{
  "aliases": null,
  "backlinks": [
    {
      "field": "about",
      "target": "PKP7RVAZK6"
    },
    {
      "field": "uses",
      "target": "PKP7RVAZK6"
    },
    {
      "field": "defines",
      "target": "WZNF6Q5JBS"
    }
  ],
  "body": "\n## Summary\n\nA pseudo-Anosov map is a homeomorphism of a surface that stretches by a factor $\\lambda \u003e 1$ along one singular [[foliation|98HTB7MSPX]].\nIt contracts by $1/\\lambda$ along a transverse one.\nIt generalizes the [[Anosov diffeomorphisms|RCQH8P6DPQ]] of the torus to surfaces of higher genus.\nThe price is a finite set of singular points where the foliations have three or more prongs.\nBy the Nielsen–Thurston classification, pseudo-Anosov maps are the typical elements of the [[mapping class group|JV5NYG913V]].\nThey have rich dynamics: positive entropy $\\log \\lambda$, dense periodic points, and they are the monodromies of hyperbolic 3-manifolds fibering over the circle.\n\n## Explanations\n\n### L1 human\n\nTake a square torus and apply the matrix $\\begin{pmatrix} 2 \u0026 1 \\\\ 1 \u0026 1 \\end{pmatrix}$: it stretches along one direction and squeezes along a perpendicular one, and wraps around.\nOn a surface with two or more holes, like a pretzel, there is no such linear map.\nBut there are maps that stretch and squeeze in the same way everywhere except at a few special points.\nNear such a point the stretching lines meet in a star with three or more arms, like the lines on a monkey saddle.\nThese maps are the pseudo-Anosov maps, and they mix the surface as thoroughly as possible.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "37E30",
      "57K20"
    ],
    "name": "Pseudo-Anosov map",
    "same_as": {
      "wikidata": "Q7254395",
      "wikipedia": "Pseudo-Anosov map"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Pseudo-Anosov_map"
      }
    ],
    "tag": "GYW7GYABD9"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/GYW7GYABD9/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/GYW7GYABD9.md",
  "msc": [
    "37E30",
    "57K20"
  ],
  "name": "Pseudo-Anosov map",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/GYW/GYW7GYABD9.md",
  "status": null,
  "tag": "GYW7GYABD9",
  "uri": "m2://GYW7GYABD9",
  "url": "https://tamnd.github.io/math2/wiki/pseudo-anosov-map/",
  "writing": {
    "tag": "GYW7GYABD9",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}