Hausdorff Dimension of a Class of Self-Affine Sets

Abstract: In this paper, exact Hausdorff dimension formulas for a class of self-affine attractors generated by affine Iterated Function Systems are derived. We consider systems containing an affine map whose $n$-th iterate is a similarity contraction, alongside standard similarities whose linear parts commute with the symmetric operator $A^\top A$, where $A$ is the linear part of the affine map. We prove that the attractor of such a system exists uniquely, and, under the Open Set Condition, we compute its exact Hausdorff dimension. We extend this framework to systems where all map compositions of some fixed length are similarities, and to systems where overlaps are exact homothetic copies of the attractor. We unify these approaches to establish dimension formulas for hybrid systems that combine multiple eventually contractive affine maps with universally aligned similarities. Finally, we conclude with a topological classification of these systems in the plane. For a two-map system comprising an affine map whose second iterate is a similarity with contraction ratio $c$, alongside an $f$-aligned similarity with ratio $r$, we prove that the precise parameter balance $c + r = 1$ acts as a strict topological bottleneck uniquely guaranteeing both the open set condition and the connectedness of the attractor.

• The paper derives an explicit formula for the Hausdorff dimension of self-affine attractors using iterated similarity and alignment conditions.
• It establishes that when at least one map’s iterate is a similarity and aligned with principal axes, the dimension can be exactly computed via scalar equations.
• The study further provides a topological classification for planar systems, linking parameter sums to attractor connectivity and separation.

Hausdorff Dimension of a Class of Self-Affine Sets: Rigorous Formulas for Aligned-Affine Systems

Introduction and Technical Context

The problem of determining the precise Hausdorff dimension of self-affine fractal sets, in contrast to the well-studied self-similar setting, remains a central challenge in geometric measure theory and fractal geometry. While the theory for self-similar sets is robust—given sufficient separation, the dimension reduces to the solution of the Moran equation—the non-uniformity inherent in self-affine dynamics typically obstructs such closed-form characterization. Existing results for the self-affine case, notably the affinity dimension available through singular value functions [Falconer], provide optimal upper bounds but rarely match the true Hausdorff dimension except under restrictive geometric or probabilistic frameworks [10, 12, 2].

This work advances the theory by rigorously classifying a class of self-affine iterated function systems (IFS) for which the Hausdorff dimension of the attractor can be exactly computed via explicit scalar equations, thus paralleling the simplicity present in the self-similar case. The key structural conditions enabling this dimension formula are:

• The presence of at least one affine map whose $n$-th iterate is a similarity contraction (a $G_n$-similarity).
• The incorporation of similarity maps whose linear parts commute with the principal axes (as encoded by the symmetric matrix $A^T A$) of the affine generator—termed $f$-aligned similarities.

The authors develop and unify several generalizations, dealing with hybrid systems and systems with explicit overlap, culminating in both algebraic and topological classification results when the generators are planar maps.

Main Results and Mathematical Mechanisms

Exact Hausdorff Dimension Formula for Aligned-Elliptic Systems

The central theorem establishes that, under the stated alignment and eventual similarity conditions (including the Open Set Condition, OSC), the Hausdorff dimension $s$ of the attractor $A$ of an IFS $\mathcal{F} = \{f, g_1, ..., g_k\}$—with $f$ a $G_n$-similarity with ratio $c$ and each $G_n$0 an $G_n$1-aligned similarity with ratio $G_n$2—is governed by the unique solution $G_n$3 to:

$G_n$4

The existence and uniqueness of the attractor, as well as the validity of the mass distribution principle for verifying $G_n$5, are addressed via careful analysis of the formal weights in the IFS tree, structural commutativity, and geometric alignment. The proof is robust to higher-dimensional settings, provided the alignment condition is interpreted in terms of operator commutativity.

Generalizations: $G_n$6-Iterate Similarity and Overlap Structure

The dimension formula generalizes in two important directions:

• $G_n$7-Iterate Similarity Systems: If every word of length $G_n$8 in the maps is a similarity and the OSC holds, then the dimension formula reduces to solving

$G_n$9

with $A^T A$0 the contraction ratios of the $A^T A$1-fold compositions.

• Explicitly Homothetic Overlaps: If the overlap sets are homothetic to the attractor, the dimension correction aligns with the expected inclusion-exclusion via the formula:

$A^T A$2

where $A^T A$3 are the scaling ratios of overlap pieces carrying multiplicity $A^T A$4.

These extensions acknowledge the classic technical barrier that arbitrary overlaps typically ruin clean dimension formulas. The explicit handling of structured overlaps with homothetic copies marks a technical strengthening.

Hybrid Systems and Universal Alignment

For unions of $A^T A$5-iterate similarities and universally aligned similarities, the dimension formula melds the contraction contributions from both classes:

$A^T A$6

provided alignment holds between all pairs and the union system satisfies the OSC. The authors construct hybrid examples and discuss distortion constants required for mass-diameter estimates in the general measure-theoretic framework.

Topological Classification: Planar Case and Connectivity

A deep contribution is the topological characterization of the attractors in planar settings for two-map systems where one generator is a strict $A^T A$7-similarity and the other is an $A^T A$8-aligned similarity (excluding axial reflections). The main findings include:

• Critical Parameter Condition: The sum of the contractive ratios $A^T A$9 (where $f$0 from $f$1 and $f$2 from the similarity) uniquely guarantees both the OSC and the global connectedness of the attractor for all translation parameters.
• Sharp Dichotomy: If $f$3, the attractor is totally disconnected (strong separation); if $f$4, it is always connected but may not satisfy the OSC unless $f$5.
• Failure for Reflections: When the alignment is via a (scaled) axial reflection, neither the OSC nor connectedness follow from the parameter sum, breaking the dichotomy.

The proofs employ an intricate combination of invariant projections, dual axes analysis, and algebraic decomposition of affine and similarity actions, cementing the sharpness of the result.

Illustrative Examples and Numerical Results

Numerous two-dimensional IFS constructions illustrate the scope and necessity of the alignment and iterate-similarity conditions:

• Systems where all conditions hold: For maps with $f$6-th iterates as similarities and all other maps $f$7-aligned, the unique solution to the dimension equations is shown to coincide with numeric computations ($f$8, $f$9, $s$0 for the various constructed systems).
• Failure cases: When $s$1-alignment is violated (e.g., similarity maps introduce rotations mixing the principal axes), naively applying the formula yields a dimension less than 1, while direct projection proofs demonstrate that the true Hausdorff dimension is at least 1, confirming the indispensability of the alignment constraint.

Implications and Future Directions

The theoretical implications are multifold:

• Structural Insight: The work elucidates when non-uniform dynamics arising from affine maps can, through iterated similarity and alignment, be reduced to the scalar contraction analysis, thus bridging a significant portion of the complexity gap between self-affine and self-similar sets.
• Rigidity and Flexibility: The algebraic and topological dichotomies derived—particularly the strict cutoff at $s$2 for the existence of connected, non-overlapping attractors—provide comprehensive classification criteria for planar systems, with potential applications in inverse fractal construction and computer graphics where robust control of connectivity is required.
• Computational Tractability: By transforming the dimension problem for a class of self-affine sets into the solution of an explicit scalar equation, the results permit effective computation of fractal dimension in contexts hitherto accessible only via bounds or statistical approximation.

These results prompt several lines of further investigation:

• Beyond Universal Alignment: Exploring generalizations where the alignment constraint is relaxed to allow controlled non-commutativity may lead to new classes of computable systems.
• Robustness to Random Perturbations: The integration with random IFS techniques [14] and higher codimensional affine systems, especially those arising in applied contexts, could broaden the scope.
• Extension to Measures and Dynamical Invariants: While the current focus is on attractor geometry, dimension formulas for natural measures (e.g., self-affine Gibbs states) on these attractors remain a rich avenue, potentially linked to the Ledrappier-Young and Furstenberg measure machinery.

Conclusion

This paper establishes a full algebraic and geometric theory for the computation of Hausdorff dimensions in a certain class of self-affine IFSs, characterized by eventual similarity under iteration and precise commutative alignment among generators. The dimension is given by explicit formulae paralleling those known for self-similar sets. The approach fully addresses both measure-theoretic and topological aspects, offering rigidity/connectedness classifications for planar systems and demonstrating the failure of such strategies in the absence of alignment or under reflections. These findings refine our fundamental understanding of which self-affine fractals admit closed-form dimension computation and reveal sharp phase transitions in their topology and measure-theoretic structure.