Hyperbolicity in dissipative polygonal billiards Joint work with P. - - PowerPoint PPT Presentation

Hyperbolicity in dissipative polygonal billiards

Jo˜ ao Lopes Dias

Departamento de Matem´ atica - ISEG e CEMAPRE Universidade T´ ecnica de Lisboa Portugal

1 / 42

Joint work with

• P. Duarte, G. del Magno, J.P. Gaiv˜

ao, D. Pinheiro

2 / 42

Outline

1

Billiard dynamics Examples of billiard tables Examples of reflection laws

2

Conservative polygons

3

Dissipative polygons Hyperbolicity SRB measures

3 / 42

Billiard dynamics

A billiard is a mechanical system consisting of a point-particle moving freely inside a planar region D (billiard table) and being reflected off the perimeter of the region ∂D according to some reflection law.

Example (Billiards in the world)

Light in mirrors Acoustics in closed rooms, echoes Lorentz gas model for electricity (small electron bounces between large molecules) games: billiard, pool, snooker, pinballs, flippers

4 / 42

Examples of billiard tables

s s’ θ θ’

Convex Non-convex Smooth billiards billiard map Φ: (s, θ) → (s′, θ′)

5 / 42

Examples of billiard tables

??

Convex Non-convex Polygonal billiards

6 / 42

More examples

Triangle Rectangle Z-shaped

7 / 42

Examples of reflection laws

θ θ Conservative classical standard specular elastic λ = 1 θ λθ Linear contraction dissipative non-elastic pinball 0 < λ < 1 θ Slap λ = 0

8 / 42

Billiard map

Singularities/Discontinuity points

V = {corners and tangencies} ×

• −π

2 , π 2

• S+ = V ∪ Φ−1(V )

Billiard map

Piecewise smooth Φ: M → M (s, θ) → (s′, θ′) Phase space M = ∂D ×

• −π

2 , π 2

• \ S+

9 / 42

Polygonal convex billiard table (it has corners)

θ π/2 −π/20 1 2 3 4 s

Phase space M

10 / 42

Standard reflection law

The billiard map Φ preserves the measure cos θ dθ ds Conservative system No attractors

Contractive reflection law

The area is contracted Dissipative system There are attractors

11 / 42

Outline

1

Billiard dynamics Examples of billiard tables Examples of reflection laws

2

Conservative polygons

3

Dissipative polygons Hyperbolicity SRB measures

12 / 42