Dynamical systems and ergodic theory Smooth and hyperbolic dynamics Example

Sinai billiard

Builds on Dynamical billiards Mixing Torus Lyapunov exponent

Example

Let $\mathbb{T}^2 = \R^2 / \Z^2$ be the flat torus, and let $D \subset \mathbb{T}^2$ be the image of a closed disc of radius $r$ with $0 < r < 1/2$ . The Sinai billiard is the billiard on the table $Q = \mathbb{T}^2 \setminus \operatorname{int} D$ . Equivalently, it is the billiard in the unit square with a disc of radius $r$ removed from its centre. A particle leaving through one side of the square comes back through the opposite side. The table fits the definition: away from $\partial D$ the particle moves in straight lines, and on the circle $\partial D$ it reflects. The boundary $\partial D$ is convex when seen from the disc, so it curves away from the table: the billiard is dispersing.

Remarks

Sinai proved in 1970 that this billiard flow is ergodic and mixing with respect to the Liouville measure. The mechanism is dispersion: a narrow beam of parallel trajectories hitting a convex obstacle spreads out after the reflection, which gives a positive Lyapunov exponent. Unfolding the torus gives the periodic Lorentz gas: a particle moving among discs placed at the points of $\Z^2$ .

Reading path

To understand this page from the ground up, read these 67 pages in order. The foundations come first.

Dynamical billiards concept
Element definition
Lyapunov exponent concept
Mixing concept
Proposition concept
Real numbers object
Set definition
Conjunction definition
Disjunction definition
Implication definition
Predicate definition
Empty set object
Family of sets definition
Ordered pair definition
Intersection definition
Union definition
Set-builder notation concept
Existential quantifier definition
Universal quantifier definition
Cartesian product definition
Subset definition
Union of a family of sets definition
Uniqueness quantifier definition
Binary relation definition
Universal set definition
Topological space definition
Reflexive relation definition
Symmetric relation definition
Transitive relation definition
Complement definition
Function definition
Basis of a topology definition
Subspace topology definition
Equivalence relation definition
Product topology definition
Preimage of a set definition
Metric space definition
Codomain definition
Function composition definition
Injective function definition
Equivalence class definition
Open ball definition
Surjective function definition
Metric topology definition
Quotient set definition
Bijective function definition
Open set definition
Quotient topology definition
Inverse function definition
Hausdorff space definition
Open cover definition
Continuous map of topological spaces definition
Closed set definition
Compact space definition
Complement of an open set is closed statement
Homeomorphism definition
Universal property of the quotient topology statement
Continuity via closed sets statement
Compact set definition
Compactness via open covers statement
Compact subsets of a Hausdorff space are closed statement
Continuous image of a compact set is compact statement
Closed subsets of compact sets are compact statement
Continuous bijection from a compact space to a Hausdorff space is a homeomorphism statement
Circle as a quotient of the real line example
Torus example
Sinai billiard example

Where this leads

1 page builds on this one.

Bunimovich stadium

Tag	`GX9Q2PYNZE`
URI	`m2://GX9Q2PYNZE`
Kind	example
Writing	W0 stub
MSC	37C83, 37A25
Instance of	Dynamical billiards
Uses	Dynamical billiards, Mixing, Torus, Lyapunov exponent

Sinai billiard

Example

Remarks

Reading path

Where this leads

Further reading

What this page needs for W1

What links here

Used by