Lalley–Gatzouras Carpets: Structure & Dimension

• Lalley–Gatzouras carpets are planar self-affine attractors defined by axis-parallel affine contractions arranged in a row/column structure with dominated anisotropy.
• Their analysis employs a symbolic organization using approximate squares to balance variable horizontal and vertical scales across rows.
• Dimension theory, invariant measures, multifractal analysis, and quantization of self-affine measures provide deep insights into their non-conformal geometric properties.

Searching arXiv for recent and foundational papers on Lalley–Gatzouras carpets and closely related results. A Lalley–Gatzouras carpet is a planar self-affine attractor generated by a finite family of axis-parallel affine contractions arranged in a carpet-like row/column structure with dominated anisotropy. In one standard formulation, the maps are

$$A_{ij}(x,y)=(a_{ij}x+c_{ij},\, b_i y + d_i), \quad (i,j)\in J,$$

with $0<a_{ij}<b_i<1$, disjoint placement conditions in each row and between rows, and attractor $S\subset[0,1]^2$ determined by

$S=\bigcup_{(i,j)\in J} A_{ij}(S).$

A closely related formulation writes

$$f_{ij}(x,y)=(x,y)\begin{pmatrix} a_{ij} & 0 \ 0 & b_j \end{pmatrix} +(c_{ij},d_j), \qquad E=\bigcup_{(i,j)\in G} f_{ij}(E),$$

under Lalley–Gatzouras separation and ordering assumptions ensuring that the images of the open unit square are disjoint and contained in the unit square. The literature uses both names “Lalley–Gatzouras carpets” and “Gatzouras–Lalley carpets” for the same class, and the coordinate convention varies across papers; the common core is a diagonally defined, dominated, non-conformal planar carpet that generalizes the more rigid Bedford–McMullen construction (Mackay, 2010, Zhu, 9 Aug 2025, Barral et al., 2011).

1. Definition, nomenclature, and geometric model

In Mackay’s formulation, a Lalley–Gatzouras carpet has $m$ horizontal rows, the $i$-th row has vertical contraction ratio $b_i$, and within that row there are $n_i$ disjoint rectangles with horizontal contraction ratios $a_{ij}$. The inequalities

$0<a_{ij}<b_i<1$0

encode the basic anisotropy, while the ordering and packing conditions on $0<a_{ij}<b_i<1$1 and $0<a_{ij}<b_i<1$2 force the first-level pieces to be pairwise disjoint rectangles inside $0<a_{ij}<b_i<1$3. Geometrically, each row is a horizontal band of height $0<a_{ij}<b_i<1$4, and the copies inside that band are strictly narrower than the band height (Mackay, 2010).

The 2025 quantization work uses the equivalent attractor picture of a finite family of diagonal affine contractions $0<a_{ij}<b_i<1$5 with

$0<a_{ij}<b_i<1$6

and emphasizes that the Lalley–Gatzouras separation and ordering assumptions (A1)–(A4) ensure that the images $0<a_{ij}<b_i<1$7 of the open unit square $0<a_{ij}<b_i<1$8 are disjoint and contained in $0<a_{ij}<b_i<1$9. By Hutchinson’s theorem there is a unique nonempty compact attractor $S\subset[0,1]^2$0 satisfying the invariance equation above (Zhu, 9 Aug 2025).

The same class is also described in more recent tangent work by a diagonal self-affine IFS

$S\subset[0,1]^2$1

with pairwise disjoint images, a column condition in the first coordinate, and domination

$S\subset[0,1]^2$2

This suggests that the apparent reversal of inequalities across papers is a matter of coordinate convention rather than a change in the geometric class (Käenmäki et al., 2024).

A standard specialization recovers Bedford–McMullen carpets: if the horizontal and vertical contractions are constant,

$S\subset[0,1]^2$3

with corresponding lattice translations, then the Lalley–Gatzouras carpet degenerates to a Bedford–McMullen carpet (Zhu, 9 Aug 2025).

2. Symbolic organization and approximate squares

The natural symbolic model uses a full shift over the digit set. For the repeller formulation, the symbolic space is

$S\subset[0,1]^2$4

with vertical factor

$S\subset[0,1]^2$5

and projection $S\subset[0,1]^2$6. The attractor is represented as

$S\subset[0,1]^2$7

and cylinders encode nested rectangles in the usual way (Reeve, 2010).

The central combinatorial difficulty is that equal symbolic depth does not imply comparable Euclidean scale: the contractions $S\subset[0,1]^2$8 and $S\subset[0,1]^2$9 vary. Both the multifractal and quantization analyses therefore replace ordinary cylinders by “approximate squares.” In Reeve’s formulation, for

$S=\bigcup_{(i,j)\in J} A_{ij}(S).$0

the approximate symbolic square

$S=\bigcup_{(i,j)\in J} A_{ij}(S).$1

corresponds to a rectangle whose diameter captures the anisotropic scaling correctly (Reeve, 2010).

The quantization paper introduces a family of approximate squares $S=\bigcup_{(i,j)\in J} A_{ij}(S).$2 whose elements have the form

$S=\bigcup_{(i,j)\in J} A_{ij}(S).$3

with $S=\bigcup_{(i,j)\in J} A_{ij}(S).$4 a word in $S=\bigcup_{(i,j)\in J} A_{ij}(S).$5 and $S=\bigcup_{(i,j)\in J} A_{ij}(S).$6 a word in the vertical alphabet $S=\bigcup_{(i,j)\in J} A_{ij}(S).$7, chosen so that the horizontal and vertical scales are balanced through

$S=\bigcup_{(i,j)\in J} A_{ij}(S).$8

The associated rectangle

$S=\bigcup_{(i,j)\in J} A_{ij}(S).$9

satisfies

$$f_{ij}(x,y)=(x,y)\begin{pmatrix} a_{ij} & 0 \ 0 & b_j \end{pmatrix} +(c_{ij},d_j), \qquad E=\bigcup_{(i,j)\in G} f_{ij}(E),$$0

and

$$f_{ij}(x,y)=(x,y)\begin{pmatrix} a_{ij} & 0 \ 0 & b_j \end{pmatrix} +(c_{ij},d_j), \qquad E=\bigcup_{(i,j)\in G} f_{ij}(E),$$1

These approximate squares are the geometric backbone of the argument, because they restore scale comparability in a genuinely non-uniform self-affine setting (Zhu, 9 Aug 2025).

3. Dimension theory and fine scaling

A basic large-scale invariant is the Assouad dimension. For the classical Lalley–Gatzouras carpet $$f_{ij}(x,y)=(x,y)\begin{pmatrix} a_{ij} & 0 \ 0 & b_j \end{pmatrix} +(c_{ij},d_j), \qquad E=\bigcup_{(i,j)\in G} f_{ij}(E),$$2, Mackay defines $$f_{ij}(x,y)=(x,y)\begin{pmatrix} a_{ij} & 0 \ 0 & b_j \end{pmatrix} +(c_{ij},d_j), \qquad E=\bigcup_{(i,j)\in G} f_{ij}(E),$$3 by

$$f_{ij}(x,y)=(x,y)\begin{pmatrix} a_{ij} & 0 \ 0 & b_j \end{pmatrix} +(c_{ij},d_j), \qquad E=\bigcup_{(i,j)\in G} f_{ij}(E),$$4

and $$f_{ij}(x,y)=(x,y)\begin{pmatrix} a_{ij} & 0 \ 0 & b_j \end{pmatrix} +(c_{ij},d_j), \qquad E=\bigcup_{(i,j)\in G} f_{ij}(E),$$5 as the maximal Hausdorff dimension of a horizontal fiber, obtained from a row $$f_{ij}(x,y)=(x,y)\begin{pmatrix} a_{ij} & 0 \ 0 & b_j \end{pmatrix} +(c_{ij},d_j), \qquad E=\bigcup_{(i,j)\in G} f_{ij}(E),$$6 for which

$$f_{ij}(x,y)=(x,y)\begin{pmatrix} a_{ij} & 0 \ 0 & b_j \end{pmatrix} +(c_{ij},d_j), \qquad E=\bigcup_{(i,j)\in G} f_{ij}(E),$$7

The main formula is

$$f_{ij}(x,y)=(x,y)\begin{pmatrix} a_{ij} & 0 \ 0 & b_j \end{pmatrix} +(c_{ij},d_j), \qquad E=\bigcup_{(i,j)\in G} f_{ij}(E),$$8

The same paper proves a conformal Assouad dimension dichotomy: $$f_{ij}(x,y)=(x,y)\begin{pmatrix} a_{ij} & 0 \ 0 & b_j \end{pmatrix} +(c_{ij},d_j), \qquad E=\bigcup_{(i,j)\in G} f_{ij}(E),$$9 while otherwise

$m$0

This identifies the Assouad dimension as the sum of a vertical projection term and a maximal horizontal fiber term (Mackay, 2010).

Recent tangent theory refines this local product picture. For a Gatzouras–Lalley carpet $m$1, writing $m$2 for projection onto the first coordinate axis and $m$3 for the symbolic slice determined by a coding $m$4, the maximal tangent size at a regular point $m$5 is

$m$6

If $m$7 satisfies the strong separation condition, this formula holds for all points. The same work proves that the set of points with tangents as large as possible has full Hausdorff measure at the critical exponent, and that for any

$m$8

the pointwise Assouad level set

$m$9

has Hausdorff dimension $i$0 (Käenmäki et al., 2024).

The Assouad spectrum gives a finer interpolation between box and Assouad dimensions. For Gatzouras–Lalley carpets, the spectrum is given explicitly by

$i$1

where $i$2 is the minimum of the column functions

$i$3

and $i$4 is a parameter change depending on

$i$5

The formula implies a piecewise-analytic spectrum that can be differentiable on all of $i$6 and strictly concave on open intervals (Banaji et al., 2024).

4. Invariant measures, full dimension, and multifractal structure

The Hausdorff dimension problem for Lalley–Gatzouras carpets is closely tied to variational principles over invariant measures. In the linear carpet construction used by Barral and Feng, Gatzouras and Lalley proved that the Hausdorff dimension of the attractor $i$7 is given by a variational quantity

$i$8

and that this dimension is attained by some Bernoulli measure. Barral and Feng then constructed a specific Gatzouras–Lalley carpet for which there are at least two distinct ergodic Bernoulli measures $i$9 such that

$b_i$0

thereby giving a negative answer to a conjecture of Gatzouras and Peres asserting uniqueness of the ergodic full-dimension measure (Barral et al., 2011).

That non-uniqueness is not universal. In a subclass $b_i$1 of non-linear Lalley–Gatzouras carpets defined by

$b_i$2

equivalently

$b_i$3

Jordan and Simon proved that there exists an open set $b_i$4 containing all non-trivial general Sierpiński carpets such that every carpet $b_i$5 has a unique ergodic measure of full dimension $b_i$6, and the map

$b_i$7

is continuous. The proof uses a relativized thermodynamic formalism, a characterization of full-dimension measures as relative equilibrium states, and strict concavity of the relevant pressure function under perturbation (Luzia, 2016).

The multifractal analysis of Birkhoff averages on Lalley–Gatzouras repellers exhibits the same measure-dependent anisotropy. For a continuous potential $b_i$8, the level set

$b_i$9

has Hausdorff dimension

$n_i$0

where $n_i$1 is the Ledrappier–Young quantity involving entropy and both Lyapunov exponents. The spectrum $n_i$2 is continuous on the full admissible interval. This is a genuinely non-conformal conditional variational principle: the dimension depends on the invariant measure through the pair of Lyapunov exponents, not through a single geometric exponent (Reeve, 2010).

5. Quantization of self-affine measures

A self-affine measure on a Lalley–Gatzouras carpet is defined from a positive probability vector

$n_i$3

by the self-affinity relation

$n_i$4

For a Borel probability measure $n_i$5 with finite $n_i$6-moment, the $n_i$7-th quantization error of order $n_i$8 is

$n_i$9

and the upper and lower $a_{ij}$0-dimensional quantization coefficients are

$a_{ij}$1

The quantization dimensions are defined from the asymptotics of $a_{ij}$2 in the usual way (Zhu, 9 Aug 2025).

For self-affine measures on a Lalley–Gatzouras carpet, the main 2025 theorem proves the existence of a unique $a_{ij}$3, defined by a pressure-type limit over approximate squares, such that

$a_{ij}$4

Equivalently, the quantization dimension exists,

$a_{ij}$5

and the quantization error has exact power-law decay,

$a_{ij}$6

The result substantially generalizes the previous Bedford–McMullen theory by allowing variable horizontal and vertical contractions and proving the exact asymptotic order of the quantization error, not merely the existence of the quantization dimension (Zhu, 9 Aug 2025).

The proof uses two new ingredients. For the lower bound, the author constructs pairwise disjoint approximate squares, special finite anti-chains, and a refined family $a_{ij}$7 so that the energies

$a_{ij}$8

are uniformly controlled and the quantization error can be bounded from below by a sum over these energies. For the upper bound, the argument constructs an auxiliary probability measure on

$a_{ij}$9

from the tight family

$0<a_{ij}<b_i<1$00

and applies Prohorov’s theorem to obtain a weak limit $0<a_{ij}<b_i<1$01 satisfying

$0<a_{ij}<b_i<1$02

This auxiliary measure controls the sums that determine the quantization coefficients (Zhu, 9 Aug 2025).

6. Separation structure, connectivity, and conformal geometry

The topology of Lalley–Gatzouras carpets is not exhausted by dimension formulas. For the classical planar self-affine sets, Xi and Xiong proved that a Lalley–Gatzouras set $0<a_{ij}<b_i<1$03 is uniformly disconnected if and only if $0<a_{ij}<b_i<1$04 is totally disconnected and there exists a row index $0<a_{ij}<b_i<1$05 such that

$0<a_{ij}<b_i<1$06

Thus the decisive separation feature is the existence of an empty row. In the same setting they showed that $0<a_{ij}<b_i<1$07 is quasisymmetrically equivalent to the Cantor ternary set if and only if $0<a_{ij}<b_i<1$08 is totally disconnected and some row is empty, and they derived the gap-sequence asymptotic

$0<a_{ij}<b_i<1$09

for totally disconnected carpets with an empty row (Xi et al., 2019).

A higher-dimensional reformulation appears in the theory of self-affine sponges of Lalley–Gatzouras type. Here the defining assumptions are the coordinate ordering condition and the neat projection condition. For a non-degenerated Lalley–Gatzouras type sponge $0<a_{ij}<b_i<1$10, the canonical Bernoulli measure $0<a_{ij}<b_i<1$11 is a component-counting measure if and only if

$0<a_{ij}<b_i<1$12

and this is equivalent to the requirement that every projection $0<a_{ij}<b_i<1$13 contains trivial points in the open cube, or equivalently that trivial points are dense in every projection. When the maximal power law holds, every cylinder $0<a_{ij}<b_i<1$14 satisfies

$0<a_{ij}<b_i<1$15

If it fails, then there exists

$0<a_{ij}<b_i<1$16

such that

$0<a_{ij}<b_i<1$17

for every cylinder (Zhang et al., 2023).

The conformal-dimension theory of the same sponge class makes the separation criterion even sharper. For a self-affine sponge $0<a_{ij}<b_i<1$18 of Lalley–Gatzouras type, the following are equivalent: $0<a_{ij}<b_i<1$19 is uniformly disconnected; all major projections $0<a_{ij}<b_i<1$20 are totally disconnected; and the attractors of all fiber IFSs are not $0<a_{ij}<b_i<1$21. The main conclusion is

$0<a_{ij}<b_i<1$22

If a fiber IFS has attractor $0<a_{ij}<b_i<1$23, then the sponge contains a line segment of the form $0<a_{ij}<b_i<1$24, which obstructs uniform disconnectedness and forces conformal dimension at least $0<a_{ij}<b_i<1$25 (Zhang et al., 29 Oct 2025).

A different topological phenomenon appears in a homogeneous self-similar Lalley–Gatzouras-type construction. Luo, Rao, and Xiong conjectured that a planar self-similar IFS with the open set condition and no rotations or reflections should have all connected components locally connected. A 24-map homogeneous counterexample of Lalley–Gatzouras type disproves this: the attractor satisfies the convex open set condition, uses only similarities

$0<a_{ij}<b_i<1$26

and still has a connected component that is not locally connected. The paper explicitly notes that this example is self-similar and homogeneous rather than a standard classical self-affine Lalley–Gatzouras carpet, but geometrically it is of Lalley–Gatzouras type in the sense of its row/column arrangement (Xiao, 2024).

7. Generalizations, weak separation, and overlapping variants

Several later developments enlarge the Lalley–Gatzouras framework without abandoning its dominated non-conformal structure. One direction replaces exact separation by weak separation. For diagonally aligned self-affine carpets whose coordinate projections satisfy the weak separation condition, the Hausdorff dimension equals the limit of the Barański formula over higher iterates, and the box-counting dimension is the limit of the Feng–Wang formula taken over the $0<a_{ij}<b_i<1$27-th level systems. In the ordered case relevant to Gatzouras–Lalley carpets, the paper gives an alternative limiting Hausdorff-dimension formula in terms of almost homogeneous $0<a_{ij}<b_i<1$28-level subsystems. Lalley–Gatzouras carpets are recovered in this framework when the system has a column structure, orientation-preserving maps, a coordinate ordering, and suitable separation assumptions (Bárány et al., 7 Jun 2025).

Another direction replaces diagonal matrices by lower triangular ones. Triangular Gatzouras–Lalley-type carpets are defined by

$0<a_{ij}<b_i<1$29

so the cylinders are parallelograms rather than rectangles, but the column structure is retained. Under Hochman’s Exponential Separation Condition for the horizontal IFS and suitable transversality assumptions controlling overlaps within columns, the paper proves that these overlaps do not cause dimension drop from the “GL brother” value in the Hausdorff-dimension formula; corresponding box-dimension results also hold under slightly different hypotheses (Kolossváry et al., 2018).

These extensions clarify the scope of the notion. In the strict sense, a Lalley–Gatzouras carpet is a diagonally defined, strongly separated, dominated planar self-affine carpet. In the broader literature, “Lalley–Gatzouras type” can also refer to higher-dimensional sponges, weakly separated diagonal carpets, non-linear skew-product perturbations, or homogeneous self-similar analogues. A plausible implication is that the enduring content of the concept is not a single normal form but a geometric paradigm: anisotropic product-like structure, symbolic organization by rows or columns, and dimension theory governed by the interaction of horizontal and vertical scaling.