bigr)$$^^^^5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5^^^^

so the same bi-zero philosophy remains in force, but the arithmetic now involves $$\mu_{M,D}(E)=\frac13\sum_{d\in D}\mu_{M,D}\bigl(M^{-1}(E-d)\bigr)$$^^^^5max_results5^^^^-divisibility rather than only congruence modulo $$\mu_{M,D}(E)=\frac13\sum_{d\in D}\mu_{M,D}\bigl(M^{-1}(E-d)\bigr)$$^^^^5query5^^^^ (&&&^^^^5id:(Liu et al., 2020) OR \5^^^^&&&).

In the diagonal case $$\mu_{M,D}(E)=\frac13\sum_{d\in D}\mu_{M,D}\bigl(M^{-1}(E-d)\bigr)$$^^^^5id:(Liu et al., 2020) OR \5^^^^, infinite orthogonal systems exist if and only if

$$\mu_{M,D}(E)=\frac13\sum_{d\in D}\mu_{M,D}\bigl(M^{-1}(E-d)\bigr)$$5

and

$$\mu_{M,D}(E)=\frac13\sum_{d\in D}\mu_{M,D}\bigl(M^{-1}(E-d)\bigr)$$6

The same paper gives sharp finite-size statements when these conditions fail partially. If $$\mu_{M,D}(E)=\frac13\sum_{d\in D}\mu_{M,D}\bigl(M^{-1}(E-d)\bigr)$$7, there can be at most $$\mu_{M,D}(E)=\frac13\sum_{d\in D}\mu_{M,D}\bigl(M^{-1}(E-d)\bigr)$$8 mutually orthogonal exponentials, and that bound is attained. If $$\mu_{M,D}(E)=\frac13\sum_{d\in D}\mu_{M,D}\bigl(M^{-1}(E-d)\bigr)$$9 with $E\subset \mathbb R^2$^^^^5query5^^^^, the maximal orthogonal size is exactly $E\subset \mathbb R^2$^^^^5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5^^^^ when $E\subset \mathbb R^2$^^^^5max_results5^^^^, whereas $E\subset \mathbb R^2$^^^^5query5^^^^ allows arbitrarily large orthogonal families, still without a basis in general (&&&^^^^5id:(Liu et al., 2020) OR \5^^^^&&&).

The full diagonal spectral criterion is sharper: $E\subset \mathbb R^2$^^^^5id:(Liu et al., 2020) OR \5^^^^ Thus infinite orthogonality, arbitrarily large orthogonal families, and complete spectrality become distinct arithmetic regimes (&&&^^^^5id:(Liu et al., 2020) OR \5^^^^&&&).

When $E\subset \mathbb R^2$5, a shear conjugation

$E\subset \mathbb R^2$6

reduces the matrix to diagonal form and transfers the arithmetic to the transformed digit set. Spectrality forces $E\subset \mathbb R^2$7 and $E\subset \mathbb R^2$8. There is then a dichotomy depending on whether $E\subset \mathbb R^2$9 lies in the exceptional set

PRESERVED_PLACEHOLDER_^^^^5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5query5query5^^^^

If PRESERVED_PLACEHOLDER_^^^^5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5query5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5^^^^, the only necessary-and-sufficient condition is PRESERVED_PLACEHOLDER_^^^^5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5query5max_results5^^^^. If PRESERVED_PLACEHOLDER_^^^^5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5query5query5^^^^ and PRESERVED_PLACEHOLDER_^^^^5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5query5id:(Liu et al., 2020) OR \5^^^^ is divisible by exactly PRESERVED_PLACEHOLDER_^^^^5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5query55^^^^ but not by PRESERVED_PLACEHOLDER_^^^^5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5query5^^^^6, then spectrality is equivalent to

PRESERVED_PLACEHOLDER_^^^^5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5query5^^^^7

The paper packages these cases into a master criterion and also gives Example 6.^^^^5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5^^^^ as a counterexample to Conjecture ^^^^5id:(Liu et al., 2020) OR \5^^^^.^^^^5max_results5^^^^ in CWZ.

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 PRESERVED_PLACEHOLDER_^^^^5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5query5^^^^8-adic or mod-PRESERVED_PLACEHOLDER_^^^^5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5query5^^^^9 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 PRESERVED_PLACEHOLDER_^^^^5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5query5^^^^ is exact-dimensional, and for PRESERVED_PLACEHOLDER_^^^^5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5^^^^-almost every PRESERVED_PLACEHOLDER_^^^^5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5max_results5^^^^,

PRESERVED_PLACEHOLDER_^^^^5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5query5^^^^

where PRESERVED_PLACEHOLDER_^^^^5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5id:(Liu et al., 2020) OR \5^^^^ is the unique zero of the pressure function PRESERVED_PLACEHOLDER_^^^^5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures55^^^^. The pressure is defined from the singular-value function PRESERVED_PLACEHOLDER_^^^^5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5^^^^6 and the Bernoulli weights PRESERVED_PLACEHOLDER_^^^^5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5^^^^7, and its zero furnishes the information dimension in this generic regime (&&&^^^^5max_results5query5^^^^&&&).

For random statistically self-affine Sierpinski sponges, the dimensional theory has a Ledrappier–Young form. Writing PRESERVED_PLACEHOLDER_^^^^5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5^^^^8, and denoting by PRESERVED_PLACEHOLDER_^^^^5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5^^^^9 the averaged projected Bernoulli measure on the PRESERVED_PLACEHOLDER_^^^^5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5max_results5query5^^^^-th factor, the Mandelbrot measure PRESERVED_PLACEHOLDER_^^^^5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5max_results5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5^^^^ is exact-dimensional and satisfies

PRESERVED_PLACEHOLDER_^^^^5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5max_results5max_results5^^^^

The same work establishes the variational principle

PRESERVED_PLACEHOLDER_^^^^5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5max_results5query5^^^^

and shows that the supremum is attained by a unique Mandelbrot measure PRESERVED_PLACEHOLDER_^^^^5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5max_results5id:(Liu et al., 2020) OR \5^^^^ (Barral et al., 2020).

Level-dependent planar models admit analogous formulas. For self-affine Moran measures PRESERVED_PLACEHOLDER_^^^^5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5max_results55^^^^, the entropy dimensions are expressed through the approximate-square entropy PRESERVED_PLACEHOLDER_^^^^5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5max_results5^^^^6: PRESERVED_PLACEHOLDER_^^^^5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5max_results5^^^^7 Under either frequency-separation (FSC) or measure-separation (MSC), one obtains for PRESERVED_PLACEHOLDER_^^^^5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5max_results5^^^^8-almost every PRESERVED_PLACEHOLDER_^^^^5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5max_results5^^^^9 the corresponding local-dimension limits, hence

PRESERVED_PLACEHOLDER_^^^^5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5query5query5^^^^

A corollary identifies settings in which one can choose PRESERVED_PLACEHOLDER_^^^^5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5query5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5^^^^ so that PRESERVED_PLACEHOLDER_^^^^5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5query5max_results5^^^^ (Gu et al., 2023).

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 (&&&^^^^5max_results5query5^^^^&&&).

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 PRESERVED_PLACEHOLDER_^^^^5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5query5query5^^^^ on a V-variable fractal PRESERVED_PLACEHOLDER_^^^^5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5query5id:(Liu et al., 2020) OR \5^^^^, each cell PRESERVED_PLACEHOLDER_^^^^5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5query55^^^^ carries a resistance scale PRESERVED_PLACEHOLDER_^^^^5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5query5^^^^6 and mass PRESERVED_PLACEHOLDER_^^^^5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5query5^^^^7, and the crossing time is

PRESERVED_PLACEHOLDER_^^^^5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5query5^^^^8

The spectral exponent PRESERVED_PLACEHOLDER_^^^^5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5query5^^^^9 is the unique solution of

PRESERVED_PLACEHOLDER_^^^^5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5id:(Liu et al., 2020) OR \5query5^^^^

and

PRESERVED_PLACEHOLDER_^^^^5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5id:(Liu et al., 2020) OR \5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5^^^^

For the flat measure in the resistance metric,

PRESERVED_PLACEHOLDER_^^^^5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5id:(Liu et al., 2020) OR \5max_results5^^^^

which is called the spectral dimension PRESERVED_PLACEHOLDER_^^^^5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5id:(Liu et al., 2020) OR \5query5^^^^. The same work proves two-sided neck-cut heat-kernel bounds and almost-sure global small-PRESERVED_PLACEHOLDER_^^^^5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5id:(Liu et al., 2020) OR \5id:(Liu et al., 2020) OR \5^^^^ fluctuations of the form

PRESERVED_PLACEHOLDER_^^^^5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5id:(Liu et al., 2020) OR \55^^^^

for the flat measure PRESERVED_PLACEHOLDER_^^^^5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5id:(Liu et al., 2020) OR \5^^^^6. This is a different notion of spectrality from the Fourier-analytic one used for PRESERVED_PLACEHOLDER_^^^^5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5id:(Liu et al., 2020) OR \5^^^^7 (&&&^^^^5max_results5id:(Liu et al., 2020) OR \5^^^^&&&).

Another line of work connects discrete address distributions to geometric measure. For a self-affine set PRESERVED_PLACEHOLDER_^^^^5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5id:(Liu et al., 2020) OR \5^^^^8 and the associated address measure

PRESERVED_PLACEHOLDER_^^^^5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5id:(Liu et al., 2020) OR \5^^^^9

the upper Beurling density is

PRESERVED_PLACEHOLDER_^^^^5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures55query5^^^^

When PRESERVED_PLACEHOLDER_^^^^5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures55(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures5^^^^, positivity of the Lebesgue measure of the self-affine tile is equivalent to the OSC, and

PRESERVED_PLACEHOLDER_^^^^5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures55max_results5^^^^

When PRESERVED_PLACEHOLDER_^^^^5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures55query5^^^^ and PRESERVED_PLACEHOLDER_^^^^5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures55id:(Liu et al., 2020) OR \5^^^^ is a similarity, the Hausdorff measure obeys

PRESERVED_PLACEHOLDER_^^^^5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures555^^^^

under the OSC. The paper gives Sierpinski-type examples including the planar Sierpinski gasket and the Sierpinski carpet (&&&^^^^5max_results55^^^^&&&).

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 PRESERVED_PLACEHOLDER_^^^^5(Liu et al., 2020) Spectrality of generalized Sierpinski-type self-affine measures55^^^^6 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.