When the conformal dimension of a self-affine sponge of Lalley type is zero

Abstract: A compact metric space $X$ is uniformly disconnected if there exists $\delta_0$ such that there is no $\delta_0$-sequence, which is a sequence of points $(x_0,x_1,\dots,x_n)$ satisfying $\rho(x_{i-1},x_{i})\leq \delta_0 \rho(x_0,x_n)$ for all $1\leq i\leq n$. We present two main results. First, we give a necessary and sufficient condition for a diagonal self-affine sponge of Lalley-Gatzouras type to be uniformly disconnected. Second, we show that $K$ is uniformly disconnected if and only if the conformal dimension of $K$ is $0$.

dynamical systemsnumber theory

Audience: researchers in the topic

( paper | slides )

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