# Continuous Chaotic Functions Are Sensitive

Theorem
If $\left(X,d\right)$ is an infinite metric space and $f:X\to X$ is continuous and chaotic, then $f$ sensitively depends on initial conditions.

This proof is from Adams and Franzosa (2008), who in turn got it from Banks, Brooks, Cairns, Davis, and Stacey (1992).

Idea: We find a pair of disjoint periodic orbits $O\left(q\right)$ and $O\left(q\prime \right)$, so any $x\in X$ will be at least a constant distance away from at least one of them. We use the first property of chaos (density of periodic points) to get a periodic point $p$ nearby $x$ (call its period $m$), and the second property of chaos (topological transitivity) to get a point $w$ nearby $x$ such that $f$ eventually brings $w$ near $q$ or $q\prime$. The continuity of $f$ lets us pick $w$ such that $m$ iterations of $f$ on $w$ stay close to $O\left(q\right)$ or $O\left(q\prime \right)$. Then for the $n$ we need to demonstrate sensitive dependence on initial conditions, we pick an integer that's a multiple of $m$ but also keeps ${f}^{n}\left(w\right)$ near $O\left(q\right)$ or $O\left(q\prime \right)$. Then $d\left({f}^{n}\left(p\right),{f}^{n}\left(w\right)\right)=d\left(p,{f}^{n}\left(w\right)\right)$, which is large because $p$ is near $x$ and $x$ is far from $O\left(q\right)$ or $O\left(q\prime \right)$. By the triangle inequality, at least one of $d\left({f}^{n}\left(x\right),{f}^{n}\left(p\right)\right)$ or $d\left({f}^{n}\left(x\right),{f}^{n}\left(w\right)\right)$ must be large, proving the theorem.

Proof: First of all, observe that $X$ contains infinitely many periodic points of $f$, since it's infinite and periodic points are dense in it. At least two of these points—call them $q$ and $q\prime$—must have disjoint orbits. Otherwise, all of the infinitely many periodic points would be part of the same cycle, prohibiting any of them from having a finite period. Now let ${\delta }_{0}=min\left\{d\left(x,y\right):x\in O\left(q\right),y\in O\left(q\prime \right)\right\}$, so that ${\delta }_{0}$ is the minimum distance between the orbits of $q$ and $q\prime$.

Define $\delta =\frac{{\delta }_{0}}{8}$. We'll prove that this is the $\delta$ needed for sensitive dependence on initial conditions. So pick an arbitrary $x\in X$ and $\epsilon >0$. Without loss of generality, we may force $\epsilon <\delta$. (We can shrink $\epsilon$ as much as we like because if we find a suitable $y\in {B}_{\epsilon }\left(x\right)$ for the new $\epsilon$, then of course $y\in {B}_{\epsilon }\left(x\right)$ for the original, larger $\epsilon$ as well.) Since periodic points are dense, there's a periodic point $p\in {B}_{\epsilon }\left(x\right)$. Let $m$ be the period of $p$.

It follows from the triangle inequality (and it's obvious from the picture) that wherever $p$ is, it must be at least $\frac{{\delta }_{0}}{2}$ units away from either every point in $O\left(q\right)$ or every point in $O\left(q\prime \right)$; suppose, without loss of generality, that the former is the case.

For each $j=0,1,\dots ,m$, let ${B}_{j}={B}_{\delta }\left({f}^{j}\left(q\right)\right)$. (This means that ${B}_{0}$ is just ${B}_{\delta }\left(q\right)$. But ${B}_{m}$ need not equal ${B}_{0}$, since $m$ is the period of $p$, not $q$.) Then for each $j$, the continuity of $f$ implies that ${f}^{j}$ is continuous, so ${\left({f}^{j}\right)}^{-1}\left({B}_{j}\right)$ is open; in fact, it's a neighborhood of $q$. Let $V$ be the intersection of all $m+1$ of these neighborhoods.

Since $f$ is topologically transitive, there exist $w\in {B}_{\epsilon }\left(x\right)$ and $k\in ℕ$ such that ${f}^{k}\left(w\right)\in V$.

Now pick $h\in ℕ$ with $k\le hm\le k+m$. Suppose we knew that

$d\left({f}^{hm}\left(p\right),{f}^{hm}\left(w\right)\right)>2\delta .$
(1)

The triangle inequality would then imply that $d\left({f}^{hm}\left(x\right),{f}^{hm}\left(p\right)\right)>\delta$ or $d\left({f}^{hm}\left(x\right),{f}^{hm}\left(w\right)\right)>\delta$, proving the theorem, since $p,w\in {B}_{\epsilon }\left(x\right)$. So all we have left to do is prove (1).

By the triangle inequality,

$d\left(x,{f}^{hm-k}\left(q\right)\right)\le d\left(x,p\right)+d\left(p,{f}^{hm}\left(w\right)\right)+d\left({f}^{hm}\left(w\right),{f}^{hm-k}\left(q\right)\right).$
(2)

Since $p\in {B}_{\epsilon }\left(x\right)$,

$d\left(x,p\right)<\epsilon <\delta .$
(3)

Since ${f}^{k}\left(w\right)\in V$,

${f}^{k}\left(w\right)\in {B}_{0},{f}^{k+1}\left(w\right)\in {B}_{1},\dots ,{f}^{k+m}\left(w\right)\in {B}_{m};$

in particular, since $k\le hm\le k+m$, ${f}^{hm}\left(w\right)\in {B}_{hm-k}={B}_{\delta }\left({f}^{hm-k}\left(q\right)\right)$, so

$d\left({f}^{hm}\left(w\right),{f}^{hm-k}\left(q\right)\right)<\delta .$
(4)

Combining (2), (3), and (4) yields

$d\left(x,{f}^{hm-k}\left(q\right)\right)\le 2\delta +d\left(p,{f}^{hm}\left(w\right)\right).$

But we assumed that $x$ is more than $\frac{{\delta }_{0}}{2}=4\delta$ units away from every point in $O\left(q\right)$, so

$4\delta

and thus

$2\delta

This implies (1), since

${f}^{hm}\left(p\right)={f}^{m}\left({f}^{m}\left(\cdots {f}^{m}\left(p\right)\cdots \right)\right)=p,$

$m$ being the period of $p$. □