Abstract
In a recent work, we proved that under natural conditions satisfied in several important examples, the rate of growth 1/sA of norms of matrices in a semigroup (respectively ) dictates the Hausdorff dimension of the attractor RA 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)sA/n is a lower bound for the Hausdorff dimension of RA.
Notes
1Recall that an IFS {f1, … , fm} on a metric space (M, d) is said to be hyperbolic if all fi are contractions with respect to d, and parabolic if all fi 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.