Displaying similar documents to “On some notions of chaos in dimension zero”

On the origin and development of some notions of entropy

Francisco Balibrea (2015)

Topological Algebra and its Applications

Similarity:

Discrete dynamical systems are given by the pair (X, f ) where X is a compact metric space and f : X → X a continuous maps. During years, a long list of results have appeared to precise and understand what is the complexity of the systems. Among them, one of the most popular is that of topological entropy. In modern applications other conditions on X and f have been considered. For example X can be non-compact or f can be discontinuous (only in a finite number of points and with bounded...

Chaotic behaviour of the map x ↦ ω(x, f)

Emma D’Aniello, Timothy Steele (2014)

Open Mathematics

Similarity:

Let K(2ℕ) be the class of compact subsets of the Cantor space 2ℕ, furnished with the Hausdorff metric. Let f ∈ C(2ℕ). We study the map ω f: 2ℕ → K(2ℕ) defined as ω f (x) = ω(x, f), the ω-limit set of x under f. Unlike the case of n-dimensional manifolds, n ≥ 1, we show that ω f is continuous for the generic self-map f of the Cantor space, even though the set of functions for which ω f is everywhere discontinuous on a subsystem is dense in C(2ℕ). The relationships between the continuity...

Transitive sensitive subsystems for interval maps

Sylvie Ruette (2005)

Studia Mathematica

Similarity:

We prove that for continuous interval maps the existence of a non-empty closed invariant subset which is transitive and sensitive to initial conditions is implied by positive topological entropy and implies chaos in the sense of Li-Yorke, and we exhibit examples showing that these three notions are distinct.

Maximal entropy measures in dimension zero

Dawid Huczek (2012)

Colloquium Mathematicae

Similarity:

We prove that an invertible zero-dimensional dynamical system has an invariant measure of maximal entropy if and only if it is an extension of an asymptotically h-expansive system of equal topological entropy.

The topological entropy versus level sets for interval maps

Jozef Bobok (2002)

Studia Mathematica

Similarity:

We answer affirmatively Coven's question [PC]: Suppose f: I → I is a continuous function of the interval such that every point has at least two preimages. Is it true that the topological entropy of f is greater than or equal to log 2?