For a one-to-one self-conformal contractive system on with attractor K and conformality dimension α, Peres et al. showed that the open set condition and strong open set condition are both equivalent to . We give a simple proof of this result as well as discuss some further properties related to the separation condition.
Denote by , , the regular tree whose vertices have valence , its boundary. Yu. A. Neretin has proposed a group of transformations of , thought of as a combinatorial analogue of the diffeomorphism group of the circle. We show that is generated by two groups: the group of tree automorphisms, and a Higman-Thompson group . We prove the simplicity of and of a family of its subgroups.
We present a new method for establishing the ‘‘gap” property for finitely generated subgroups of , providing an elementary solution of Ruziewicz problem on as well as giving many new examples of finitely generated subgroups of with an explicit gap. The distribution of the eigenvalues of the elements of the group ring in the -th irreducible representation of is also studied. Numerical experiments indicate that for a generic (in measure) element of , the “unfolded” consecutive spacings...
Let (G,X) be a transformation group, where X is a locally compact Hausdorff space and G is a compact group. We investigate the stable rank and the real rank of the transformation group C*-algebra C₀(X)⋊ G. Explicit formulae are given in the case where X and G are second countable and X is locally of finite G-orbit type. As a consequence, we calculate the ranks of the group C*-algebra C*(ℝⁿ ⋊ G), where G is a connected closed subgroup of SO(n) acting on ℝⁿ by rotation.
We show that the automorphism group Aut([0,1],λ) of the Lebesgue measure has no non-trivial subgroups of index .