Hyperbolicity in dissipative polygonal billiards Joint work with P. Duarte, G. del Magno, J.P. Gaiv˜ ao, D. Pinheiro Jo˜ ao Lopes Dias Departamento de Matem´ atica - ISEG e CEMAPRE Universidade T´ ecnica de Lisboa Portugal 1 / 42 2 / 42 Outline Billiard dynamics A billiard is a mechanical system consisting of a point-particle moving Billiard dynamics 1 freely inside a planar region D ( billiard table ) and being reflected off the Examples of billiard tables perimeter of the region ∂D according to some reflection law . Examples of reflection laws Example (Billiards in the world) Light in mirrors Conservative polygons 2 Acoustics in closed rooms, echoes Lorentz gas model for electricity (small electron bounces between Dissipative polygons 3 large molecules) Hyperbolicity games: billiard, pool, snooker, pinballs, flippers SRB measures 3 / 42 4 / 42
Examples of billiard tables Examples of billiard tables s ?? θ θ’ s ’ Convex Non-convex Convex Non-convex Smooth billiards Polygonal billiards billiard map Φ: ( s, θ ) �→ ( s ′ , θ ′ ) 5 / 42 6 / 42 More examples Examples of reflection laws λθ θ θ θ θ Triangle Rectangle Conservative Linear contraction Slap classical dissipative λ = 0 standard non-elastic specular pinball elastic 0 < λ < 1 λ = 1 Z-shaped 7 / 42 8 / 42
Billiard map Singularities/Discontinuity points − π 2 , π � � V = { corners and tangencies } × π/ 2 2 S + = V ∪ Φ − 1 ( V ) 0 θ Billiard map Piecewise smooth − π/ 20 1 2 3 4 s Polygonal convex billiard table (it has Φ: M → M corners) Phase space M ( s, θ ) �→ ( s ′ , θ ′ ) Phase space − π 2 , π � � \ S + M = ∂D × 2 9 / 42 10 / 42 Outline Standard reflection law Billiard dynamics 1 The billiard map Φ preserves the measure cos θ dθ ds Examples of billiard tables Conservative system Examples of reflection laws No attractors Contractive reflection law Conservative polygons 2 The area is contracted Dissipative system Dissipative polygons There are attractors 3 Hyperbolicity SRB measures 11 / 42 12 / 42
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