---
tag: TAFJV2YWKR
kind: concept
name: Expansive homeomorphism
msc: [37B05]
specializes: [0QT00PB5G2]
sources:
  - url: https://en.wikipedia.org/wiki/Expansive_homeomorphism
same_as:
  wikidata: Q5420801
  wikipedia: Expansive homeomorphism
---

## Summary

An expansive homeomorphism is a [[homeomorphism|0QT00PB5G2]] of a metric space that separates any two distinct points by a fixed distance at some time.
This time may lie in the past or in the future.
The fixed distance is the expansivity constant.
Shift spaces and Anosov diffeomorphisms are the basic examples.
On a compact metric space, an expansive homeomorphism has finite [[topological entropy|BDZYBXBFDE]] and at most finitely many periodic points of each period.

## Explanations

### L1 human

Two different starting points can look the same for a while, but an expansive map never lets them stay close forever.
At some moment, forward or backward in time, they are at least a fixed distance apart.
So with a fixed precision one can tell every orbit apart from every other one by watching the whole history.
In a shift space of two-sided sequences, two different sequences differ at some position, and shifting that position to the centre makes them far apart.
A rotation of the circle is not expansive, because it keeps all distances unchanged.