---
tag: GYW7GYABD9
kind: concept
name: Pseudo-Anosov map
msc: ["37E30", "57K20"]
sources:
  - url: https://en.wikipedia.org/wiki/Pseudo-Anosov_map
same_as:
  wikidata: Q7254395
  wikipedia: Pseudo-Anosov map
---

## Summary

A pseudo-Anosov map is a homeomorphism of a surface that stretches by a factor $\lambda > 1$ along one singular [[foliation|98HTB7MSPX]].
It contracts by $1/\lambda$ along a transverse one.
It generalizes the [[Anosov diffeomorphisms|RCQH8P6DPQ]] of the torus to surfaces of higher genus.
The price is a finite set of singular points where the foliations have three or more prongs.
By the Nielsen–Thurston classification, pseudo-Anosov maps are the typical elements of the [[mapping class group|JV5NYG913V]].
They have rich dynamics: positive entropy $\log \lambda$, dense periodic points, and they are the monodromies of hyperbolic 3-manifolds fibering over the circle.

## Explanations

### L1 human

Take a square torus and apply the matrix $\begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix}$: it stretches along one direction and squeezes along a perpendicular one, and wraps around.
On a surface with two or more holes, like a pretzel, there is no such linear map.
But there are maps that stretch and squeeze in the same way everywhere except at a few special points.
Near such a point the stretching lines meet in a star with three or more arms, like the lines on a monkey saddle.
These maps are the pseudo-Anosov maps, and they mix the surface as thoroughly as possible.