Topological mixing, dense periodic points, sensitive dependence, and the stated chaos list

Let $(X,d)$ be a metric space (Metric space: $d(x,y)=0$ iff $x=y$, symmetry, and the triangle inequality; pseudometric and ultrametric) with its metric topology (The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement) and let $f:X\to X$ be a map. For $n\ge 1$ write $f^n$ for the $n$-fold iterate, $f^1:=f$ and $f^{n+1}:=f\circ f^n$.

1. $f$ is topologically mixing when for all nonempty open sets $U,V\subseteq X$ there is an integer $N$ such that

$f^n(U)\cap V\mathrm{\ne }\mathrm{\varnothing }\text{for every }n\ge N\mathrm{.}$

2. A point $x\in X$ is a periodic point of $f$ when $f^n(x)=x$ for some $n\ge 1$. The set of periodic points of $f$ is written $\mathrm{Per}(f)$. The periodic points are dense in $X$ when $\mathrm{Per}(f)$ is dense in $X$, that is (Dense, nowhere dense and codense subsets of a topological space, and the criterion by basic open sets) when every nonempty open subset of $X$ meets $\mathrm{Per}(f)$.

3. $f$ has sensitive dependence on initial conditions when there is a real number $\delta >0$ such that for every $x\in X$ and every $\epsilon >0$ there are $y\in X$ and $n\ge 1$ with

$d(x,y)<\epsilon \text{and}d(f^n(x),f^n(y))\ge \delta \mathrm{.}$

Such a number $\delta$ is called a sensitivity constant for $f$.

Finally, say that $f$ is chaotic in the stated sense when $f$ is continuous (Continuity of a map between metric spaces, at a point and globally, in the $\epsilon$-$\delta$ form), topologically mixing, has dense periodic points, and has sensitive dependence on initial conditions.

• The list is a definition of a chosen package of properties, not a definition of the word "chaos" in all of its uses. Devaney's closely related list replaces topological mixing by topological transitivity; mixing implies transitivity, while this definition makes no converse claim. Positive topological entropy, Li-Yorke chaos, and measure-theoretic mixing are further distinct properties that this definition neither implies nor assumes.

• Topological mixing immediately implies the open-set form of topological transitivity: for nonempty open $U,V$ the definition supplies $N$ with $f^N(U)\cap V\mathrm{\ne }\mathrm{\varnothing }$. Sensitivity is required with one constant $\delta$ uniform over all points and all radii $\epsilon$; it is a statement about the family of iterates, not about a single pair of orbits.

• The definition applies to non-invertible maps; only forward iterates are used. When $f$ is a bijection, applying the same definitions to $f^{-1}$ gives a separate statement, and nothing here asserts two-sided mixing.