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Dynamics of the Pentagram Map Supervised by Dr Introduction Ergodicity Parametrization Unifying the Maps Conclusion Dynamics of the Pentagram Map Supervised by Dr. Xingping Sun H. Dinkins, E. Pavlechko, K. Williams Missouri State University July 28th, 2016 H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 1 / 45 Finite Fourier Transforms Coefficients of ergodicity Elementary geometry Introduction Ergodicity Parametrization Unifying the Maps Conclusion Review Midpoint Iteration Problem of midpoint iterations FIGURE – Basic midpoint iteration on a heptagon H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 2 / 45 Coefficients of ergodicity Elementary geometry Introduction Ergodicity Parametrization Unifying the Maps Conclusion Review Midpoint Iteration Problem of midpoint iterations Finite Fourier Transforms FIGURE – Basic midpoint iteration on a heptagon H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 2 / 45 Elementary geometry Introduction Ergodicity Parametrization Unifying the Maps Conclusion Review Midpoint Iteration Problem of midpoint iterations Finite Fourier Transforms Coefficients of ergodicity FIGURE – Basic midpoint iteration on a heptagon H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 2 / 45 Introduction Ergodicity Parametrization Unifying the Maps Conclusion Review Midpoint Iteration Problem of midpoint iterations Finite Fourier Transforms Coefficients of ergodicity Elementary geometry FIGURE – Basic midpoint iteration on a heptagon H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 2 / 45 Solved using Brouwer’s fixed point theorem Projective geometry Introduction Ergodicity Parametrization Unifying the Maps Conclusion Review Variation First proposed in 1800’s FIGURE – Pentagram mapping on a pentagon H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 3 / 45 Projective geometry Introduction Ergodicity Parametrization Unifying the Maps Conclusion Review Variation First proposed in 1800’s Solved using Brouwer’s fixed point theorem FIGURE – Pentagram mapping on a pentagon H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 3 / 45 Introduction Ergodicity Parametrization Unifying the Maps Conclusion Review Variation First proposed in 1800’s Solved using Brouwer’s fixed point theorem Projective geometry FIGURE – Pentagram mapping on a pentagon H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 3 / 45 Shown that the max rate of area decrease for any pentagon is 14/15 Proved convergence for any regular n-gon Introduction Ergodicity Parametrization Unifying the Maps Conclusion Review Our Work Looked into the generalizations to any n-gon FIGURE – Pentagram mapping on a pentagon H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 4 / 45 Proved convergence for any regular n-gon Introduction Ergodicity Parametrization Unifying the Maps Conclusion Review Our Work Looked into the generalizations to any n-gon Shown that the max rate of area decrease for any pentagon is 14/15 FIGURE – Pentagram mapping on a pentagon H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 4 / 45 Introduction Ergodicity Parametrization Unifying the Maps Conclusion Review Our Work Looked into the generalizations to any n-gon Shown that the max rate of area decrease for any pentagon is 14/15 Proved convergence for any regular n-gon FIGURE – Pentagram mapping on a pentagon H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 4 / 45 Show the area method converges to a point Use the matrices to prove convergence of vertices Explore connections to projective geometry Introduction Ergodicity Parametrization Unifying the Maps Conclusion Review Midterm Goals Generalize the pentagon’s area method to n-gons H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 5 / 45 Use the matrices to prove convergence of vertices Explore connections to projective geometry Introduction Ergodicity Parametrization Unifying the Maps Conclusion Review Midterm Goals Generalize the pentagon’s area method to n-gons Show the area method converges to a point H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 5 / 45 Explore connections to projective geometry Introduction Ergodicity Parametrization Unifying the Maps Conclusion Review Midterm Goals Generalize the pentagon’s area method to n-gons Show the area method converges to a point Use the matrices to prove convergence of vertices H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 5 / 45 Introduction Ergodicity Parametrization Unifying the Maps Conclusion Review Midterm Goals Generalize the pentagon’s area method to n-gons Show the area method converges to a point Use the matrices to prove convergence of vertices Explore connections to projective geometry H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 5 / 45 Introduction Ergodicity Parametrization Unifying the Maps Conclusion Review Contents 1 Introduction Review Extending Previous Results 2 Ergodicity Coefficients of Ergodicity 3 Parametrization Set-up Linear Transformation Python Program 4 Unifying the Maps Construction Basic Properties of the Map Regular Polygon Case General Polygons The Pentagon Case 5 Conclusion H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 6 / 45 Definition The cross ratio of collinear points A; B; C; D 2 R2 is defined as jA − Cj · jB − Dj χ(A; B; C; D) = jA − Bj · jC − Dj where j · j denotes the Euclidean distance. If the four points are ordered A; B; C; D, then χ(A; B; C; D) ≥ 1. Introduction Ergodicity Parametrization Unifying the Maps Conclusion Extending Previous Results Schwartz’s Proof Richard Schwartz [4] proved that the pentagram map converges on any convex polygon. H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 7 / 45 If the four points are ordered A; B; C; D, then χ(A; B; C; D) ≥ 1. Introduction Ergodicity Parametrization Unifying the Maps Conclusion Extending Previous Results Schwartz’s Proof Richard Schwartz [4] proved that the pentagram map converges on any convex polygon. Definition The cross ratio of collinear points A; B; C; D 2 R2 is defined as jA − Cj · jB − Dj χ(A; B; C; D) = jA − Bj · jC − Dj where j · j denotes the Euclidean distance. H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 7 / 45 Introduction Ergodicity Parametrization Unifying the Maps Conclusion Extending Previous Results Schwartz’s Proof Richard Schwartz [4] proved that the pentagram map converges on any convex polygon. Definition The cross ratio of collinear points A; B; C; D 2 R2 is defined as jA − Cj · jB − Dj χ(A; B; C; D) = jA − Bj · jC − Dj where j · j denotes the Euclidean distance. If the four points are ordered A; B; C; D, then χ(A; B; C; D) ≥ 1. H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 7 / 45 The pentagram map preserves the vertex invariants of a pentagon. The pentagram map preserves the product of the vertex invariants for any polygon. Introduction Ergodicity Parametrization Unifying the Maps Conclusion Extending Previous Results Schwartz’s Proof Definition Let v be a vertex of a polygon Π. The vertex invariant of v, written χ(v) is defined by χ(v) = χ(A; B; C; D) where A; B; C; and D are the points defined in the diagram. H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 8 / 45 The pentagram map preserves the product of the vertex invariants for any polygon. Introduction Ergodicity Parametrization Unifying the Maps Conclusion Extending Previous Results Schwartz’s Proof Definition Let v be a vertex of a polygon Π. The vertex invariant of v, written χ(v) is defined by χ(v) = χ(A; B; C; D) where A; B; C; and D are the points defined in the diagram. The pentagram map preserves the vertex invariants of a pentagon. H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 8 / 45 Introduction Ergodicity Parametrization Unifying the Maps Conclusion Extending Previous Results Schwartz’s Proof Definition Let v be a vertex of a polygon Π. The vertex invariant of v, written χ(v) is defined by χ(v) = χ(A; B; C; D) where A; B; C; and D are the points defined in the diagram. The pentagram map preserves the vertex invariants of a pentagon. The pentagram map preserves the product of the vertex invariants for any polygon. H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 8 / 45 Introduction Ergodicity Parametrization Unifying the Maps Conclusion Extending Previous Results Schwartz’s Proof Definition Let A; B 2 S where S is a convex subset of R2. Let x and y be the intersection points of the line through A and B with the boundary of S, where the points are ordered x; A; B; y. Then the Hilbert Distance between A and B in S is defined as δS(A; B) = log(χ(x; A; B; y)) H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 9 / 45 k 1 Assume there exists a line L that intersects each Π in a nontrivial segment. T k T k−1 2 The endpoints of L Π become arbitrarily close to the endpoints of L Π as k ! 1 (Cauchy Sequence w.r.t. Hausdorff Distance). T k k−1 3 The Hilbert Distance between the endpoints of L Π inside Π becomes infinite as k ! 1. k k−1 4 By the triangle inequality, the Hilbert Perimeter of Π in Π becomes infinite as k ! 1. k k−1 5 But we can show that the Hilbert Perimeter of Π in Π is the log of the product of the vertex invariants of Πk , so it is invariant with respect to the pentagram map. 6 Contradiction ! Introduction Ergodicity Parametrization Unifying the Maps Conclusion Extending Previous Results Schwartz’s Proof Proof Sketch (by contradiction) : H.
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