Homogeneous Self-Similar Measures

• Homogeneous self-similar measures are invariant Borel probability measures generated by equicontractive IFS with a canonical Bernoulli coding structure.
• They are analyzed using exact dimensionality, entropy–Lyapunov formulas, and L^q-dimension methods to link overlap phenomena with Fourier decay and absolute continuity.
• Applications span fractal geometry, projection and slicing behavior, random models, and dynamical systems, highlighting interplay between symbolic regularity and arithmetic resonance.

Homogeneous self-similar measures are invariant Borel probability measures generated by an equicontractive iterated function system, typically of the form

$S_i(x)=r x+t_i \quad \text{on } \mathbb{R},$

or

$S_i(x)=r R x+t_i \quad \text{on } \mathbb{R}^2,$

with a common contraction factor, or more generally a common linear part. They satisfy a Hutchinson-type fixed-point equation

$\mu=\sum_i p_i\,\mu\circ S_i^{-1},$

and may also be realized as laws of random series such as $\sum_{n\ge 0} r^n \xi_n$. Within fractal geometry they constitute a basic class for the study of exact dimensionality, $L^q$-spectra, Fourier decay, projections, slices, absolute continuity, and random fractal models. At the same time, recent work shows that high dimension, strong separation, and dense rotations do not by themselves enforce regular projection or slicing behavior (Rapaport, 2017, Solomyak, 2019).

1. Classical framework and terminology

In the classical one-dimensional setting, a homogeneous self-similar measure is associated to an IFS

$S_i(x)=r x+t_i,\qquad i=1,\dots,m,$

with a common contraction ratio $r\in(0,1)$ and a probability vector $p=(p_1,\dots,p_m)$. The invariant measure $\mu$ is the unique Borel probability measure satisfying

$\mu=\sum_{i=1}^m p_i\,\mu\circ S_i^{-1}.$

Equivalent formulations identify $S_i(x)=r R x+t_i \quad \text{on } \mathbb{R}^2,$0 as the law of the random series

$S_i(x)=r R x+t_i \quad \text{on } \mathbb{R}^2,$1

where the $S_i(x)=r R x+t_i \quad \text{on } \mathbb{R}^2,$2 are i.i.d. and take values in the digit set $S_i(x)=r R x+t_i \quad \text{on } \mathbb{R}^2,$3 with probabilities $S_i(x)=r R x+t_i \quad \text{on } \mathbb{R}^2,$4 (Solomyak, 2019, Simon et al., 2017).

In the planar setting, homogeneous usually means that all similarities share the same linear part. Rapaport formulates this as

$S_i(x)=r R x+t_i \quad \text{on } \mathbb{R}^2,$5

where $S_i(x)=r R x+t_i \quad \text{on } \mathbb{R}^2,$6 is fixed and $S_i(x)=r R x+t_i \quad \text{on } \mathbb{R}^2,$7 is fixed, so every branch has the same contraction and rotation. The strong separation condition (SSC) is the requirement that the first-level pieces are pairwise disjoint subsets of the attractor (Rapaport, 2017).

The term is not entirely uniform across subfields. In graph-directed systems on limit spaces of self-similar groups, the canonical measure on the tile is called homogeneous because the inverse branches carry equal weights $S_i(x)=r R x+t_i \quad \text{on } \mathbb{R}^2,$8, yielding the invariance relation

$S_i(x)=r R x+t_i \quad \text{on } \mathbb{R}^2,$9

for Borel $\mu=\sum_i p_i\,\mu\circ S_i^{-1},$0 (Bondarenko et al., 2010). In random homogeneous models, the word “homogeneous” refers to the fact that at each level a single environment is chosen and applied across all branches simultaneously, rather than independently branch by branch (Banerjee et al., 22 Dec 2025).

This variation of terminology suggests that two structural features recur throughout the literature: a common geometric scaling mechanism, and a canonical symbolic coding with a Bernoulli-type product structure.

2. Dimension, entropy, and overlap structure

For exact-dimensional self-similar measures under SSC, the entropy–Lyapunov formula takes the form

$\mu=\sum_i p_i\,\mu\circ S_i^{-1},$1

In the homogeneous case with equal weights this simplifies to

$\mu=\sum_i p_i\,\mu\circ S_i^{-1},$2

On the line, the same quantity is usually written as the similarity dimension

$\mu=\sum_i p_i\,\mu\circ S_i^{-1},$3

(Rapaport, 2017, Simon et al., 2017).

The expected formula

$\mu=\sum_i p_i\,\mu\circ S_i^{-1},$4

is known to hold in broad one-parameter families when there are no exact overlaps, or more generally no super-exponential concentration of cylinders. In the language of homogeneous line IFS, exact overlaps mean that distinct words of the same length generate the same cylinder map, equivalently the same translation parameter. Hochman’s dimension theorem is invoked in this form in the study of singularity versus exact overlaps (Simon et al., 2017).

For homogeneous systems of three maps on $\mu=\sum_i p_i\,\mu\circ S_i^{-1},$5, the dimension problem becomes subtler. After affine normalization one may write

$\mu=\sum_i p_i\,\mu\circ S_i^{-1},$6

and the paper on three-map systems develops an algebraic approximation theory adapted to this setting. A key feature is the appearance of families of exact-overlap curves

$\mu=\sum_i p_i\,\mu\circ S_i^{-1},$7

with $\mu=\sum_i p_i\,\mu\circ S_i^{-1},$8 drawn from a class of ratios of power series with coefficients in $\mu=\sum_i p_i\,\mu\circ S_i^{-1},$9. In this model, the exceptional parameter set where the expected dimension formula fails but there are no exact overlaps has Hausdorff dimension zero (Rapaport et al., 2020).

Beyond Hausdorff dimension, the $\sum_{n\ge 0} r^n \xi_n$0-dimension admits an explicit formula under exponential separation. For homogeneous self-similar measures on $\sum_{n\ge 0} r^n \xi_n$1,

$\sum_{n\ge 0} r^n \xi_n$2

In the equal-weight case $\sum_{n\ge 0} r^n \xi_n$3, this reduces to the similarity dimension whenever it is at most $\sum_{n\ge 0} r^n \xi_n$4 (Shmerkin, 2016).

A recurrent theme is that dimension loss is strongly linked to overlap phenomena, but not all regularity failures reduce to exact overlaps. This distinction becomes especially sharp in projection problems and in supercritical regimes.

3. Fourier structure, decay, and absolute continuity

Homogeneity yields a decisive harmonic-analytic simplification. If

$\sum_{n\ge 0} r^n \xi_n$5

then the Fourier transform satisfies

$\sum_{n\ge 0} r^n \xi_n$6

and hence

$\sum_{n\ge 0} r^n \xi_n$7

This Riesz-product structure is the basis of the homogeneous Fourier decay theory, including the Erdős–Kahane mechanism and its later extensions (Solomyak, 2019).

A central consequence is that, outside a zero Hausdorff dimension exceptional set of parameters, homogeneous self-similar measures on the line have power Fourier decay,

$\sum_{n\ge 0} r^n \xi_n$8

for some $\sum_{n\ge 0} r^n \xi_n$9. In the Bernoulli-convolution case, Pisot arithmetic yields the classical obstruction: if $L^q$0 is Pisot, then $L^q$1 need not even tend to zero at infinity. By contrast, for non-Pisot parameters one has Rajchman decay, and outside a zero-dimensional exceptional set one gets quantitative power decay (Solomyak, 2019).

A quantitative Kaufman–Tsujii-type refinement for homogeneous measures on $L^q$2 shows that the set of large frequencies where $L^q$3 fails to decay can be covered by comparatively few short intervals. This sparse-frequency control yields several consequences. First, non-linear smooth images $L^q$4 with $L^q$5 have power Fourier decay. Second, convolution with a homogeneous self-similar measure increases correlation dimension by a quantitative amount. Third, the dimension and Frostman exponent of biased Bernoulli convolutions tend to $L^q$6 as the contraction ratio tends to $L^q$7, with explicit quantitative rates (Mosquera et al., 2017).

The same product structure also feeds into density results. For Bernoulli convolutions, outside a zero-dimensional exceptional set in $L^q$8, the measure belongs to $L^q$9 for every finite $S_i(x)=r x+t_i,\qquad i=1,\dots,m,$0; outside a zero-dimensional exceptional set in $S_i(x)=r x+t_i,\qquad i=1,\dots,m,$1 it has a continuous density. More generally, the $S_i(x)=r x+t_i,\qquad i=1,\dots,m,$2-dimension formalism for dynamically driven self-similar measures extends Hochman’s entropy-based methods to $S_i(x)=r x+t_i,\qquad i=1,\dots,m,$3 norms by means of an inverse theorem for convolutions of discrete measures (Shmerkin, 2016).

These results establish a broad dichotomy. Arithmetic resonance can obstruct decay and promote singularity, but away from small exceptional sets the homogeneous convolution structure strongly favors Fourier decay and, in supercritical regimes, absolute continuity.

4. Projections, slices, and dimension conservation

The planar theory exhibits both the most striking pathologies and some of the strongest positive results. For planar self-similar measures with SSC and dense rotations, Hochman–Shmerkin’s projection theorem gives

$S_i(x)=r x+t_i,\qquad i=1,\dots,m,$4

where $S_i(x)=r x+t_i,\qquad i=1,\dots,m,$5 is the unit circle and

$S_i(x)=r x+t_i,\qquad i=1,\dots,m,$6

Thus, if $S_i(x)=r x+t_i,\qquad i=1,\dots,m,$7, every orthogonal projection has full Hausdorff dimension $S_i(x)=r x+t_i,\qquad i=1,\dots,m,$8 (Rapaport, 2017).

Rapaport constructed a homogeneous planar self-similar measure with SSC, dense rotations, and $S_i(x)=r x+t_i,\qquad i=1,\dots,m,$9 such that the set of directions with singular projections contains a dense $r\in(0,1)$0 subset of $r\in(0,1)$1. In the same residual set of directions, typical slices perpendicular to the projection direction are discrete, and dimension conservation fails:

$r\in(0,1)$2

The mechanism is arithmetic. The construction uses a complex Pisot number $r\in(0,1)$3, writes $r\in(0,1)$4, and exploits the near-integrality estimate

$r\in(0,1)$5

to force non-decay of Fourier transforms in residual sets of directions (Rapaport, 2017).

This establishes an important correction to a common heuristic: large dimension and dense rotations do not imply absolute continuity of every projection, nor do they imply dimension conservation in every direction.

A complementary body of results shows that these irregularities are not generic in parameter space. Outside a Hausdorff-dimension-zero exceptional set of contraction parameters, every planar homogeneous self-similar measure with SSC, dense rotations, and dimension greater than $r\in(0,1)$6 has projections in all directions belonging to some common $r\in(0,1)$7, $r\in(0,1)$8. In that regime, the map

$r\in(0,1)$9

is continuous in the weak topology of $p=(p_1,\dots,p_m)$0, and $p=(p_1,\dots,p_m)$1 is dimension conserving in each direction. Moreover, for every direction $p=(p_1,\dots,p_m)$2, the projected measure is equivalent to Lebesgue measure restricted to the projected attractor (Rapaport, 2018).

A related one-parameter family of homogeneous self-similar measures on the line arises from projecting the natural measure of the Sierpiński carpet. In that setting the singular parameter set is a dense $p=(p_1,\dots,p_m)$3 but has Hausdorff dimension zero, and singularity is not caused by exact overlaps: exact overlaps occur only for countably many angles, whereas singular parameters form a residual set. This gives a clear instance where topological largeness and fractal smallness diverge sharply (Simon et al., 2017).

5. Random, graph-directed, and group-theoretic extensions

Homogeneous self-similar measures extend naturally beyond deterministic Euclidean IFS. In the framework of contracting self-similar groups, the limit $p=(p_1,\dots,p_m)$4-space carries a canonical measure obtained by pushing forward the uniform Bernoulli measure on $p=(p_1,\dots,p_m)$5. Restricted to the tile $p=(p_1,\dots,p_m)$6, this measure satisfies the equal-weight self-similarity relation

$p=(p_1,\dots,p_m)$7

so it is homogeneous in the sense of equal probabilities across inverse branches. The associated limit dynamical system $p=(p_1,\dots,p_m)$8 is conjugate to the one-sided Bernoulli $p=(p_1,\dots,p_m)$9-shift, and the tile measure $\mu$0 is an integer computable from the nucleus automaton (Bondarenko et al., 2010).

Random homogeneous IFS provide another major extension. In the model studied by Hare–Hare–Troscheit, an environment $\mu$1 is sampled from a Bernoulli measure, and at level $\mu$2 the IFS indexed by $\mu$3 is applied uniformly across all cylinders. The random measure satisfies

$\mu$4

Under the uniform strong separation condition, the multifractal formalism parallels the deterministic homogeneous case: the attainable local dimensions form an interval, and the spectrum is obtained from a pressure function $\mu$5. In overlapping equicontractive finite-type systems, local dimensions are described by Lyapunov exponents of transition matrices, and an essential class governs almost all points. In the commuting subcase, the attainable local dimensions form almost surely a closed interval (Hare et al., 2017).

Quantization theory has recently been developed for random homogeneous self-similar measures as well. For $\mu$6, the almost sure quantization dimension is determined by a critical exponent $\mu$7 derived from the expected pressure equation

$\mu$8

Under the uniform extra strong separation condition, one has

$\mu$9

With an additional strong uniform open set condition, the lower $\mu=\sum_{i=1}^m p_i\,\mu\circ S_i^{-1}.$0-dimensional quantization coefficient is almost surely positive; without any separation assumption, a cocycle boundedness condition suffices for almost sure finiteness of the upper coefficient (Banerjee et al., 22 Dec 2025).

These extensions show that the homogeneous paradigm is not confined to a single deterministic Euclidean model. It persists in symbolic, group-directed, and random settings, but the relevant regularity statements are then mediated by ergodic theory, automata, transition matrices, and pressure.

6. Higher-dimensional, dynamical, and current directions

In the plane, a general absolute continuity theorem for self-similar measures on $\mu=\sum_{i=1}^m p_i\,\mu\circ S_i^{-1}.$1 states that for fixed distinct translations and positive probabilities, and for Lebesgue-almost every contraction parameter in the supercritical region

$\mu=\sum_{i=1}^m p_i\,\mu\circ S_i^{-1}.$2

the corresponding measure is absolutely continuous. In the homogeneous case this recovers the typical-parameter paradigm for measures of the form

$\mu=\sum_{i=1}^m p_i\,\mu\circ S_i^{-1}.$3

and embeds it into a broader framework involving random self-similar measures, power Fourier decay, and dimension alternatives (Solomyak et al., 2023).

For dimensions $\mu=\sum_{i=1}^m p_i\,\mu\circ S_i^{-1}.$4, recent work treats homogeneous self-similar measures of the form

$\mu=\sum_{i=1}^m p_i\,\mu\circ S_i^{-1}.$5

with $\mu=\sum_{i=1}^m p_i\,\mu\circ S_i^{-1}.$6. If the digit set $\mu=\sum_{i=1}^m p_i\,\mu\circ S_i^{-1}.$7 spans $\mu=\sum_{i=1}^m p_i\,\mu\circ S_i^{-1}.$8, then for each fixed $\mu=\sum_{i=1}^m p_i\,\mu\circ S_i^{-1}.$9 and positive probability vector $S_i(x)=r R x+t_i \quad \text{on } \mathbb{R}^2,$00, one has power Fourier decay for all but a zero-Hausdorff-dimension set of $S_i(x)=r R x+t_i \quad \text{on } \mathbb{R}^2,$01. In even dimensions $S_i(x)=r R x+t_i \quad \text{on } \mathbb{R}^2,$02, assuming only affine irreducibility, power Fourier decay holds for Lebesgue-almost every $S_i(x)=r R x+t_i \quad \text{on } \mathbb{R}^2,$03 and Haar-almost every $S_i(x)=r R x+t_i \quad \text{on } \mathbb{R}^2,$04. Combined with recent $S_i(x)=r R x+t_i \quad \text{on } \mathbb{R}^2,$05-dimension results, this yields absolute continuity with $S_i(x)=r R x+t_i \quad \text{on } \mathbb{R}^2,$06 density in the supercritical parameter regime for almost all such homogeneous systems (Solomyak, 20 Aug 2025).

Homogeneous self-similar measures also arise naturally in dynamics. For a homogeneous SSC self-similar set $S_i(x)=r R x+t_i \quad \text{on } \mathbb{R}^2,$07 with natural equal-weight measure $S_i(x)=r R x+t_i \quad \text{on } \mathbb{R}^2,$08 and shift-induced map $S_i(x)=r R x+t_i \quad \text{on } \mathbb{R}^2,$09, the quantitative recurrence set

$S_i(x)=r R x+t_i \quad \text{on } \mathbb{R}^2,$10

satisfies the sharp dichotomy

$S_i(x)=r R x+t_i \quad \text{on } \mathbb{R}^2,$11

where

$S_i(x)=r R x+t_i \quad \text{on } \mathbb{R}^2,$12

A parallel zero-full law holds for Hausdorff $S_i(x)=r R x+t_i \quad \text{on } \mathbb{R}^2,$13-measures under a doubling hypothesis on $S_i(x)=r R x+t_i \quad \text{on } \mathbb{R}^2,$14 (Chang et al., 2018).

Several open directions are explicit in the current literature. Rapaport’s construction shows that under dense rotations it remains essential to exclude arithmetic resonances if one wants absolute continuity of all projections or dimension conservation in all directions (Rapaport, 2017). In the non-homogeneous planar theory, typical absolute continuity is known, but the approach used there does not directly extend to $S_i(x)=r R x+t_i \quad \text{on } \mathbb{R}^2,$15 because of non-commutative growth issues in $S_i(x)=r R x+t_i \quad \text{on } \mathbb{R}^2,$16 (Solomyak et al., 2023). In random quantization, weakening UESSC or clarifying the relation between UOSC and SUOSC remains open (Banerjee et al., 22 Dec 2025). A plausible implication is that the modern theory of homogeneous self-similar measures is increasingly organized around a tension between symbolic regularity and arithmetic resonance: the former drives exact formulas and typical smoothing, while the latter produces the sharpest counterexamples.