Devaney’s chaos implies existence of $s$-scrambled sets

Vague Existence Implies Vague Identity

The Existence of Level Sets in a Free Group Implies the Axiom of Choice

Since all known proofs of the NIELSEN-SCHREIER theorem (every subgroup of a free group is free, henceforth denoted by (NS)) make use of the axiom of choice (AC) it is reasonable to conjecture that some form of (AC) is necessary to prove (NS). (See [ l ] , [3], [5] and [S] for several proofs of (NS).) This conjecture was verified by LAUCHLI in [4], where it was shown that the negation of (NS) is...

Scrambled Sets of Continuous Maps of 1-dimensional Polyhedra

Let K be a 1-dimensional simplicial complex in R3 without isolated vertexes, X = |K| be the polyhedron of K with the metric dK induced by K, and f : X → X be a continuous map. In this paper we prove that if K is finite, then the interior of every scrambled set of f in X is empty. We also show that if K is an infinite complex, then there exist continuous maps from X to itself having scrambled se...

Motion of vortices implies chaos in Bohmian mechanics

– Bohmian mechanics is a causal interpretation of quantum mechanics in which particles describe trajectories guided by the wave function. The dynamics in the vicinity of nodes of the wave function, usually called vortices, is regular if they are at rest. However, vortices generically move during time evolution of the system. We show that this movement is the origin of chaotic behavior of quantu...

Existence theorems in linear chaos

Chaotic linear dynamics deals primarily with various topological ergodic properties of semigroups of continuous linear operators acting on a topological vector space. We treat the questions of the following type. Characterize which of the spaces from a given class support a semigroup of prescribed shape satisfying a given topological ergodic property. In particular, we characterize LBS-spaces (...