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Register a new userOn the Measure of the Midpoints of the Cantor Set in $mathbb{R}$
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In this paper, we are going to discuss the following problem: Let $T$ be a fixed set in $mathbb{R}^n$. And let $S$ and $B$ he two subsets in $mathbb{R}^n$ such that for any $x$ in $S$, there exists an $r$ such that $x+ r T$ is a subset of $B$. How small can be $B$ be if we know the size of $S$? Stein proved that for $n$ is greater than or equal to 3 and $T$ is a sphere centered at origin, then $S$ has positive measure implies $B$ has positive measure using spherical maximal operator. Later, Bourgain and Marstrand proved the similar result for $n =2$. And we found an example for why the result fails for $n=1$.
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Consider an arbitrary extension of a free $mathbb Z^d$-action on the Cantor set. It is shown that it has dynamical asymptotic dimension at most $3^d - 1$.
By a Cantor-like measure we mean the unique self-similar probability measure $mu $ satisfying $mu =sum_{i=0}^{m-1}p_{i}mu circ S_{i}^{-1}$ where $% S_{i}(x)=frac{x}{d}+frac{i}{d}cdot frac{d-1}{m-1}$ for integers $2leq d<mle 2d-1$ and probabilities $p_{i}>0$, $sum p_{i}=1$. In the uniform case ($p_{i}=1/m$ for all $i$) we show how one can compute the entropy and Hausdorff dimension to arbitrary precision. In the non-uniform case we find bounds on the entropy.
We describe a numerical technique to compute the equilibrium measure, in logarithmic potential theory, living on the attractor of Iterated Function Systems composed of one-dimensional affine maps. This measure is obtained as the limit of a sequence of equilibrium measures on finite unions of intervals. Although these latter are known analytically, their computation requires the evaluation of a number of integrals and the solution of a non-linear set of equations. We unveil the potential numerical dangers hiding in these problems and we propose detailed solutions to all of them. Convergence of the procedure is illustrated in specific examples and is gauged by computing the electrostatic potential.
We examine topological dynamical systems on the Cantor set from the point of view of the continuous model theory of commutative C*-algebras. After some general remarks we focus our attention on the generic homeomorphism of the Cantor set, as constructed by Akin, Glasner, and Weiss. We show that this homeomorphism is the prime model of its theory. We also show that the notion of generic used by Akin, Glasner, and Weiss is distinct from the notion of generic encountered in Fraisse theory.
Let $M$ be a hyperbolic 3-manifold with no rank two cusps admitting an embedding in $mathbb S^3$. Then, if $M$ admits an exhaustion by $pi_1$-injective sub-manifolds there exists cantor sets $C_nsubset mathbb S^3$ such that $N_n=mathbb S^3setminus C_n$ is hyperbolic and $N_nrightarrow M$ geometrically.
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