A Conjecture on the Hausdorff Dimension of Attractors of Real Self-Projective Iterated Function Systems

Abstract

In a recent work, we proved that under natural conditions satisfied in several important examples, the rate of growth 1/s_A of norms of matrices in a semigroup (respectively ) dictates the Hausdorff dimension of the attractor R_A of the corresponding semigroups of projective transformations on (respectively ). In the present work, we begin a study of the higher-dimensional case. In particular, we introduce a certain family of semigroups , and we study numerically concrete cases for n = 3 and n = 4. Our results suggest that for n ≥ 3, (n − 1)s_A/n is a lower bound for the Hausdorff dimension of R_A.

2000 AMS Subject Classification::

• 28A80
• 28A78
• 20M20
• 53A20

Keywords::

• real self-projective iterated function systems
• self-affine iterated function systems
• Apollonian gasket
• Sierpiński gasket
• Hausdorff dimension

Notes

¹Recall that an IFS {f₁, … , f_m} on a metric space (M, d) is said to be hyperbolic if all f_i are contractions with respect to d, and parabolic if all f_i are nonexpanding maps.

²By this we mean that N₁ = ‖C₁‖, N₂ = ‖C₂‖, N₃ = ‖C₃‖, N₄ = ‖C₁C₁‖, N₅ = ‖C₁C₂‖, … , N₁₄ = ‖C₁C₁C₁‖, and so on.

³In Soddy’s honor, the two new circles are called Soddy’s circles.

⁴Since all norms are equivalent in finite dimension, this is true for every norm.