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Billiards in right triangles are unstable W. Patrick Hooper Northwestern University Geometry, Dynamics and Topology Day Eastern Illinois University October 27, 2007 W. Patrick Hooper (Northwestern University) Billiards in right triangles are unstable EIU 2007 1 / 27 Comparing billiard paths in different triangles Mark the edges of our triangles by {1, 2, 3}. The orbit-type O(γ) of a periodic billiard path γ is the bi-inﬁnite periodic sequence of markings corresponding to the edges hit. 2 3 1 A periodic billiard path γ with orbit type O(γ) = 123123. W. Patrick Hooper (Northwestern University) Billiards in right triangles are unstable EIU 2007 2 / 27 Comparing billiard paths in different triangles Open Question Does every triangle have a periodic billiard path? W. Patrick Hooper (Northwestern University) Billiards in right triangles are unstable EIU 2007 3 / 27 Comparing billiard paths in different triangles Open Question Does every triangle have a periodic billiard path? Let T be the space of marked triangles up to similarities preserving the markings. We coordinatize T by the angles of the triangles: T = {(α1 , α2 , α3 ) ∈ R3 | α1 + α2 + α3 = π and each αi > 0}. W. Patrick Hooper (Northwestern University) Billiards in right triangles are unstable EIU 2007 3 / 27 Comparing billiard paths in different triangles Open Question Does every triangle have a periodic billiard path? Let T be the space of marked triangles up to similarities preserving the markings. We coordinatize T by the angles of the triangles: T = {(α1 , α2 , α3 ) ∈ R3 | α1 + α2 + α3 = π and each αi > 0}. The tile of a periodic billiard path γ is the subset tile(γ) ⊂ T consisting of all triangles ∆ ∈ T with periodic billiard paths η with ˆ the same orbit type as γ. W. Patrick Hooper (Northwestern University) Billiards in right triangles are unstable EIU 2007 3 / 27 Comparing billiard paths in different triangles Open Question Does every triangle have a periodic billiard path? Let T be the space of marked triangles up to similarities preserving the markings. We coordinatize T by the angles of the triangles: T = {(α1 , α2 , α3 ) ∈ R3 | α1 + α2 + α3 = π and each αi > 0}. The tile of a periodic billiard path γ is the subset tile(γ) ⊂ T consisting of all triangles ∆ ∈ T with periodic billiard paths η with ˆ the same orbit type as γ. The question above becomes equivalent to “Can T be covered by tiles?” W. Patrick Hooper (Northwestern University) Billiards in right triangles are unstable EIU 2007 3 / 27 Theorem (Classiﬁcation of tiles) Let γ be a periodic billiard path in a triangle ∆. Then either 1 tile(γ) is an open subset of T , or 2 tile(γ) is an open subset of a rational line of the form {(α1 , α2 , α3 ) ∈ T | n1 α1 + n2 α2 + n3 α3 = 0} for some integers n1 , n2 , n3 (not all zero). W. Patrick Hooper (Northwestern University) Billiards in right triangles are unstable EIU 2007 4 / 27 Theorem (Classiﬁcation of tiles) Let γ be a periodic billiard path in a triangle ∆. Then either 1 tile(γ) is an open subset of T , or 2 tile(γ) is an open subset of a rational line of the form {(α1 , α2 , α3 ) ∈ T | n1 α1 + n2 α2 + n3 α3 = 0} for some integers n1 , n2 , n3 (not all zero). In the ﬁrst case, γ is called stable. In any sufﬁciently small perturbation of ∆, we can ﬁnd a periodic billiard path with the same orbit-type. W. Patrick Hooper (Northwestern University) Billiards in right triangles are unstable EIU 2007 4 / 27 Theorem (Classiﬁcation of tiles) Let γ be a periodic billiard path in a triangle ∆. Then either 1 tile(γ) is an open subset of T , or 2 tile(γ) is an open subset of a rational line of the form {(α1 , α2 , α3 ) ∈ T | n1 α1 + n2 α2 + n3 α3 = 0} for some integers n1 , n2 , n3 (not all zero). In the ﬁrst case, γ is called stable. In any sufﬁciently small perturbation of ∆, we can ﬁnd a periodic billiard path with the same orbit-type. Almost every triangle only has stable periodic billiard paths! But for example, right triangles may have unstable periodic billiard paths. Since if α1 = π , then 2 α1 − α2 − α3 = 0. W. Patrick Hooper (Northwestern University) Billiards in right triangles are unstable EIU 2007 4 / 27 Example of an unstable periodic billiard path A periodic billiard path γ with orbit type 2 O(γ) = 1323 is unstable. By similar triangles, tile(γ) is the 3 collection of isosceles triangles with base marked ‘3’. 1 tile(γ) = {(α1 , α2 , α3 ) ∈ T | α1 − α2 = 0}. W. Patrick Hooper (Northwestern University) Billiards in right triangles are unstable EIU 2007 5 / 27 Theorems on stability in right triangles The following settles a conjecture of Vorobets, Galperin, and Stepin. Theorem (H) Right triangles do not have stable periodic billiard paths. W. Patrick Hooper (Northwestern University) Billiards in right triangles are unstable EIU 2007 6 / 27 Theorems on stability in right triangles The following settles a conjecture of Vorobets, Galperin, and Stepin. Theorem (H) Right triangles do not have stable periodic billiard paths. α2 Let R ⊂ T denote the collection of right α1 + α2 < π triangles. α1 W. Patrick Hooper (Northwestern University) Billiards in right triangles are unstable EIU 2007 6 / 27 Theorems on stability in right triangles The following settles a conjecture of Vorobets, Galperin, and Stepin. Theorem (H) Right triangles do not have stable periodic billiard paths. α2 Let R ⊂ T denote the collection of right α1 + α2 < π triangles. α1 Theorem (H) If γ is a stable periodic billiard path in a triangle, then tile(γ) is contained in one of the connected components of T R. W. Patrick Hooper (Northwestern University) Billiards in right triangles are unstable EIU 2007 6 / 27 When are billiard paths stable? (1 of 4) Given a triangle ∆ we can build a Euclidean cone surface D(∆) by doubling the triangle across its boundary. (A triangular pillowcase.) Let Σ denote the collection of cone singularities on D(∆). i ii i ii W. Patrick Hooper (Northwestern University) Billiards in right triangles are unstable EIU 2007 7 / 27 When are billiard paths stable? (1 of 4) Given a triangle ∆ we can build a Euclidean cone surface D(∆) by doubling the triangle across its boundary. (A triangular pillowcase.) Let Σ denote the collection of cone singularities on D(∆). i ii i ii There is a natural folding (or ironing) map D(∆) → ∆. W. Patrick Hooper (Northwestern University) Billiards in right triangles are unstable EIU 2007 7 / 27 When are billiard paths stable? (1 of 4) Given a triangle ∆ we can build a Euclidean cone surface D(∆) by doubling the triangle across its boundary. (A triangular pillowcase.) Let Σ denote the collection of cone singularities on D(∆). i ii i ii There is a natural folding (or ironing) map D(∆) → ∆. The billiard ﬂow on ∆ can be pulled back to the geodesic ﬂow on D(∆) Σ. W. Patrick Hooper (Northwestern University) Billiards in right triangles are unstable EIU 2007 7 / 27 When are billiard paths stable? (1 of 4) Given a triangle ∆ we can build a Euclidean cone surface D(∆) by doubling the triangle across its boundary. (A triangular pillowcase.) Let Σ denote the collection of cone singularities on D(∆). i ii i ii There is a natural folding (or ironing) map D(∆) → ∆. The billiard ﬂow on ∆ can be pulled back to the geodesic ﬂow on D(∆) Σ. Each periodic billiard path γ on ∆ pulls back to a closed geodesic γ on D(∆). W. Patrick Hooper (Northwestern University) Billiards in right triangles are unstable EIU 2007 7 / 27 When are billiard paths stable? (2 of 4) Let θ : T1 R2 → R/2πZ be the function which measures angle. W. Patrick Hooper (Northwestern University) Billiards in right triangles are unstable EIU 2007 8 / 27 When are billiard paths stable? (2 of 4) Let θ : T1 R2 → R/2πZ be the function which measures angle. The closed 1-form dθ on T1 R2 is invariant under the action of Isom+ (R2 ). W. Patrick Hooper (Northwestern University) Billiards in right triangles are unstable EIU 2007 8 / 27 When are billiard paths stable? (2 of 4) Let θ : T1 R2 → R/2πZ be the function which measures angle. The closed 1-form dθ on T1 R2 is invariant under the action of Isom+ (R2 ). It pulls back to closed 1-form on the unit tangent bundle of any locally Euclidean surface. W. Patrick Hooper (Northwestern University) Billiards in right triangles are unstable EIU 2007 8 / 27 When are billiard paths stable? (2 of 4) Let θ : T1 R2 → R/2πZ be the function which measures angle. The closed 1-form dθ on T1 R2 is invariant under the action of Isom+ (R2 ). It pulls back to closed 1-form on the unit tangent bundle of any locally Euclidean surface. If γ is a closed geodesic on a locally Euclidean surface then dθ = 0. γ W. Patrick Hooper (Northwestern University) Billiards in right triangles are unstable EIU 2007 8 / 27 When are billiard paths stable? (2 of 4) Let θ : T1 R2 → R/2πZ be the function which measures angle. The closed 1-form dθ on T1 R2 is invariant under the action of Isom+ (R2 ). It pulls back to closed 1-form on the unit tangent bundle of any locally Euclidean surface. If γ is a closed geodesic on a locally Euclidean surface then dθ = 0. γ Because dθ is closed, this is a homological invariant of the curve γ in the unit tangent bundle. W. Patrick Hooper (Northwestern University) Billiards in right triangles are unstable EIU 2007 8 / 27 When are billiard paths stable? (3 of 4) In our case, the surface is the double of a triangle D(∆). i β3 ii β2 β1 β3 i ii The homology group of the unit tangent bundle, H1 T1 D(∆) Σ, Z ∼ Z3 , is generated by derivatives of curves = which travel around the punctures. W. Patrick Hooper (Northwestern University) Billiards in right triangles are unstable EIU 2007 9 / 27 When are billiard paths stable? (3 of 4) In our case, the surface is the double of a triangle D(∆). i β3 ii β2 β1 β3 i ii The homology group of the unit tangent bundle, H1 T1 D(∆) Σ, Z ∼ Z3 , is generated by derivatives of curves = which travel around the punctures. If x = n1 β1 + n2 β2 + n3 β3 ∈ H1 (T1 D(∆) Σ, Z) then dθ = 2n1 α1 + 2n2 α2 + 2n3 α3 . x W. Patrick Hooper (Northwestern University) Billiards in right triangles are unstable EIU 2007 9 / 27 When are billiard paths stable? (4 of 4) Consequently, in order for a homology class x = n1 β1 + n2 β2 + n3 β3 ∈ H1 (T1 D(∆) Σ, Z) to contain the derivative of a closed geodesic, it must be n1 α1 + n2 α2 + n3 α3 = 0. W. Patrick Hooper (Northwestern University) Billiards in right triangles are unstable EIU 2007 10 / 27 When are billiard paths stable? (4 of 4) Consequently, in order for a homology class x = n1 β1 + n2 β2 + n3 β3 ∈ H1 (T1 D(∆) Σ, Z) to contain the derivative of a closed geodesic, it must be n1 α1 + n2 α2 + n3 α3 = 0. Suppose, γ is a stable periodic billiard path. Let γ be the pull back to D(∆). Then γ must be homologous to zero in T1 D(∆) Σ. W. Patrick Hooper (Northwestern University) Billiards in right triangles are unstable EIU 2007 10 / 27 When are billiard paths stable? (4 of 4) Consequently, in order for a homology class x = n1 β1 + n2 β2 + n3 β3 ∈ H1 (T1 D(∆) Σ, Z) to contain the derivative of a closed geodesic, it must be n1 α1 + n2 α2 + n3 α3 = 0. Suppose, γ is a stable periodic billiard path. Let γ be the pull back to D(∆). Then γ must be homologous to zero in T1 D(∆) Σ. Remark: In fact, this is a sufﬁcient condition for stability. This can be seen by checking that all remaining conditions for a homotopy class in D(∆) Σ to contain a geodesic are open conditions. W. Patrick Hooper (Northwestern University) Billiards in right triangles are unstable EIU 2007 10 / 27 When are billiard paths stable? (4 of 4) Consequently, in order for a homology class x = n1 β1 + n2 β2 + n3 β3 ∈ H1 (T1 D(∆) Σ, Z) to contain the derivative of a closed geodesic, it must be n1 α1 + n2 α2 + n3 α3 = 0. Suppose, γ is a stable periodic billiard path. Let γ be the pull back to D(∆). Then γ must be homologous to zero in T1 D(∆) Σ. Remark: In fact, this is a sufﬁcient condition for stability. This can be seen by checking that all remaining conditions for a homotopy class in D(∆) Σ to contain a geodesic are open conditions. Theorem (Classiﬁcation of tiles) Let γ be a periodic billiard path in a triangle ∆. Then tile(γ) is stable iff γ is null homologous. Otherwise, tile(γ) is an open subset of a rational line. W. Patrick Hooper (Northwestern University) Billiards in right triangles are unstable EIU 2007 10 / 27 Translation surfaces (1 of 2) A translation surface is a Euclidean cone surface whose cone angles are all in 2πN ∪ {∞}. These surfaces appear naturally from the point of view of the previous discussion. W. Patrick Hooper (Northwestern University) Billiards in right triangles are unstable EIU 2007 11 / 27 Translation surfaces (1 of 2) A translation surface is a Euclidean cone surface whose cone angles are all in 2πN ∪ {∞}. These surfaces appear naturally from the point of view of the previous discussion. Consider the group homomorphism Θ : π1 (T1 D(∆) Σ) → R : [x] → dθ. x Let φ : T1 D(∆) Σ → D(∆) Σ. Let G = φ∗ (ker Θ) ⊂ π1 (D(∆) Σ). W. Patrick Hooper (Northwestern University) Billiards in right triangles are unstable EIU 2007 11 / 27 Translation surfaces (1 of 2) A translation surface is a Euclidean cone surface whose cone angles are all in 2πN ∪ {∞}. These surfaces appear naturally from the point of view of the previous discussion. Consider the group homomorphism Θ : π1 (T1 D(∆) Σ) → R : [x] → dθ. x Let φ : T1 D(∆) Σ → D(∆) Σ. Let G = φ∗ (ker Θ) ⊂ π1 (D(∆) Σ). A homotopy class [γ] in D(∆) Σ must lie in G in order to contain a geodesic. W. Patrick Hooper (Northwestern University) Billiards in right triangles are unstable EIU 2007 11 / 27 Translation surfaces (1 of 2) A translation surface is a Euclidean cone surface whose cone angles are all in 2πN ∪ {∞}. These surfaces appear naturally from the point of view of the previous discussion. Consider the group homomorphism Θ : π1 (T1 D(∆) Σ) → R : [x] → dθ. x Let φ : T1 D(∆) Σ → D(∆) Σ. Let G = φ∗ (ker Θ) ⊂ π1 (D(∆) Σ). A homotopy class [γ] in D(∆) Σ must lie in G in order to contain a geodesic. The cover of D(∆) branched over Σ associated to G is a translation surface MT (∆). W. Patrick Hooper (Northwestern University) Billiards in right triangles are unstable EIU 2007 11 / 27 Translation surfaces (2 of 2) We call MT (∆) the minimal translation surface cover of D(∆). W. Patrick Hooper (Northwestern University) Billiards in right triangles are unstable EIU 2007 12 / 27 Translation surfaces (2 of 2) We call MT (∆) the minimal translation surface cover of D(∆). We will now describe a less technical deﬁnition of MT (∆). W. Patrick Hooper (Northwestern University) Billiards in right triangles are unstable EIU 2007 12 / 27 Translation surfaces (2 of 2) We call MT (∆) the minimal translation surface cover of D(∆). We will now describe a less technical deﬁnition of MT (∆). Let H be the group Z2 ∗ Z2 ∗ Z2 generated by r1 , r2 , r3 . Let ρ : H → Isom R2 which sends each generator ri to reﬂection in the i-th side of ∆. MT (∆) is {h(∆) | h ∈ H} with some identiﬁcations: W. Patrick Hooper (Northwestern University) Billiards in right triangles are unstable EIU 2007 12 / 27 Translation surfaces (2 of 2) We call MT (∆) the minimal translation surface cover of D(∆). We will now describe a less technical deﬁnition of MT (∆). Let H be the group Z2 ∗ Z2 ∗ Z2 generated by r1 , r2 , r3 . Let ρ : H → Isom R2 which sends each generator ri to reﬂection in the i-th side of ∆. MT (∆) is {h(∆) | h ∈ H} with some identiﬁcations: −1 1 Identify h1 (∆) and h2 (∆) along the edge i if h1 ◦ h2 = ri . W. Patrick Hooper (Northwestern University) Billiards in right triangles are unstable EIU 2007 12 / 27 Translation surfaces (2 of 2) We call MT (∆) the minimal translation surface cover of D(∆). We will now describe a less technical deﬁnition of MT (∆). Let H be the group Z2 ∗ Z2 ∗ Z2 generated by r1 , r2 , r3 . Let ρ : H → Isom R2 which sends each generator ri to reﬂection in the i-th side of ∆. MT (∆) is {h(∆) | h ∈ H} with some identiﬁcations: −1 1 Identify h1 (∆) and h2 (∆) along the edge i if h1 ◦ h2 = ri . −1 2 Identify triangles h1 (∆) and h2 (∆) if ρ(h1 ◦ h2 ) is a translation. W. Patrick Hooper (Northwestern University) Billiards in right triangles are unstable EIU 2007 12 / 27 Example minimal translation surfaces II I II III III I II I IV I III III IV II The minimal translation surface for the 30-60-90 triangle is a hexagonal torus. The minimal translation surface of a right triangle whose other angles are not rational multiples of π is an inﬁnite union of rombi. W. Patrick Hooper (Northwestern University) Billiards in right triangles are unstable EIU 2007 13 / 27 Closed geodesics on translation surfaces Every closed geodesic on a Euclidean cone surface D(∆) lifts to the minimal translation surface cover of MT (∆). W. Patrick Hooper (Northwestern University) Billiards in right triangles are unstable EIU 2007 14 / 27 Closed geodesics on translation surfaces Every closed geodesic on a Euclidean cone surface D(∆) lifts to the minimal translation surface cover of MT (∆). The direction map θ : T1 R2 → R/2πZ lifts to a map θ : T1 MT (∆) → R/2πZ. W. Patrick Hooper (Northwestern University) Billiards in right triangles are unstable EIU 2007 14 / 27 Closed geodesics on translation surfaces Every closed geodesic on a Euclidean cone surface D(∆) lifts to the minimal translation surface cover of MT (∆). The direction map θ : T1 R2 → R/2πZ lifts to a map θ : T1 MT (∆) → R/2πZ. The direction map θ is invariant under the geodesic ﬂow. Thus, closed geodesics on MT(∆) never intersect. W. Patrick Hooper (Northwestern University) Billiards in right triangles are unstable EIU 2007 14 / 27 Closed geodesics on translation surfaces Every closed geodesic on a Euclidean cone surface D(∆) lifts to the minimal translation surface cover of MT (∆). The direction map θ : T1 R2 → R/2πZ lifts to a map θ : T1 MT (∆) → R/2πZ. The direction map θ is invariant under the geodesic ﬂow. Thus, closed geodesics on MT(∆) never intersect. Moreover, two geodesics which travel in the same direction can not intersect. W. Patrick Hooper (Northwestern University) Billiards in right triangles are unstable EIU 2007 14 / 27 Closed geodesics on translation surfaces Every closed geodesic on a Euclidean cone surface D(∆) lifts to the minimal translation surface cover of MT (∆). The direction map θ : T1 R2 → R/2πZ lifts to a map θ : T1 MT (∆) → R/2πZ. The direction map θ is invariant under the geodesic ﬂow. Thus, closed geodesics on MT(∆) never intersect. Moreover, two geodesics which travel in the same direction can not intersect. This is the main idea behind the proof of Theorem Right triangles don’t have stable periodic billiard paths. We will now discuss the proof. W. Patrick Hooper (Northwestern University) Billiards in right triangles are unstable EIU 2007 14 / 27 No stable trajectories (1 of 4) It is sufﬁcient to prove that a right triangle ∆ whose other angles are not rational multiples of π has no stable periodic billiard paths. W. Patrick Hooper (Northwestern University) Billiards in right triangles are unstable EIU 2007 15 / 27 No stable trajectories (1 of 4) It is sufﬁcient to prove that a right triangle ∆ whose other angles are not rational multiples of π has no stable periodic billiard paths. A periodic billiard path γ in ∆ lifts to a closed geodesic γ in D(∆). W. Patrick Hooper (Northwestern University) Billiards in right triangles are unstable EIU 2007 15 / 27 No stable trajectories (1 of 4) It is sufﬁcient to prove that a right triangle ∆ whose other angles are not rational multiples of π has no stable periodic billiard paths. A periodic billiard path γ in ∆ lifts to a closed geodesic γ in D(∆). γ lifts to a closed geodesic γ in MT (∆). ˜ W. Patrick Hooper (Northwestern University) Billiards in right triangles are unstable EIU 2007 15 / 27 No stable trajectories (1 of 4) It is sufﬁcient to prove that a right triangle ∆ whose other angles are not rational multiples of π has no stable periodic billiard paths. A periodic billiard path γ in ∆ lifts to a closed geodesic γ in D(∆). γ lifts to a closed geodesic γ in MT (∆). ˜ i iii R2 R−1 R1 R0 ii ii iii i W. Patrick Hooper (Northwestern University) Billiards in right triangles are unstable EIU 2007 15 / 27 No stable trajectories (2 of 4) i R2 ii MT (∆) supports a rotation by π, i ρ, which preserves each R1 rhombus. This rotation by π, ρ, is an automorphism of the cover iii MT(∆) → D(∆) R0 ii R−1 iii W. Patrick Hooper (Northwestern University) Billiards in right triangles are unstable EIU 2007 16 / 27 No stable trajectories (3 of 4) i ii Sketch of proof: Right triangles 0000000 1111111 1111111 0000000 0000000 1111111 i only have unstable periodic 0000000 1111111 C1 1111111 0000000 1111111 0000000 billiard paths. 0000000 1111111 γ 0000000 1111111 1111111 0000000 1111111 0000000 We will show that the pair of 1111111 0000000 0000000 1111111 111 000 curves γ and ρ(˜ ) bound a ˜ γ 1111111 0000000 111 000 1111111 0000000 000 111 1111111 0000000 000 iii 111 0000000 1111111 111 000 cylinder in MT(∆) containing two 0000000 1111111 000 111 1111111 0000000 111 000 0000000 ii 1111111 ρ(γ) centers of rhombi, C1 and C2 . 0000000 1111111 0000000 1111111 C2 0000000 1111111 0000000 1111111 1111111 0000000 0000000 1111111 0000000 1111111 iii W. Patrick Hooper (Northwestern University) Billiards in right triangles are unstable EIU 2007 17 / 27 No stable trajectories (4 of 4) i Then in homology on MT∆ , ii γ + ρ(γ) = ∂C1 + ∂C2 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000∂C1 i 0000000 1111111 0000000 1111111 1111111 0000000 γ 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 0000000 1111111 111 000 1111111 0000000 111 000 1111111 0000000 000 111 1111111 0000000 111 000 iii 1111111 0000000 111 000 0000000 1111111 ∂C2 111 000 1111111 0000000 111 000 1111111 0000000 ii ρ(γ) 1111111 0000000 0000000 1111111 1111111 0000000 1111111 0000000 0000000 1111111 1111111 0000000 1111111 0000000 iii W. Patrick Hooper (Northwestern University) Billiards in right triangles are unstable EIU 2007 18 / 27 No stable trajectories (4 of 4) i Then in homology on MT∆ , ii γ + ρ(γ) = ∂C1 + ∂C2 0000000 1111111 1111111 0000000 0000000 1111111 0000000 1111111∂C1 i Under the covering map 0000000 1111111 0000000 1111111 0000000 1111111 γ 0000000 1111111 1111111 0000000 φ : MT∆ → D∆ , 1111111 0000000 1111111 0000000 0000000 1111111 φ( γ ) = φ( ρ(γ) ) = γ 1111111 0000000 111 000 1111111 0000000 111 000 0000000 1111111 111 000 0000000 1111111 111 000 iii 1111111 0000000 000 111 ∂C2 0000000 1111111 111 000 1111111 111 000 0000000 ii ρ(γ) 1111111 0000000 1111111 0000000 0000000 1111111 1111111 0000000 1111111 0000000 1111111 0000000 0000000 1111111 iii W. Patrick Hooper (Northwestern University) Billiards in right triangles are unstable EIU 2007 18 / 27 No stable trajectories (4 of 4) i Then in homology on MT∆ , ii γ + ρ(γ) = ∂C1 + ∂C2 0000000 1111111 1111111 0000000 0000000 1111111 0000000 1111111∂C1 i Under the covering map 0000000 1111111 1111111 0000000 0000000 1111111 γ 0000000 1111111 0000000 1111111 φ : MT∆ → D∆ , 0000000 1111111 1111111 0000000 1111111 0000000 φ( γ ) = φ( ρ(γ) ) = γ 1111111 0000000 000 111 1111111 0000000 111 000 φ( ∂C1 ) = φ( ∂C2 ) 1111111 0000000 000 111 1111111 0000000 000 iii 111 0000000 1111111 111 000 ∂C2 0000000 1111111 111 000 1111111 000 111 0000000 ii ρ(γ) 1111111 0000000 0000000 1111111 1111111 0000000 0000000 1111111 1111111 0000000 0000000 1111111 0000000 1111111 iii W. Patrick Hooper (Northwestern University) Billiards in right triangles are unstable EIU 2007 18 / 27 No stable trajectories (4 of 4) i Then in homology on MT∆ , ii γ + ρ(γ) = ∂C1 + ∂C2 0000000 1111111 1111111 0000000 0000000 1111111 0000000 1111111∂C1 i Under the covering map 0000000 1111111 1111111 0000000 0000000 1111111 γ 0000000 1111111 0000000 1111111 φ : MT∆ → D∆ , 0000000 1111111 1111111 0000000 1111111 0000000 φ( γ ) = φ( ρ(γ) ) = γ 1111111 0000000 000 111 1111111 0000000 111 000 φ( ∂C1 ) = φ( ∂C2 ) 1111111 0000000 000 111 1111111 0000000 000 iii 111 Thus, 1111111 0000000 111 000 ∂C2 1111111 0000000 000 111 1111111 000 111 0000000 ii ρ(γ) 1111111 0000000 γ = φ( ∂C1 ) = 0 1111111 0000000 1111111 0000000 0000000 1111111 1111111 0000000 0000000 1111111 0000000 1111111 iii W. Patrick Hooper (Northwestern University) Billiards in right triangles are unstable EIU 2007 18 / 27 No stable trajectories (4 of 4) i Then in homology on MT∆ , ii γ + ρ(γ) = ∂C1 + ∂C2 0000000 1111111 1111111 0000000 0000000 1111111 0000000 1111111∂C1 i Under the covering map 0000000 1111111 1111111 0000000 0000000 1111111 γ 0000000 1111111 1111111 0000000 φ : MT∆ → D∆ , 0000000 1111111 1111111 0000000 0000000 1111111 φ( γ ) = φ( ρ(γ) ) = γ 0000000 1111111 111 000 1111111 0000000 000 111 φ( ∂C1 ) = φ( ∂C2 ) 0000000 1111111 000 111 0000000 1111111 111 000 iii Thus, 0000000 1111111 000 111 ∂C2 0000000 1111111 111 000 1111111 111 000 0000000 ii ρ(γ) 1111111 0000000 γ = φ( ∂C1 ) = 0 0000000 1111111 1111111 0000000 0000000 1111111 0000000 1111111 0000000 1111111 The geodesic γ is unstable! 0000000 1111111 iii W. Patrick Hooper (Northwestern University) Billiards in right triangles are unstable EIU 2007 18 / 27 Finding the cylinder (1 of 3) Claim 1: γ ∪ ρ(γ) intersects each edge of each rhombus an even num- ber of times. Proof: W. Patrick Hooper (Northwestern University) Billiards in right triangles are unstable EIU 2007 19 / 27 Finding the cylinder (1 of 3) Claim 1: γ ∪ ρ(γ) intersects each edge of each rhombus an even num- ber of times. Proof: Fixing the direction γ travels, there are only 3 possible ways γ can cross each rhombus Rk . W. Patrick Hooper (Northwestern University) Billiards in right triangles are unstable EIU 2007 19 / 27 Finding the cylinder (1 of 3) Claim 1: γ ∪ ρ(γ) intersects each edge of each rhombus an even num- ber of times. Proof: Fixing the direction γ travels, there are only 3 possible ways γ can cross each rhombus Rk . The claim is equivalent to showing that the number of type A crossings of γ equals the number of type C crossings of γ. W. Patrick Hooper (Northwestern University) Billiards in right triangles are unstable EIU 2007 19 / 27 Finding the cylinder (1 of 3) Claim 1: γ ∪ ρ(γ) intersects each edge of each rhombus an even num- ber of times. Proof: Fixing the direction γ travels, there are only 3 possible ways γ can cross each rhombus Rk . The claim is equivalent to showing that the number of type A crossings of γ equals the number of type C crossings of γ. But, γ must close up. So, each time it passes from Rk +1 to Rk −1 it must later pass from Rk −1 to Rk +1 . W. Patrick Hooper (Northwestern University) Billiards in right triangles are unstable EIU 2007 19 / 27 Finding the cylinder (2 of 3) Claim 2: γ ∪ρ(γ) disconnects MT(∆). At least one component contains no singularities with inﬁnite cone angle. Proof: W. Patrick Hooper (Northwestern University) Billiards in right triangles are unstable EIU 2007 20 / 27 Finding the cylinder (2 of 3) Claim 2: γ ∪ρ(γ) disconnects MT(∆). At least one component contains no singularities with inﬁnite cone angle. Proof: By claim 1, each rhombus Rk (γ ∪ ρ(γ)) may be colored so that each inﬁnite cone point is blue, and colors alternate blue/green by adjacency. W. Patrick Hooper (Northwestern University) Billiards in right triangles are unstable EIU 2007 20 / 27 Finding the cylinder (2 of 3) Claim 2: γ ∪ρ(γ) disconnects MT(∆). At least one component contains no singularities with inﬁnite cone angle. Proof: By claim 1, each rhombus Rk (γ ∪ ρ(γ)) may be colored so that each inﬁnite cone point is blue, and colors alternate blue/green by adjacency. The colorings of each rhombus agree along the boundaries of the rhombi. So, the green and blue components are distinct. W. Patrick Hooper (Northwestern University) Billiards in right triangles are unstable EIU 2007 20 / 27 Finding the cylinder (3 of 3) The only oriented Euclidean surface with one or two geodesic boundary components is a cylinder. W. Patrick Hooper (Northwestern University) Billiards in right triangles are unstable EIU 2007 21 / 27 Finding the cylinder (3 of 3) The only oriented Euclidean surface with one or two geodesic boundary components is a cylinder. We still must show that the cylinder contains two centers of rhombi. W. Patrick Hooper (Northwestern University) Billiards in right triangles are unstable EIU 2007 21 / 27 Finding the cylinder (3 of 3) The only oriented Euclidean surface with one or two geodesic boundary components is a cylinder. We still must show that the cylinder contains two centers of rhombi. By construction, the rotation by π must preserve the cylinder. A rotation by π of a cylinder has 2 ﬁxed points. The only ﬁxed points of the rotation by π of MT(∆) are the centers. W. Patrick Hooper (Northwestern University) Billiards in right triangles are unstable EIU 2007 21 / 27 The generalization The right triangles consist of three lines 1, 2, and 3 in the space of triangles T . α2 α1 Theorem (H) If γ is a stable periodic billiard path in a triangle, then tile(γ) is contained in one of the four components of T ( 1 ∪ 2 ∪ 3 ). W. Patrick Hooper (Northwestern University) Billiards in right triangles are unstable EIU 2007 22 / 27 The argument in action (1 of 5) Here is a stable periodic billiard path in a slightly obtuse triangle. 2 Let’s prove that a periodic billiard 3 path with the same orbit type can not appear in a triangle where this obtuse angle becomes right or acute. 1 W. Patrick Hooper (Northwestern University) Billiards in right triangles are unstable EIU 2007 23 / 27 The argument in action (2 of 5) The proof follows from the “general principle” that intersections between geodesics on locally Euclidean surfaces are “essential.” A a W. Patrick Hooper (Northwestern University) Billiards in right triangles are unstable EIU 2007 24 / 27 The argument in action (2 of 5) The proof follows from the “general principle” that intersections between geodesics on locally Euclidean surfaces are “essential.” For every triangle ∆ with a geodesic in this homotopy A a class on D(∆), we can ﬁnd an intersection A with similar topological properties. W. Patrick Hooper (Northwestern University) Billiards in right triangles are unstable EIU 2007 24 / 27 The argument in action (3 of 5) This angle a must satisfy 0 < a < π. W. Patrick Hooper (Northwestern University) Billiards in right triangles are unstable EIU 2007 25 / 27 The argument in action (3 of 5) This angle a must satisfy 0 < a < π. We compute this angle using a "detecting curve" η on the unit tangent bundle. a= dθ η W. Patrick Hooper (Northwestern University) Billiards in right triangles are unstable EIU 2007 25 / 27 The argument in action (3 of 5) This angle a must satisfy 0 < a < π. We compute this angle using a "detecting curve" η on the unit tangent bundle. a= dθ η Thus, for all ∆ ∈ tile(γ), 0< dθ∆ < π η W. Patrick Hooper (Northwestern University) Billiards in right triangles are unstable EIU 2007 25 / 27 The argument in action (4 of 5) α2 We compute a = η dθ = 2α3 − 2α1 − 2α2 = 4α3 − 2π α3 α1 W. Patrick Hooper (Northwestern University) Billiards in right triangles are unstable EIU 2007 26 / 27 The argument in action (4 of 5) α2 We compute a = η dθ = 2α3 − 2α1 − 2α2 = 4α3 − 2π For all ∆ ∈ tile(γ), α3 0<a<π So, π 3π < α3 < α1 2 4 W. Patrick Hooper (Northwestern University) Billiards in right triangles are unstable EIU 2007 26 / 27 The argument in action (5 of 5) Iterating over all intersections gives a convex bounding box for the tile. W. Patrick Hooper (Northwestern University) Billiards in right triangles are unstable EIU 2007 27 / 27

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