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Register a new userOn the dynamical asymptotic dimension of a free $mathbb Z^d$-action on the Cantor set
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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 $3^d - 1$.
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Consider a minimal free topological dynamical system $(X, T, mathbb{Z}^d)$. It is shown that the comparison radius of the crossed product C*-algebra $mathrm{C}(X) rtimes mathbb{Z}^d$ is at most the half of the mean topological dimension of $(X, T, mathbb{Z}^d)$. As a consequence, the C*-algebra $mathrm{C}(X) rtimes mathbb{Z}^d$ is classifiable if $(X, T, mathbb{Z}^d)$ has zero mean dimension.
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.
In this paper, we prove a power-law version dynamical localization for a random operator $mathrm{H}_{omega}$ on $mathbb{Z}^d$ with long-range hopping. In breif, for the linear Schrodinger equation $$mathrm{i}partial_{t}u=mathrm{H}_{omega}u, quad u in ell^2(mathbb{Z}^d), $$ the Sobolev norm of the solution with well localized initial state is bounded for any $tgeq 0$.
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 $n=1$.
We know that $mathbb{Z}_n$ is a finite field for a prime number $n$. Let $m,n$ be arbitrary natural numbers and let $mathbb{Z}^m_n= mathbb{Z}_n timesmathbb{Z}_ntimes...timesmathbb{Z}_n$ be the Cartesian product of $m$ rings $mathbb{Z}_n$. In this note, we present the action of $SL(m, mathbb{Z}_n)={A in mathbb{Z}^{m,m}_{n} : det A equiv 1 (modsimn)}$, where $SL(m, mathbb{Z}_n)$ for $ngeq 2$ is a group under matrix multiplication modulo $n$, on the ring $mathbb{Z}^m_n$ as a right multiplication of a row vector of $mathbb{Z}^m_n$ by a matrix of $SL(m, mathbb{Z}_n)$ to determine the orbits of the ring $mathbb{Z}^m_n$. This work is an extension of [1]
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