{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "YYB1JP7TWJ"
    }
  ],
  "body": "\n## Example\n\nLet $\\mathbb{T}^2 = \\R^2 / \\Z^2$ be the flat [[torus|03NM7DZXXC]], and let $D \\subset \\mathbb{T}^2$ be the image of a closed disc of radius $r$ with $0 \u003c r \u003c 1/2$.\nThe *Sinai billiard* is the [[billiard|XWMKYE038F]] on the table $Q = \\mathbb{T}^2 \\setminus \\operatorname{int} D$.\nEquivalently, it is the billiard in the unit square with a disc of radius $r$ removed from its centre.\nA particle leaving through one side of the square comes back through the opposite side.\nThe table fits the definition: away from $\\partial D$ the particle moves in straight lines, and on the circle $\\partial D$ it reflects.\nThe boundary $\\partial D$ is convex when seen from the disc, so it curves away from the table: the billiard is *dispersing*.\n\n## Remarks\n\nSinai proved in 1970 that this billiard flow is ergodic and [[mixing|S3RJQ9CX2P]] with respect to the Liouville measure.\nThe mechanism is dispersion: a narrow beam of parallel trajectories hitting a convex obstacle spreads out after the reflection, which gives a positive [[Lyapunov exponent|NJ1FJSNK6A]].\nUnfolding the torus gives the periodic Lorentz gas: a particle moving among discs placed at the points of $\\Z^2$.\n\n## Explanations\n\n### L1 human\n\nPut a round post in the middle of a square billiard table whose sides wrap around, as in a video game.\nEach time the ball hits the post, nearby paths are pushed apart, like light reflected by a convex mirror.\nAfter many bounces the ball forgets where it started, and in the long run it spends equal time in regions of equal area.\nIt was one of the first billiards proved to be chaotic in this strong sense.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "instance_of": [
      "XWMKYE038F"
    ],
    "kind": "example",
    "msc": [
      "37C83",
      "37A25"
    ],
    "name": "Sinai billiard",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Dynamical_billiards"
      }
    ],
    "tag": "GX9Q2PYNZE",
    "uses": [
      "XWMKYE038F",
      "S3RJQ9CX2P",
      "03NM7DZXXC",
      "NJ1FJSNK6A"
    ]
  },
  "kind": "example",
  "link": "https://tamnd.github.io/math2/node/GX9Q2PYNZE/",
  "links": [
    {
      "field": "instance_of",
      "target": "XWMKYE038F"
    },
    {
      "field": "uses",
      "target": "XWMKYE038F"
    },
    {
      "field": "uses",
      "target": "S3RJQ9CX2P"
    },
    {
      "field": "uses",
      "target": "03NM7DZXXC"
    },
    {
      "field": "uses",
      "target": "NJ1FJSNK6A"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/GX9Q2PYNZE.md",
  "msc": [
    "37C83",
    "37A25"
  ],
  "name": "Sinai billiard",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/GX9/GX9Q2PYNZE.md",
  "status": null,
  "tag": "GX9Q2PYNZE",
  "uri": "m2://GX9Q2PYNZE",
  "url": "https://tamnd.github.io/math2/wiki/sinai-billiard/",
  "writing": {
    "tag": "GX9Q2PYNZE",
    "kind": "example",
    "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
      }
    ]
  }
}