Ergodic Theory and Dynamical Systems最新文献

英文中文

Shifts of finite type on locally finite groups 局部有限群上有限类型的移动

IF 0.9 3区数学 Q2 Mathematics

Ergodic Theory and Dynamical Systems

Pub Date : 2024-02-26 DOI: 10.1017/etds.2024.14

JADE RAYMOND

In this work we prove that every shift of finite type (SFT), sofic shift, and strongly irreducible shift on locally finite groups has strong dynamical properties. These properties include that every sofic shift is an SFT, every SFT is strongly irreducible, every strongly irreducible shift is an SFT, every SFT is entropy minimal, and every SFT has a unique measure of maximal entropy, among others. In addition, we show that if every SFT on a group is strongly irreducible, or if every sofic shift is an SFT, then the group must be locally finite, and this extends to all of the properties we explore. These results are collected in two main theorems which characterize the local finiteness of groups by purely dynamical properties. In pursuit of these results, we present a formal construction of free extension shifts on a group G, which takes a shift on a subgroup H of G, and naturally extends it to a shift on all of G.

在这项工作中，我们证明了局部有限群上的每一个有限型移位（SFT）、sofic 移位和强不可还原移位都具有强动力学性质。这些性质包括：每个sofic shift 都是一个SFT，每个SFT 都是强不可还原的，每个强不可还原的 shift 都是一个SFT，每个SFT 都是熵最小的，每个SFT 都有一个唯一的最大熵量，等等。此外，我们还证明，如果一个群上的每一个 SFT 都是强不可还原的，或者每一个sofic shift 都是一个 SFT，那么这个群一定是局部有限的，这也扩展到了我们探索的所有性质。这些结果集合在两个主要定理中，它们通过纯粹的动力学性质描述了群的局部有限性。为了追寻这些结果，我们提出了群 G 上自由扩展位移的形式构造，它将 G 的一个子群 H 上的位移，自然地扩展为 G 全部上的位移。

引用次数: 0

On the stochastic bifurcations regarding random iterations of polynomials of the form 关于形式为多项式的随机迭代的随机分岔

IF 0.9 3区数学 Q2 Mathematics

Ergodic Theory and Dynamical Systems

Pub Date : 2024-02-26 DOI: 10.1017/etds.2024.17

TAKAYUKI WATANABE

In this paper, we consider random iterations of polynomial maps

$z^{2} + c_{n}$

, where

$c_{n}$

are complex-valued independent random variables following the uniform distribution on the closed disk with center c and radius r. The aim of this paper is twofold. First, we study the (dis)connectedness of random Julia sets. Here, we reveal the relationships between the bifurcation radius and connectedness of random Julia sets. Second, we investigate the bifurcation of our random iterations and give quantitative estimates of bifurcation parameters. In particular, we prove that for the central parameter

$c = -1$

, almost every random Julia set is totally disconnected with much smaller radial parameters r than expected. We also introduce several open questions worth discussing.

在本文中，我们考虑多项式映射 $z^{2} + c_{n}$ 的随机迭代。+ c_{n}$ ，其中 $c_{n}$ 是复值独立随机变量，在以 c 为圆心、r 为半径的封闭圆盘上服从均匀分布。首先，我们研究随机 Julia 集的（不）连通性。在这里，我们揭示了随机 Julia 集的分岔半径和连通性之间的关系。其次，我们研究了随机迭代的分岔，并给出了分岔参数的定量估计。特别是，我们证明了对于中心参数 $c = -1$ ，几乎每个随机 Julia 集都是完全断开的，其径向参数 r 比预期的要小得多。我们还介绍了几个值得讨论的开放问题。

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引用次数: 0

Upper, down, two-sided Lorenz attractor, collisions, merging, and switching 上、下、双面洛伦兹吸引子、碰撞、合并和切换

IF 0.9 3区数学 Q2 Mathematics

Ergodic Theory and Dynamical Systems

Pub Date : 2024-02-21 DOI: 10.1017/etds.2024.8

DIEGO BARROS, CHRISTIAN BONATTI, MARIA JOSÉ PACIFICO

We present a modified version of the well-known geometric Lorenz attractor. It consists of a

$C^1$

open set

${mathcal O}$

of vector fields in

${mathbb R}^3$

having an attracting region

${mathcal U}$

satisfying three properties. Namely, a unique singularity

$sigma $

; a unique attractor

$Lambda $

including the singular point and the maximal invariant in

${mathcal U}$

has at most two chain recurrence classes, which are

$Lambda $

and (at most) one hyperbolic horseshoe. The horseshoe and the singular attractor have a collision along with the union of

$2$

codimension

$1$

我们提出了著名的几何洛伦兹吸引子的改进版本。它由${mathbb R}^3$ 中向量场的$C^1$开集${mathcal O}$组成，该向量场有一个满足三个性质的吸引区域${mathcal U}$。即，一个唯一的奇异点 $sigma $ ；一个唯一的吸引子 $Lambda $ 包括奇异点和最大不变量，在 ${mathcal U}$ 中最多有两个链递归类，分别是 $Lambda $ 和（最多）一个双曲马蹄形。马蹄形和奇异吸引子有一个碰撞点，碰撞点是 2 元codimension 1 元的子满足的结合点，它将 ${mathcal O}$ 分割成三个区域。越过这个碰撞点，吸引子和马蹄形可能合并成双面洛伦兹吸引子，也可能交换性质：洛伦兹吸引子驱逐奇异点 $sigma $，变成马蹄形，而马蹄形吸收 $sigma $，变成洛伦兹吸引子。

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引用次数: 0

Measures of maximal entropy of bounded density shifts 有界密度移动的最大熵的度量

IF 0.9 3区数学 Q2 Mathematics

Ergodic Theory and Dynamical Systems

Pub Date : 2024-02-20 DOI: 10.1017/etds.2024.6

FELIPE GARCÍA-RAMOS, RONNIE PAVLOV, CARLOS REYES

We find sufficient conditions for bounded density shifts to have a unique measure of maximal entropy. We also prove that every measure of maximal entropy of a bounded density shift is fully supported. As a consequence of this, we obtain that bounded density shifts are surjunctive.

我们找到了有界密度移动具有唯一最大熵量的充分条件。我们还证明，有界密度移动的每个最大熵量都是完全支持的。由此，我们得出有界密度移动是附加的。

引用次数: 0

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