Displaying similar documents to “Topological entropy of Cournot-Puu duopoly.”

On some notions of chaos in dimension zero

Rafał Pikuła (2007)

Colloquium Mathematicae

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We compare four different notions of chaos in zero-dimensional systems (subshifts). We provide examples showing that in that case positive topological entropy does not imply strong chaos, strong chaos does not imply complicated dynamics at all, and ω-chaos does not imply Li-Yorke chaos.

On the origin and development of some notions of entropy

Francisco Balibrea (2015)

Topological Algebra and its Applications

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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...

The topological entropy versus level sets for interval maps

Jozef Bobok (2002)

Studia Mathematica

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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?

Topological sequence entropy for maps of the circle

Roman Hric (2000)

Commentationes Mathematicae Universitatis Carolinae

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A continuous map f of the interval is chaotic iff there is an increasing sequence of nonnegative integers T such that the topological sequence entropy of f relative to T , h T ( f ) , is positive ([FS]). On the other hand, for any increasing sequence of nonnegative integers T there is a chaotic map f of the interval such that h T ( f ) = 0 ([H]). We prove that the same results hold for maps of the circle. We also prove some preliminary results concerning topological sequence entropy for maps of general compact...

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

Emma D’Aniello, Timothy Steele (2014)

Open Mathematics

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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

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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.