Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2507.02102

Turbulent Closed Relations

Judy Kennedy·Christopher Mouron·Van Nall
View original source ↗Read source document ↗Export metadata ↓

Abstract

This paper generalizes the classical notion of turbulence from dynamical systems generated by continuous functions to those defined by closed relations on compact metric spaces. Using the Mahavier product and the associated shift map, we introduce and explore CR-turbulence and reverse CR-turbulence, analyzing their relationship to topological entropy. A key focus is understanding when turbulence implies entropy and vice versa, with results showing that for finite closed relations, these properties are equivalent. However, examples are provided to demonstrate that this equivalence can fail for more general relations. We also construct a large class of explicit turbulent closed relations on the unit interval that are dynamically rich yet structurally simple. Additionally, since homeomorphisms cannot admit turbulence, we investigate a weakened notion of turbulence: separated, continuum-wise semi-turbulence for homeomorphisms, and then prove that smooth fans cannot support even this weaker form of turbulent dynamics. The paper includes new examples, counterexamples, and open questions that deepen the understanding of turbulence in non-classical settings.

Explore related subjects

math.DS

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Judy Kennedy, Christopher Mouron, Van Nall. 2025-07-02. Turbulent Closed Relations. https://arxiv.org/abs/2507.02102

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Effective equidistribution of orbits under semisimple groups on congruence quotients

We prove an effective equidistribution result for periodic orbits of semisimple groups on congruence quotients of an ambient semisimple group.This extends a previous work of Einsiedler, Margulis and Venkatesh. The main new feature is that we allow for periodic orbits of semisimple groups with nontrivial centralizer in the ambient group. Our proof uses crucially an effective closing lemma from work of the author with Lindenstrauss, Margulis,Mohammadi, and Shah.

math.DS↗
arXiv2026

Generalized entropy of measure-induced maps

A classical result by E. Glasner and B. Weiss states that the topological entropy of a map $f$ is zero if and only if the topological entropy of its measure-induced map $f_*$ is zero, where $f_*$ is defined as the push-forward of a measure. In this work, we use generalized entropy to distinguish the complexity of these maps and prove that the measure-induced map is much more complex than the original map. Moreover, we introduce the generalized mean dimension, an invariant that is useful for distinguishing dynamical systems with zero mean dimension, including those with the small-boundary property, and we show a relationship between this new invariant and generalized entropy.

math.DS↗
arXiv2026

The endpoint problem for $\varepsilon$-hypercyclicity

For a fixed $0<\varepsilon<1$, F. Bayart asked in 2024 whether there exists an operator $T$ such that, for every $0<δ<1$, $T$ is $δ$-hypercyclic if and only if $δ\in[\varepsilon,1)$. We answer this question affirmatively by constructing a weighted backward shift on $\ell_2(\mathbb N_0,\ell_2(\mathbb N_0))$ with this property.

math.DS↗