Tag Archives: expansive

There is no positively expansive homeomorphism

Let f be a homeomorphism on a compact metric space (X,d). Then f is said to be \mathbb{Z}-expansive, if there exists \delta>0 such that for any two points x,y\in X, if d(f^nx,f^ny)<\delta for all n\in\mathbb{Z}, then x=y. The constant \delta is called the expansive constant of f.

Similarly one can define \mathbb{N}-expansiveness if f is not invertible. An interesting phenomenon observed by Schwartzman states that

Theorem. A homeomorphism f cannot be \mathbb{N}-expansive (unless X is finite).

This result was reported in Gottschalk–Hedlund’s book Topological Dynamics (1955), and a proof was given in King’s paper A map with topological minimal self-joinings in the sense of del Junco (1990). Below we copied the proof from King’s paper.

Proof. Suppose on the contrary that there is a homeo f on (X,d) that is \mathbb{N}-expansive. Let \delta>0 be the \mathbb{N}-expansive constant of f, and d_n(x,y)=\max\{d(f^k x, f^k y): 1\le k\le n\}.

It follows from the \mathbb{N}-expansiveness that N:=\sup\{n\ge 1: d_n(x,y)\le\delta \text{ for some } d(x,y)\ge\delta\} is a finite number. Pick \epsilon\in(0,\delta) such that d_N(x,y)<\delta whenever d(x,y)<\epsilon.

Claim. If d(x,y)<\epsilon, then d(f^{-n} x, f^{-n}y)<\delta for any n\ge 1.

Proof of Claim. If not, we can prolong the N-string since f^{k}=f^{k+n}\circ f^{-n}.

Recall that a pair (x,y) is said to be \epsilon-proximal, if d(f^{n_i}x, f^{n_i}y)<\epsilon for some n_i\to\infty. The upshot for the above claim is that any \epsilon-proximal pair is \delta-indistinguishable: d(f^{n}x, f^{n}y)<\delta for all n.

Cover X by open sets of radius < \epsilon, and pick a finite subcover, say \{B_i:1\le i\le I\}. Let E=\{x_j:1\le j\le I+1\} be a subset consisting of I+1 distinct points. Then for each n\ge 0, there are two points in f^n E share the room B_{i(n)}, say f^nx_{a(n)}, and f^nx_{b(n)}. Pick a subsequence n_i such that a(n_i)\equiv a and b(n_i)\equiv b. Clearly x_a\neq x_b, and d(f^{n_i}x_a,f^{n_i}x_b)<\epsilon. Hence the pair (x_a,x_b) is \epsilon-proximal and \delta-indistinguishable. This contradicts the \mathbb{N}-expansiveness assumption on f. QED.