Dynamical systems and ergodic theory Smooth and hyperbolic dynamics Example

Bunimovich stadium

Builds on Dynamical billiards Sinai billiard

Example

Let $a > 0$ and $r > 0$ . The Bunimovich stadium is the table $Q \subset \R^2$ made of the rectangle $[-a, a] \times [-r, r]$ and two half-discs of radius $r$ . The half-discs are centred at $(-a, 0)$ and $(a, 0)$ and sit on the short sides of the rectangle. Its boundary consists of two parallel segments of length $2a$ and two semicircles of radius $r$ . These pieces have a common tangent at the four junctions, so the boundary is a $C^1$ curve. The table is compact with piecewise smooth boundary, so it carries a billiard flow.

Remarks

Bunimovich proved that the billiard in the stadium is ergodic for every $a > 0$ . This was a surprise, because the boundary has no dispersing part: the semicircles are focusing and the segments are flat, unlike the Sinai billiard. The mechanism is defocusing: a beam focused by a semicircle passes through a focal point and spreads out again before the next reflection, if the free path is long enough. For $a = 0$ the table is a disc and the billiard is integrable, so the length $a > 0$ of the straight parts is essential. The stadium also has a family of periodic orbits bouncing between the two flat sides; they fill a set of zero measure, so they do not contradict ergodicity.

Reading path

To understand this page from the ground up, read these 68 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
Bunimovich stadium example

Tag	`YYB1JP7TWJ`
URI	`m2://YYB1JP7TWJ`
Kind	example
Writing	W0 stub
MSC	37C83, 37A25
Instance of	Dynamical billiards
Uses	Dynamical billiards, Sinai billiard

Bunimovich stadium

Example

Remarks

Reading path

Further reading

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