On Definitions of Discrete Topological Chaos and Their Relations on Intervals

In this paper, we distinguish three groups of definition of chaos: • the Devaney’s type of chaos: we will introduce three definitions which are equivalent on Baire Hausdorff spaces with countable base, before noting a defect in Devaney’s definition, leading us to give a new definition of chaos. We will also examine how the functor “Stone-Čech compactification” preserves and reflects chaos. • the entropy type of chaos: we will extend the definition of entropy of a map to the case of completely regular Hausdorff spaces, using the StoneČech compactification again. • the Li and Yorke’s type of chaos. A complete study of the relations between these three types will be held on intervals, introducing some other definitions of chaos. Finally, we are going to give a non-trivial example of chaotic map. In this paper, X is a topological space, most of the time Hausdorff and perfect, but those conditions will be stated when needed, and f is a continuous map from X into itself, the continuity of f being a essential hypothesis. When X is an interval, it will be denoted I. 1. Basic definitions If (X, d) is a metric space, f has sensitive dependence on initial conditions if: ∃δ > 0 : ∀x ∈ X,∀ > 0,∃n ≥ 0,∃y ∈ X, 0 < d(x, y) < : d(f(x), f(y)) > δ. If A is a subset of X, the backward, respectively forward, orbit of A (under f) is defined by: O− f (A) := ⋃ n≥0 f −n(A), respectively O f (A) := ⋃ n≥0 f (A). When A is a singleton {x}, we write O f (x) instead of O + f ({x}). We say that f is topologically transitive if for every non-empty open U , O− f (U) (which is open) is dense in X. Equivalently, for all non-empty open U and V , ∃n ≥ 0 such that: f(V ) ∩ U 6= ∅. Equivalently, for every non-empty open V , O f (V ) (not necessarily open) is dense in X. The ω-limit set of a point x ∈ X by f is the set of limits of all convergent subsequences of the sequence (f(x))n≥0, ie ωf (x) := ⋂ N≥0 cl(O + f (f N (x))) Notations. Ω := {x ∈ X : ωf (x) = X} and for N ≥ 0, ∆N := {x ∈ X : cl(O f (f (x))) = X}. Lemma 1.1. Ω = ⋂ n≥0 ∆n ⊆ · · · ⊆ ∆N ⊆ · · · ⊆ ∆2 ⊆ ∆1 ⊆ ∆0 and if X is perfect and Hausdorff, Ω = · · · = ∆N = · · · = ∆2 = ∆1 = ∆