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 ∈ R2 is deﬁned as |A − C| · |B − D| χ(A, B, C, D) = |A − B| · |C − D|

where | · | 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 ∈ R2 is deﬁned as |A − C| · |B − D| χ(A, B, C, D) = |A − B| · |C − D|

where | · | 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 ∈ R2 is deﬁned as |A − C| · |B − D| χ(A, B, C, D) = |A − B| · |C − D|

where | · | 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 ∈ 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 → ∞ (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 → ∞. k k−1 4 By the triangle inequality, the Hilbert Perimeter of Π in Π becomes inﬁnite as k → ∞. 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. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 10 / 45 T k T k−1 2 The endpoints of L Π become arbitrarily close to the endpoints of L Π as k → ∞ (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 → ∞. k k−1 4 By the triangle inequality, the Hilbert Perimeter of Π in Π becomes inﬁnite as k → ∞. 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) : k 1 Assume there exists a line L that intersects each Π in a nontrivial segment.

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 10 / 45 T k k−1 3 The Hilbert Distance between the endpoints of L Π inside Π becomes inﬁnite as k → ∞. k k−1 4 By the triangle inequality, the Hilbert Perimeter of Π in Π becomes inﬁnite as k → ∞. 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

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 10 / 45 k k−1 4 By the triangle inequality, the Hilbert Perimeter of Π in Π becomes inﬁnite as k → ∞. 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

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 10 / 45 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) : 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 → ∞ (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 → ∞. k k−1 4 By the triangle inequality, the Hilbert Perimeter of Π in Π becomes inﬁnite as k → ∞.

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 10 / 45 6 Contradiction !

Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Extending Previous Results Schwartz’s Proof

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 10 / 45 Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Extending Previous Results Schwartz’s Proof

6 Contradiction !

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 10 / 45 By symmetry |BC| < (ki − 1)|CD|. =⇒ |AC| < ki |AB| =⇒ 2|BC| < (ki − 1)(|AB| + |CD|) =⇒ |AD| − |CD| < ki |AB| =⇒ 2|BC| < (ki − 1)(|AD| − |BC|) =⇒ |AD| < (ki − 1)|AB| + |AB| + |CD| =⇒ |BC| < ki −1 |AD| =⇒ |AD| < (ki − 1)|AB| + |AD| − |BC| ki +1 =⇒ |BC| < (ki − 1)|AB|

Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Extending Previous Results Convergence for a Restricted Class of Pentagons

|AC||BD| |AC| Let ki = χ(vi ) = |AB||CD| > |AB|

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 11 / 45 By symmetry |BC| < (ki − 1)|CD|. =⇒ 2|BC| < (ki − 1)(|AB| + |CD|) =⇒ |AD| − |CD| < ki |AB| =⇒ 2|BC| < (ki − 1)(|AD| − |BC|) =⇒ |AD| < (ki − 1)|AB| + |AB| + |CD| =⇒ |BC| < ki −1 |AD| =⇒ |AD| < (ki − 1)|AB| + |AD| − |BC| ki +1 =⇒ |BC| < (ki − 1)|AB|

Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Extending Previous Results Convergence for a Restricted Class of Pentagons

|AC||BD| |AC| Let ki = χ(vi ) = |AB||CD| > |AB| =⇒ |AC| < ki |AB|

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 11 / 45 By symmetry |BC| < (ki − 1)|CD|. =⇒ 2|BC| < (ki − 1)(|AB| + |CD|) =⇒ 2|BC| < (ki − 1)(|AD| − |BC|) =⇒ |AD| < (ki − 1)|AB| + |AB| + |CD| =⇒ |BC| < ki −1 |AD| =⇒ |AD| < (ki − 1)|AB| + |AD| − |BC| ki +1 =⇒ |BC| < (ki − 1)|AB|

Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Extending Previous Results Convergence for a Restricted Class of Pentagons

|AC||BD| |AC| Let ki = χ(vi ) = |AB||CD| > |AB| =⇒ |AC| < ki |AB| =⇒ |AD| − |CD| < ki |AB|

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 11 / 45 By symmetry |BC| < (ki − 1)|CD|. =⇒ 2|BC| < (ki − 1)(|AB| + |CD|) =⇒ 2|BC| < (ki − 1)(|AD| − |BC|) =⇒ |BC| < ki −1 |AD| =⇒ |AD| < (ki − 1)|AB| + |AD| − |BC| ki +1 =⇒ |BC| < (ki − 1)|AB|

Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Extending Previous Results Convergence for a Restricted Class of Pentagons

|AC||BD| |AC| Let ki = χ(vi ) = |AB||CD| > |AB| =⇒ |AC| < ki |AB| =⇒ |AD| − |CD| < ki |AB| =⇒ |AD| < (ki − 1)|AB| + |AB| + |CD|

Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Extending Previous Results Convergence for a Restricted Class of Pentagons

|AC||BD| |AC| Let ki = χ(vi ) = |AB||CD| > |AB| =⇒ |AC| < ki |AB| =⇒ |AD| − |CD| < ki |AB| =⇒ |AD| < (ki − 1)|AB| + |AB| + |CD| =⇒ |AD| < (ki − 1)|AB| + |AD| − |BC|

Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Extending Previous Results Convergence for a Restricted Class of Pentagons

|AC||BD| |AC| Let ki = χ(vi ) = |AB||CD| > |AB| =⇒ |AC| < ki |AB| =⇒ |AD| − |CD| < ki |AB| =⇒ |AD| < (ki − 1)|AB| + |AB| + |CD| =⇒ |AD| < (ki − 1)|AB| + |AD| − |BC| =⇒ |BC| < (ki − 1)|AB|

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 11 / 45 =⇒ 2|BC| < (ki − 1)(|AB| + |CD|) =⇒ 2|BC| < (ki − 1)(|AD| − |BC|) =⇒ |BC| < ki −1 |AD| ki +1

Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Extending Previous Results Convergence for a Restricted Class of Pentagons

|AC||BD| |AC| Let ki = χ(vi ) = |AB||CD| > |AB| By symmetry |BC| < (ki − 1)|CD|. =⇒ |AC| < ki |AB| =⇒ |AD| − |CD| < ki |AB| =⇒ |AD| < (ki − 1)|AB| + |AB| + |CD| =⇒ |AD| < (ki − 1)|AB| + |AD| − |BC| =⇒ |BC| < (ki − 1)|AB|

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 11 / 45 =⇒ 2|BC| < (ki − 1)(|AD| − |BC|) =⇒ |BC| < ki −1 |AD| ki +1

Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Extending Previous Results Convergence for a Restricted Class of Pentagons

|AC||BD| |AC| Let ki = χ(vi ) = |AB||CD| > |AB| By symmetry |BC| < (ki − 1)|CD|. =⇒ |AC| < ki |AB| =⇒ 2|BC| < (ki − 1)(|AB| + |CD|) =⇒ |AD| − |CD| < ki |AB| =⇒ |AD| < (ki − 1)|AB| + |AB| + |CD| =⇒ |AD| < (ki − 1)|AB| + |AD| − |BC| =⇒ |BC| < (ki − 1)|AB|

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 11 / 45 =⇒ |BC| < ki −1 |AD| ki +1

Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Extending Previous Results Convergence for a Restricted Class of Pentagons

|AC||BD| |AC| Let ki = χ(vi ) = |AB||CD| > |AB| By symmetry |BC| < (ki − 1)|CD|. =⇒ |AC| < ki |AB| =⇒ 2|BC| < (ki − 1)(|AB| + |CD|) =⇒ |AD| − |CD| < ki |AB| =⇒ 2|BC| < (ki − 1)(|AD| − |BC|) =⇒ |AD| < (ki − 1)|AB| + |AB| + |CD| =⇒ |AD| < (ki − 1)|AB| + |AD| − |BC| =⇒ |BC| < (ki − 1)|AB|

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 11 / 45 Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Extending Previous Results Convergence for a Restricted Class of Pentagons

|AC||BD| |AC| Let ki = χ(vi ) = |AB||CD| > |AB| By symmetry |BC| < (ki − 1)|CD|. =⇒ |AC| < ki |AB| =⇒ 2|BC| < (ki − 1)(|AB| + |CD|) =⇒ |AD| − |CD| < ki |AB| =⇒ 2|BC| < (ki − 1)(|AD| − |BC|) =⇒ |AD| < (ki − 1)|AB| + |AB| + |CD| =⇒ |BC| < ki −1 |AD| =⇒ |AD| < (ki − 1)|AB| + |AD| − |BC| ki +1 =⇒ |BC| < (ki − 1)|AB|

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 11 / 45 The ki values are invariant under the 1 P4 ki −1 pentagram map. P(Π ) < i=0 di ki +1 k k P(Π ) kmax −1 kmax −1 0 So < 2 < 2 P(Π ) P(Π0) kmax +1 kmax +1 1 If k < 3, then the pentagram =⇒ P(Π ) < 2 kmax −1 max P(Π0) kmax +1 iteration converges to a point and we have a bound for the rate.

Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Extending Previous Results Convergence for a Restricted Class of Pentagons

This applies for any side of Π1.

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 12 / 45 The ki values are invariant under the pentagram map. k k P(Π ) kmax −1 kmax −1 0 So < 2 < 2 P(Π ) P(Π0) kmax +1 kmax +1 1 If k < 3, then the pentagram =⇒ P(Π ) < 2 kmax −1 max P(Π0) kmax +1 iteration converges to a point and we have a bound for the rate.

Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Extending Previous Results Convergence for a Restricted Class of Pentagons

This applies for any side of Π1. 1 P4 ki −1 P(Π ) < di i=0 ki +1

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 12 / 45 The ki values are invariant under the pentagram map. k k So P(Π ) < 2 kmax −1 P(Π0) kmax +1 1 If k < 3, then the pentagram =⇒ P(Π ) < 2 kmax −1 max P(Π0) kmax +1 iteration converges to a point and we have a bound for the rate.

Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Extending Previous Results Convergence for a Restricted Class of Pentagons

This applies for any side of Π1. 1 P4 ki −1 P(Π ) < di i=0 ki +1 < 2 kmax −1 P(Π0) kmax +1

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 12 / 45 The ki values are invariant under the pentagram map. k k So P(Π ) < 2 kmax −1 P(Π0) kmax +1

If kmax < 3, then the pentagram iteration converges to a point and we have a bound for the rate.

Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Extending Previous Results Convergence for a Restricted Class of Pentagons

This applies for any side of Π1. 1 P4 ki −1 P(Π ) < di i=0 ki +1 < 2 kmax −1 P(Π0) kmax +1 1 =⇒ P(Π ) < 2 kmax −1 P(Π0) kmax +1

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 12 / 45 k k So P(Π ) < 2 kmax −1 P(Π0) kmax +1

If kmax < 3, then the pentagram iteration converges to a point and we have a bound for the rate.

Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Extending Previous Results Convergence for a Restricted Class of Pentagons

This applies for any side of Π1. The ki values are invariant under the 1 P4 ki −1 pentagram map. P(Π ) < di i=0 ki +1 < 2 kmax −1 P(Π0) kmax +1 1 =⇒ P(Π ) < 2 kmax −1 P(Π0) kmax +1

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 12 / 45 If kmax < 3, then the pentagram iteration converges to a point and we have a bound for the rate.

Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Extending Previous Results Convergence for a Restricted Class of Pentagons

This applies for any side of Π1. The ki values are invariant under the 1 P4 ki −1 pentagram map. P(Π ) < i=0 di ki +1 k k P(Π ) kmax −1 kmax −1 0 So < 2 < 2 P(Π ) P(Π0) kmax +1 kmax +1 1 =⇒ P(Π ) < 2 kmax −1 P(Π0) kmax +1

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 12 / 45 Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Extending Previous Results Convergence for a Restricted Class of Pentagons

This applies for any side of Π1. The ki values are invariant under the 1 P4 ki −1 pentagram map. P(Π ) < i=0 di ki +1 k k P(Π ) kmax −1 kmax −1 0 So < 2 < 2 P(Π ) P(Π0) kmax +1 kmax +1 1 If k < 3, then the pentagram =⇒ P(Π ) < 2 kmax −1 max P(Π0) kmax +1 iteration converges to a point and we have a bound for the rate.

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 12 / 45 Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Extending Previous Results A Conjecture

1 Explorations in geogebra indicate that P(Π ) < kmax −1 holds in general for any P(Π0) kmax +1 polygon, regardless of the number of sides, but we have not been able to prove this.

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 13 / 45 Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Coefﬁcients of Ergodicity Representation by matrices

The Pentagram map can be represented by an n × n circulant-patterned matrix.

α0 0 1 − α0 0 ... 0   0 α1 0 1 − α1 ... 0    M =  . . .   . .. .  0 1 − αn−1 0 0 . . . αn−1

th c where αi is a proportion along the i diagonal, or α = d

Note : In a regular n-gon α0 = α1 = ... = αn−1 = sin(π/n) 2 sin(2π/n)

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 14 / 45 We can then express the vertices of Πk as

k 0 Π = Mk Mk−1 ... M0Π

Our project’s main goal is to show that the vertices of Πk converge as k → ∞

Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Coefﬁcients of Ergodicity Matrices continued

Multiplying M by a vector of vertices will result in a column vector of the next polygon’s vertices

k+1 k vertices of Π = Mk (vertices of Π )  k+1    k  v0 α0 0 1 − α0 0 ... 0 v0 v k+1 0 α 0 1 − α ... 0  v k   1   1 1   1   .  =  . . .   .   .   . . .   .   .   . . .   .  k+1 0 1 − α 0 0 . . . α v k vn−1 n−1 n−1 n−1

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 15 / 45 Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Coefﬁcients of Ergodicity Matrices continued

Multiplying M by a vector of vertices will result in a column vector of the next polygon’s vertices

We can then express the vertices of Πk as

k 0 Π = Mk Mk−1 ... M0Π

Our project’s main goal is to show that the vertices of Πk converge as k → ∞

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 15 / 45 Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Coefﬁcients of Ergodicity Past Uses

Eric Hintikka [1] used coefﬁcients of ergodicity to prove that any polygon derived from a series of stochastic circulant-patterned matrices will converge. Stochastic : All entries in each row will add to one and be non-negative. Circulant- patterned : Each matrix has the same zero pattern, which repeats through each row while shifting one column each time.

α0 0 1 − α0 0 ... 0   0 α1 0 1 − α1 ... 0    M =  . . .   . .. .  0 1 − αn−1 0 0 . . . αn−1

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 16 / 45 Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Coefﬁcients of Ergodicity Coefﬁcients of Ergodicity

Generally, ergodicity coefﬁcients estimate the rate of convergence for stochastic matrices [2]. We’ll use some key properties of one coefﬁcient, τ1 :

1 0 ≤ τ1(M) ≤ 1, and 0 = τ1(M) ⇔ M is a rank one matrix Pn 2 τ1(M) = 1 − k=1 min{mik , mjk } 3 τ1(M1M2) ≤ τ1(M1)τ1(M2)

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 17 / 45 Each group will multiply to create a positive, stochastic matrix, with Pn τ1(M) = 1 − k=1 min{mik , mjk }. Then we know that τ1 < 1 for each group (n−1) speciﬁcally, we have τ1(Mg ) ≤ 1 − n where is the smallest entry in any M matrix that is greater than zero. When we multiply each of the groups together, we have

k (n−1) k lim τ1(M ) ≤ lim (1 − n ) k→∞ k→∞

Which will equal zero when we have a bound on , the smallest possible α value. Which implies Mk is a rank one matrix, say L. Thus, the polygon converges, as

lim Πk = LΠ0 k→∞

Which is simply a point.

Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Coefﬁcients of Ergodicity Proving Convergence

Scheme : For a sequence of k stochastic matrices, divide them into groups of n. Call one such group Mg .

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 18 / 45 When we multiply each of the groups together, we have

k (n−1) k lim τ1(M ) ≤ lim (1 − n ) k→∞ k→∞

Which will equal zero when we have a bound on , the smallest possible α value. Which implies Mk is a rank one matrix, say L. Thus, the polygon converges, as

lim Πk = LΠ0 k→∞

Which is simply a point.

Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Coefﬁcients of Ergodicity Proving Convergence

Scheme : For a sequence of k stochastic matrices, divide them into groups of n. Call one such group Mg . Each group will multiply to create a positive, stochastic matrix, with Pn τ1(M) = 1 − k=1 min{mik , mjk }. Then we know that τ1 < 1 for each group (n−1) speciﬁcally, we have τ1(Mg ) ≤ 1 − n where is the smallest entry in any M matrix that is greater than zero.

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 18 / 45 Which will equal zero when we have a bound on , the smallest possible α value. Which implies Mk is a rank one matrix, say L. Thus, the polygon converges, as

lim Πk = LΠ0 k→∞

Which is simply a point.

Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Coefﬁcients of Ergodicity Proving Convergence

k (n−1) k lim τ1(M ) ≤ lim (1 − n ) k→∞ k→∞

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 18 / 45 Which implies Mk is a rank one matrix, say L. Thus, the polygon converges, as

lim Πk = LΠ0 k→∞

Which is simply a point.

Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Coefﬁcients of Ergodicity Proving Convergence

k (n−1) k lim τ1(M ) ≤ lim (1 − n ) k→∞ k→∞

Which will equal zero when we have a bound on , the smallest possible α value.

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 18 / 45 Thus, the polygon converges, as

lim Πk = LΠ0 k→∞

Which is simply a point.

Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Coefﬁcients of Ergodicity Proving Convergence

k (n−1) k lim τ1(M ) ≤ lim (1 − n ) k→∞ k→∞

Which will equal zero when we have a bound on , the smallest possible α value. Which implies Mk is a rank one matrix, say L.

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 18 / 45 Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Coefﬁcients of Ergodicity Proving Convergence

k (n−1) k lim τ1(M ) ≤ lim (1 − n ) k→∞ k→∞

Which will equal zero when we have a bound on , the smallest possible α value. Which implies Mk is a rank one matrix, say L. Thus, the polygon converges, as

lim Πk = LΠ0 k→∞

Which is simply a point.

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 18 / 45 Method only works for polygons with odd number of sides :

 α0 0 1 − α0 0 0 0   0 α1 0 1 − α1 0 0     0 0 α2 0 1 − α2 0  M =    0 0 0 α3 0 1 − α3 1 − α4 0 0 0 α4 0  0 1 − α5 0 0 0 α5

γ0 0 γ1 0 γ2 2   0 β0 0 β1 0 β2    k φ2 0 φ0 0 φ1 0  M =    0 ψ2 0 ψ0 0 ψ1 ρ1 0 ρ2 0 ρ0 0  0 ζ1 0 ζ2 0 ζ0

Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Coefﬁcients of Ergodicity Limitations

The matrices to represent the pentagram mapping are made up α values that we 1 have no control over. Eric bounded his matrices with entries (0 < δ < 2 ) and (1 − δ) so there was control over the entries in his matrix.

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 19 / 45 Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Coefﬁcients of Ergodicity Limitations

 α0 0 1 − α0 0 0 0   0 α1 0 1 − α1 0 0     0 0 α2 0 1 − α2 0  M =    0 0 0 α3 0 1 − α3 1 − α4 0 0 0 α4 0  0 1 − α5 0 0 0 α5

γ0 0 γ1 0 γ2 2   0 β0 0 β1 0 β2    k φ2 0 φ0 0 φ1 0  M =    0 ψ2 0 ψ0 0 ψ1 ρ1 0 ρ2 0 ρ0 0  0 ζ1 0 ζ2 0 ζ0

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 19 / 45 Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Coefﬁcients of Ergodicity Possibilities

If we consider α in terms of the cross-ratio, we would have CD α = AD If a bound exists on this α in terms of k, even a restricted case of k, then we are able to use the coefﬁcients of ergodicity

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 20 / 45 Deﬁne a loop f :[0, 1] → R2 which maps parameter t 7→ (x(t), y(t)). The x-coordinates in the loop representing the pentagon  (1 − 5t)a0 + 5ta1 0 ≤ t ≤ 1/5  (2 − 5t)a + (5t − 1)a 1/5 ≤ t ≤ 2/5  1 2 x(t) = (3 − 5t)a2 + (5t − 2)a3 2/5 ≤ t ≤ 3/5  (4 − 5t)a3 + (5t − 3)a4 3/5 ≤ t ≤ 4/5  (5 − 5t)a4 + (5t − 4)a0 4/5 ≤ t ≤ 1

The y-coordinate parametrization has the same form, except all a’s are replaced with b.

Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Set-up Set-up

Represent the pentagon as a loop : 0 The ﬁve points of pentagon Π are represented as (ai , bi ) where i = 0,..., 4.

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 21 / 45 The x-coordinates in the loop representing the pentagon  (1 − 5t)a0 + 5ta1 0 ≤ t ≤ 1/5  (2 − 5t)a + (5t − 1)a 1/5 ≤ t ≤ 2/5  1 2 x(t) = (3 − 5t)a2 + (5t − 2)a3 2/5 ≤ t ≤ 3/5  (4 − 5t)a3 + (5t − 3)a4 3/5 ≤ t ≤ 4/5  (5 − 5t)a4 + (5t − 4)a0 4/5 ≤ t ≤ 1

The y-coordinate parametrization has the same form, except all a’s are replaced with b.

Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Set-up Set-up

Represent the pentagon as a loop : 0 The ﬁve points of pentagon Π are represented as (ai , bi ) where i = 0,..., 4. Deﬁne a loop f :[0, 1] → R2 which maps parameter t 7→ (x(t), y(t)).

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 21 / 45 The y-coordinate parametrization has the same form, except all a’s are replaced with b.

Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Set-up Set-up

Represent the pentagon as a loop : 0 The ﬁve points of pentagon Π are represented as (ai , bi ) where i = 0,..., 4. Deﬁne a loop f :[0, 1] → R2 which maps parameter t 7→ (x(t), y(t)). The x-coordinates in the loop representing the pentagon  (1 − 5t)a0 + 5ta1 0 ≤ t ≤ 1/5  (2 − 5t)a + (5t − 1)a 1/5 ≤ t ≤ 2/5  1 2 x(t) = (3 − 5t)a2 + (5t − 2)a3 2/5 ≤ t ≤ 3/5  (4 − 5t)a3 + (5t − 3)a4 3/5 ≤ t ≤ 4/5  (5 − 5t)a4 + (5t − 4)a0 4/5 ≤ t ≤ 1

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 21 / 45 Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Set-up Set-up

The y-coordinate parametrization has the same form, except all a’s are replaced with b.

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 21 / 45 Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Set-up Loop

t (x(t), y(t)) 0 , 1 (a0, b0) 1/5 (a1, b1) 2/5 (a2, b2) 3/5 (a3, b3) 4/5 (a4, b4)

FIGURE – The parametrization of the pentagon

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 22 / 45 Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Linear Transformation Moving Segments

Paramterization of the ﬁrst pentagon allows for a simple linear translation of segments when deﬁning the second pentagon.

Requires all segments to be deﬁned by a unique linear transformation.

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 23 / 45 Once we know the position of all ﬁve points in the original pentagon, we can determine the intersections of the diagonals using analytic techniques.

So assuming we know (x(t), y(t)) and can ﬁnd (x1(t), y1(t)), we can solve a system of equations to determine the matrix :

 0 − 0 0 − 0  ai bi+1 ai+1bi ai ai+1 ai+1ai AB ai bi+1−ai+1bi ai bi+1−ai+1bi =  0 0 0 0  CD bi bi+1−bi+1bi ai bi+1−ai+1bi ai bi+1−ai+1bi ai bi+1−ai+1bi

One downside is that a matrix needs to be found for each segment’s transformation.

Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Linear Transformation Matrix Multiplication

Because the transformation is a linear one, we can represent the points through matrix multiplication.

x (t) AB x(t) i (i + 1) 1 = ≤ t ≤ y1(t) CD y(t) 5 5

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 24 / 45 So assuming we know (x(t), y(t)) and can ﬁnd (x1(t), y1(t)), we can solve a system of equations to determine the matrix :

 0 − 0 0 − 0  ai bi+1 ai+1bi ai ai+1 ai+1ai AB ai bi+1−ai+1bi ai bi+1−ai+1bi =  0 0 0 0  CD bi bi+1−bi+1bi ai bi+1−ai+1bi ai bi+1−ai+1bi ai bi+1−ai+1bi

One downside is that a matrix needs to be found for each segment’s transformation.

Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Linear Transformation Matrix Multiplication

Because the transformation is a linear one, we can represent the points through matrix multiplication.

x (t) AB x(t) i (i + 1) 1 = ≤ t ≤ y1(t) CD y(t) 5 5

Once we know the position of all ﬁve points in the original pentagon, we can determine the intersections of the diagonals using analytic techniques.

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 24 / 45 One downside is that a matrix needs to be found for each segment’s transformation.

Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Linear Transformation Matrix Multiplication

Because the transformation is a linear one, we can represent the points through matrix multiplication.

x (t) AB x(t) i (i + 1) 1 = ≤ t ≤ y1(t) CD y(t) 5 5

Once we know the position of all ﬁve points in the original pentagon, we can determine the intersections of the diagonals using analytic techniques.

So assuming we know (x(t), y(t)) and can ﬁnd (x1(t), y1(t)), we can solve a system of equations to determine the matrix :

 0 − 0 0 − 0  ai bi+1 ai+1bi ai ai+1 ai+1ai AB ai bi+1−ai+1bi ai bi+1−ai+1bi =  0 0 0 0  CD bi bi+1−bi+1bi ai bi+1−ai+1bi ai bi+1−ai+1bi ai bi+1−ai+1bi

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 24 / 45 Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Linear Transformation Matrix Multiplication

Because the transformation is a linear one, we can represent the points through matrix multiplication.

x (t) AB x(t) i (i + 1) 1 = ≤ t ≤ y1(t) CD y(t) 5 5

Once we know the position of all ﬁve points in the original pentagon, we can determine the intersections of the diagonals using analytic techniques.

So assuming we know (x(t), y(t)) and can ﬁnd (x1(t), y1(t)), we can solve a system of equations to determine the matrix :

 0 − 0 0 − 0  ai bi+1 ai+1bi ai ai+1 ai+1ai AB ai bi+1−ai+1bi ai bi+1−ai+1bi =  0 0 0 0  CD bi bi+1−bi+1bi ai bi+1−ai+1bi ai bi+1−ai+1bi ai bi+1−ai+1bi

One downside is that a matrix needs to be found for each segment’s transformation.

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 24 / 45 2 Show the Perimeter of the new pentagon is smaller due to this transformation.

3 Repeat the process over and over until we get a point.

However, this is a lot of calculation to do by hand. BRING IN THE PYTHON !

Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Linear Transformation Finding Convergence

Our goal : AB 1 Find the matrix for each segment’s transformation CD

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 25 / 45 3 Repeat the process over and over until we get a point.

However, this is a lot of calculation to do by hand. BRING IN THE PYTHON !

Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Linear Transformation Finding Convergence

Our goal : AB 1 Find the matrix for each segment’s transformation CD

2 Show the Perimeter of the new pentagon is smaller due to this transformation.

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 25 / 45 However, this is a lot of calculation to do by hand. BRING IN THE PYTHON !

Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Linear Transformation Finding Convergence

Our goal : AB 1 Find the matrix for each segment’s transformation CD

2 Show the Perimeter of the new pentagon is smaller due to this transformation.

3 Repeat the process over and over until we get a point.

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 25 / 45 Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Linear Transformation Finding Convergence

Our goal : AB 1 Find the matrix for each segment’s transformation CD

2 Show the Perimeter of the new pentagon is smaller due to this transformation.

3 Repeat the process over and over until we get a point.

However, this is a lot of calculation to do by hand. BRING IN THE PYTHON !

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 25 / 45 k k The ratio of P(Π ) , which allows for the easy calculation of P(Π ) P(Πk−1) P(Π0) Draws a picture of the n iterations Gives the vertices of Πk

Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Python Program The Game

Give it 5 points, going in counter-clockwise order. It ﬁnds : The intersection of the diagonals.

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 26 / 45 Draws a picture of the n iterations Gives the vertices of Πk

Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Python Program The Game

Give it 5 points, going in counter-clockwise order. It ﬁnds : The intersection of the diagonals. k k The ratio of P(Π ) , which allows for the easy calculation of P(Π ) P(Πk−1) P(Π0)

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 26 / 45 Gives the vertices of Πk

Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Python Program The Game

Give it 5 points, going in counter-clockwise order. It ﬁnds : The intersection of the diagonals. k k The ratio of P(Π ) , which allows for the easy calculation of P(Π ) P(Πk−1) P(Π0) Draws a picture of the n iterations

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 26 / 45 Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Python Program The Game

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 26 / 45 Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Construction Two Types of Maps

Can we unify these two maps ?

The midpoint map The pentagram map

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 27 / 45 Let Π be an n-gon with vertices v 0, v 0,..., v 0 . 0 1 n−1 1 Call the intersection vi . Choose ai , bi ∈ [0, 1] with ai ≤ bi for i = 0, 1,..., n − 1. Apply this process to each edge to form the vertices of an n-gon T (Π). Construct points Ai , Bi such that v0A v0B Applying this process k times gives us i i i i k ai = 0 0 and bi = 0 0 . T (Π). vi vi+1 vi vi+1 0 0 Connect vi−1 to Bi and vi+2 to Ai .

Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Construction The Generalized Pentagram Map (GPM)

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 28 / 45 1 Call the intersection vi . Choose ai , bi ∈ [0, 1] with ai ≤ bi for i = 0, 1,..., n − 1. Apply this process to each edge to form the vertices of an n-gon T (Π). Construct points Ai , Bi such that v0A v0B Applying this process k times gives us i i i i k ai = 0 0 and bi = 0 0 . T (Π). vi vi+1 vi vi+1 0 0 Connect vi−1 to Bi and vi+2 to Ai .

Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Construction The Generalized Pentagram Map (GPM)

Let Π be an n-gon with vertices 0 0 0 v0 , v1 ,..., vn−1.

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 28 / 45 1 Call the intersection vi . Apply this process to each edge to form the vertices of an n-gon T (Π). Construct points Ai , Bi such that v0A v0B Applying this process k times gives us i i i i k ai = 0 0 and bi = 0 0 . T (Π). vi vi+1 vi vi+1 0 0 Connect vi−1 to Bi and vi+2 to Ai .

Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Construction The Generalized Pentagram Map (GPM)

Let Π be an n-gon with vertices 0 0 0 v0 , v1 ,..., vn−1.

Choose ai , bi ∈ [0, 1] with ai ≤ bi for i = 0, 1,..., n − 1.

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 28 / 45 1 Call the intersection vi . Apply this process to each edge to form the vertices of an n-gon T (Π). Applying this process k times gives us T k (Π).

0 0 Connect vi−1 to Bi and vi+2 to Ai .

Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Construction The Generalized Pentagram Map (GPM)

Let Π be an n-gon with vertices 0 0 0 v0 , v1 ,..., vn−1.

Choose ai , bi ∈ [0, 1] with ai ≤ bi for i = 0, 1,..., n − 1.

Construct points Ai , Bi such that 0 0 vi Ai vi Bi ai = 0 0 and bi = 0 0 . vi vi+1 vi vi+1

Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Construction The Generalized Pentagram Map (GPM)

Let Π be an n-gon with vertices 0 0 0 v0 , v1 ,..., vn−1.

Choose ai , bi ∈ [0, 1] with ai ≤ bi for i = 0, 1,..., n − 1.

Construct points Ai , Bi such that 0 0 vi Ai vi Bi ai = 0 0 and bi = 0 0 . vi vi+1 vi vi+1 0 0 Connect vi−1 to Bi and vi+2 to Ai .

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 28 / 45 Apply this process to each edge to form the vertices of an n-gon T (Π). Applying this process k times gives us T k (Π).

Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Construction The Generalized Pentagram Map (GPM)

Let Π be an n-gon with vertices v 0, v 0,..., v 0 . 0 1 n−1 1 Call the intersection vi . Choose ai , bi ∈ [0, 1] with ai ≤ bi for i = 0, 1,..., n − 1.

Construct points Ai , Bi such that 0 0 vi Ai vi Bi ai = 0 0 and bi = 0 0 . vi vi+1 vi vi+1 0 0 Connect vi−1 to Bi and vi+2 to Ai .

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 28 / 45 Applying this process k times gives us T k (Π).

Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Construction The Generalized Pentagram Map (GPM)

Let Π be an n-gon with vertices v 0, v 0,..., v 0 . 0 1 n−1 1 Call the intersection vi . Choose ai , bi ∈ [0, 1] with ai ≤ bi for i = 0, 1,..., n − 1. Apply this process to each edge to form the vertices of an n-gon T (Π). Construct points Ai , Bi such that 0 0 vi Ai vi Bi ai = 0 0 and bi = 0 0 . vi vi+1 vi vi+1 0 0 Connect vi−1 to Bi and vi+2 to Ai .

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 28 / 45 Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Construction The Generalized Pentagram Map (GPM)

Let Π be an n-gon with vertices v 0, v 0,..., v 0 . 0 1 n−1 1 Call the intersection vi . Choose ai , bi ∈ [0, 1] with ai ≤ bi for i = 0, 1,..., n − 1. Apply this process to each edge to form the vertices of an n-gon T (Π). Construct points Ai , Bi such that v0A v0B Applying this process k times gives us i i i i k ai = 0 0 and bi = 0 0 . T (Π). vi vi+1 vi vi+1 0 0 Connect vi−1 to Bi and vi+2 to Ai .

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 28 / 45 A GPM T is uniquely determined by the ai and bi values.

Let f (T ) = (a0, b0, a1, b1,..., an−1, bn−1).

Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Construction Representing the Map

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 29 / 45 Let f (T ) = (a0, b0, a1, b1,..., an−1, bn−1).

Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Construction Representing the Map

A GPM T is uniquely determined by the ai and bi values.

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 29 / 45 Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Construction Representing the Map

A GPM T is uniquely determined by the ai and bi values.

Let f (T ) = (a0, b0, a1, b1,..., an−1, bn−1).

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 29 / 45 T is the identity f (T ) = (1, 1,..., 1) =⇒ T is a relabeling map. of the vertices.

Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Basic Properties of the Map Some Examples

f (T ) = (0, 0,..., 0) =⇒

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 30 / 45 f (T ) = (1, 1,..., 1) =⇒ T is a relabeling of the vertices.

Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Basic Properties of the Map Some Examples

f (T ) = (0, 0,..., 0) =⇒ T is the identity map.

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 30 / 45 T is a relabeling of the vertices.

Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Basic Properties of the Map Some Examples

f (T ) = (0, 0,..., 0) =⇒ T is the identity f (T ) = (1, 1,..., 1) =⇒ map.

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 30 / 45 Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Basic Properties of the Map Some Examples

f (T ) = (0, 0,..., 0) =⇒ T is the identity f (T ) = (1, 1,..., 1) =⇒ T is a relabeling map. of the vertices.

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 30 / 45 T is the f (T ) = (0, 1, 0, 1,..., 0, 1) =⇒ T is the midpoint map. pentagram map.

Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Basic Properties of the Map Some Examples

1 1 1 f (T ) = ( 2 , 2 ,..., 2 ) =⇒

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 31 / 45 f (T ) = (0, 1, 0, 1,..., 0, 1) =⇒ T is the pentagram map.

Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Basic Properties of the Map Some Examples

1 1 1 f (T ) = ( 2 , 2 ,..., 2 ) =⇒ T is the midpoint map.

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 31 / 45 T is the pentagram map.

Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Basic Properties of the Map Some Examples

1 1 1 f (T ) = ( 2 , 2 ,..., 2 ) =⇒ T is the f (T ) = (0, 1, 0, 1,..., 0, 1) =⇒ midpoint map.

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 31 / 45 Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Basic Properties of the Map Some Examples

1 1 1 f (T ) = ( 2 , 2 ,..., 2 ) =⇒ T is the f (T ) = (0, 1, 0, 1,..., 0, 1) =⇒ T is the midpoint map. pentagram map.

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 31 / 45 Gray regions are the overlap of two consecutive vertex triangles. The vertices of T (Π) lie inside separate gray regions. Each vertex of T (Π) can lie anywhere in its corresponding region without affecting the conﬁguration of the other vertices.

Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Basic Properties of the Map Intuition for the Map on Convex Polygons

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 32 / 45 The vertices of T (Π) lie inside separate gray regions. Each vertex of T (Π) can lie anywhere in its corresponding region without affecting the conﬁguration of the other vertices.

Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Basic Properties of the Map Intuition for the Map on Convex Polygons

Gray regions are the overlap of two consecutive vertex triangles.

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 32 / 45 Each vertex of T (Π) can lie anywhere in its corresponding region without affecting the conﬁguration of the other vertices.

Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Basic Properties of the Map Intuition for the Map on Convex Polygons

Gray regions are the overlap of two consecutive vertex triangles. The vertices of T (Π) lie inside separate gray regions.

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 32 / 45 Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Basic Properties of the Map Intuition for the Map on Convex Polygons

Gray regions are the overlap of two consecutive vertex triangles. The vertices of T (Π) lie inside separate gray regions. Each vertex of T (Π) can lie anywhere in its corresponding region without affecting the conﬁguration of the other vertices.

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 32 / 45 Do all GPMs preserve convexity ? Unfortunately, no.

Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Basic Properties of the Map Convexity and the GPM

All the maps we’ve looked at previously preserve convexity.

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 33 / 45 Unfortunately, no.

Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Basic Properties of the Map Convexity and the GPM

All the maps we’ve looked at previously preserve convexity. Do all GPMs preserve convexity ?

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 33 / 45 Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Basic Properties of the Map Convexity and the GPM

All the maps we’ve looked at previously preserve convexity. Do all GPMs preserve convexity ? Unfortunately, no.

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 33 / 45 0 π π s = s cos n − (1 − 2m) tan(z) sin n

Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Regular Polygon Case A Special Type of GPM

Let Π be a regular n-gon and let T be a GPM such that 1 f (T ) = (m, 1 − m, m, 1 − m,..., m, 1 − m) for some m ∈ [0, 2 ].

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 34 / 45 Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Regular Polygon Case A Special Type of GPM

Let Π be a regular n-gon and let T be a GPM such that 1 f (T ) = (m, 1 − m, m, 1 − m,..., m, 1 − m) for some m ∈ [0, 2 ].

0 π π s = s cos n − (1 − 2m) tan(z) sin n

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 34 / 45 Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Regular Polygon Case A Special Type of GPM

Let Π be a regular n-gon and let T be a GPM such that 1 f (T ) = (m, 1 − m, m, 1 − m,..., m, 1 − m) for some m ∈ [0, 2 ].

0 π π s = s cos n − (1 − 2m) tan(z) sin n

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 34 / 45 k+1 cos( 2π ) Plugging in m = 0 reduces this equation to P(T (Π)) = n which is what P(T k (Π)) π cos( n ) we obtained previously. So T is a convexity preserving GPM on regular polygons and T k (Π) converges to a point.

Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Regular Polygon Case A Special Type of GPM

Found using : Multiple Law of Sines applications Similar triangles Symmetry of the regular polygon

P(T k+1(Π)) π π = cos − (1 − 2m) sin tan(z) P(T k (Π)) n n

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 35 / 45 So T is a convexity preserving GPM on regular polygons and T k (Π) converges to a point.

Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Regular Polygon Case A Special Type of GPM

Found using : Multiple Law of Sines applications Similar triangles Symmetry of the regular polygon

P(T k+1(Π)) π π = cos − (1 − 2m) sin tan(z) P(T k (Π)) n n

k+1 cos( 2π ) Plugging in m = 0 reduces this equation to P(T (Π)) = n which is what P(T k (Π)) π cos( n ) we obtained previously.

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 35 / 45 Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Regular Polygon Case A Special Type of GPM

Found using : Multiple Law of Sines applications Similar triangles Symmetry of the regular polygon

P(T k+1(Π)) π π = cos − (1 − 2m) sin tan(z) P(T k (Π)) n n

k+1 cos( 2π ) Plugging in m = 0 reduces this equation to P(T (Π)) = n which is what P(T k (Π)) π cos( n ) we obtained previously. So T is a convexity preserving GPM on regular polygons and T k (Π) converges to a point.

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 35 / 45 It is a nontrivial convexity-preserving GPM on regular polygons. This very "normal" type of GPM preserves regularity and decreases side length in a predictable way.

Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Regular Polygon Case A Special Type of GPM

What makes this map important ?

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 36 / 45 This very "normal" type of GPM preserves regularity and decreases side length in a predictable way.

Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Regular Polygon Case A Special Type of GPM

What makes this map important ? It is a nontrivial convexity-preserving GPM on regular polygons.

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 36 / 45 Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Regular Polygon Case A Special Type of GPM

What makes this map important ? It is a nontrivial convexity-preserving GPM on regular polygons. This very "normal" type of GPM preserves regularity and decreases side length in a predictable way.

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 36 / 45 T2(Π) is convex at vertex 0. T2(Π) is not convex at vertex 0

Introduction Ergodicity Parametrization Unifying the Maps Conclusion

General Polygons GPM Properties

Proposition

Let T1 and T2 be GPMs on a convex n-gon Π such that f (T1) = (a0, b0,..., an−1, bn−1) and f (T2) = (x, b0,..., an−1, bn−1) where a0 ≤ x. Then A(T1(Π)) ≤ A(T2(Π)).

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 37 / 45 T2(Π) is not convex at vertex 0

Introduction Ergodicity Parametrization Unifying the Maps Conclusion

General Polygons GPM Properties

Proposition

Let T1 and T2 be GPMs on a convex n-gon Π such that f (T1) = (a0, b0,..., an−1, bn−1) and f (T2) = (x, b0,..., an−1, bn−1) where a0 ≤ x. Then A(T1(Π)) ≤ A(T2(Π)).

T2(Π) is convex at vertex 0.

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 37 / 45 Introduction Ergodicity Parametrization Unifying the Maps Conclusion

General Polygons GPM Properties

Proposition

Let T1 and T2 be GPMs on a convex n-gon Π such that f (T1) = (a0, b0,..., an−1, bn−1) and f (T2) = (x, b0,..., an−1, bn−1) where a0 ≤ x. Then A(T1(Π)) ≤ A(T2(Π)).

T2(Π) is convex at vertex 0. T2(Π) is not convex at vertex 0

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 37 / 45 The process seen in the previous proposition terminates with the pentagram map. A(T k+1(Π)) Recall that last time we proved P < 14 where Π is a pentagon. ( k (Π)) 15 A TP We can use this corollary to obtain a better bound on the rate of area reduction for the pentagram map on pentagons.

Introduction Ergodicity Parametrization Unifying the Maps Conclusion

General Polygons GPM Properties

Corollary

Let TP be the pentagram map and T be any other GPM on a convex polygon Π. Then A(TP (Π)) < A(T (Π)).

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 38 / 45 A(T k+1(Π)) Recall that last time we proved P < 14 where Π is a pentagon. ( k (Π)) 15 A TP We can use this corollary to obtain a better bound on the rate of area reduction for the pentagram map on pentagons.

Introduction Ergodicity Parametrization Unifying the Maps Conclusion

General Polygons GPM Properties

Corollary

Let TP be the pentagram map and T be any other GPM on a convex polygon Π. Then A(TP (Π)) < A(T (Π)).

The process seen in the previous proposition terminates with the pentagram map.

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 38 / 45 We can use this corollary to obtain a better bound on the rate of area reduction for the pentagram map on pentagons.

Introduction Ergodicity Parametrization Unifying the Maps Conclusion

General Polygons GPM Properties

Corollary

Let TP be the pentagram map and T be any other GPM on a convex polygon Π. Then A(TP (Π)) < A(T (Π)).

The process seen in the previous proposition terminates with the pentagram map. A(T k+1(Π)) Recall that last time we proved P < 14 where Π is a pentagon. ( k (Π)) 15 A TP

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 38 / 45 Introduction Ergodicity Parametrization Unifying the Maps Conclusion

General Polygons GPM Properties

Corollary

Let TP be the pentagram map and T be any other GPM on a convex polygon Π. Then A(TP (Π)) < A(T (Π)).

The process seen in the previous proposition terminates with the pentagram map. A(T k+1(Π)) Recall that last time we proved P < 14 where Π is a pentagon. ( k (Π)) 15 A TP We can use this corollary to obtain a better bound on the rate of area reduction for the pentagram map on pentagons.

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 38 / 45 By the proposition on the previous slide, A(T k+1(Π)) A(T (T k (Π))) P < m P < 1 − m(1 − m). ( k (Π)) ( k (Π)) A TP A TP 3 On the interval [0, 1], the function g(x) = 1 − x(1 − x) is attains a minimum of 4 1 at x = 2 . A(T k+1(Π)) A(T k (Π)) k So P < 3 =⇒ P < 3 . ( k (Π)) 4 A(Π) 4 A TP

Introduction Ergodicity Parametrization Unifying the Maps Conclusion

The Pentagon Case A Better Bound

The following proposition is due to Dan Ismailescu et al. [3].

Proposition

Let Tm be a GPM on a convex pentagon Π such that f (Tm) = (m, m,..., m). Then k+1 A(Tm (Π)) k < 1 − m(1 − m). A(Tm(Π))

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 39 / 45 3 On the interval [0, 1], the function g(x) = 1 − x(1 − x) is attains a minimum of 4 1 at x = 2 . A(T k+1(Π)) A(T k (Π)) k So P < 3 =⇒ P < 3 . ( k (Π)) 4 A(Π) 4 A TP

Introduction Ergodicity Parametrization Unifying the Maps Conclusion

The Pentagon Case A Better Bound

The following proposition is due to Dan Ismailescu et al. [3].

Proposition

Let Tm be a GPM on a convex pentagon Π such that f (Tm) = (m, m,..., m). Then k+1 A(Tm (Π)) k < 1 − m(1 − m). A(Tm(Π))

By the proposition on the previous slide, A(T k+1(Π)) A(T (T k (Π))) P < m P < 1 − m(1 − m). ( k (Π)) ( k (Π)) A TP A TP

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 39 / 45 A(T k+1(Π)) A(T k (Π)) k So P < 3 =⇒ P < 3 . ( k (Π)) 4 A(Π) 4 A TP

Introduction Ergodicity Parametrization Unifying the Maps Conclusion

The Pentagon Case A Better Bound

The following proposition is due to Dan Ismailescu et al. [3].

Proposition

Let Tm be a GPM on a convex pentagon Π such that f (Tm) = (m, m,..., m). Then k+1 A(Tm (Π)) k < 1 − m(1 − m). A(Tm(Π))

By the proposition on the previous slide, A(T k+1(Π)) A(T (T k (Π))) P < m P < 1 − m(1 − m). ( k (Π)) ( k (Π)) A TP A TP 3 On the interval [0, 1], the function g(x) = 1 − x(1 − x) is attains a minimum of 4 1 at x = 2 .

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 39 / 45 Introduction Ergodicity Parametrization Unifying the Maps Conclusion

The Pentagon Case A Better Bound

The following proposition is due to Dan Ismailescu et al. [3].

Proposition

Let Tm be a GPM on a convex pentagon Π such that f (Tm) = (m, m,..., m). Then k+1 A(Tm (Π)) k < 1 − m(1 − m). A(Tm(Π))

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 39 / 45 Proposition

Let T1 and T2 be GPMs on a convex n-gon Π such that f (T1) = (a0, b0,..., an−1, bn−1) and f (T2) = (x, b0,..., an−1, bn−1) where a0 ≤ x. Then A(T2(Π)) ≤ A(T1(Π)).

Proposition Let Π be a convex pentagon and let T be a convexity preserving GPM on Π such that k f (T ) = (a0, b0, a1, b1,..., a4, b4) with ai ≤ m ≤ bi for i = 0, 1,..., 4. Then T (Π) shrinks to a region of zero area. In particular,

A(T k (Π)) ≤ (1 − m(1 − m))k A(Π)

Proof : Apply the top proposition to each coordinate of f (T ).

Introduction Ergodicity Parametrization Unifying the Maps Conclusion

The Pentagon Case A More General Result

What else can we say about different types of GPMs on convex pentagons ?

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 40 / 45 Proposition Let Π be a convex pentagon and let T be a convexity preserving GPM on Π such that k f (T ) = (a0, b0, a1, b1,..., a4, b4) with ai ≤ m ≤ bi for i = 0, 1,..., 4. Then T (Π) shrinks to a region of zero area. In particular,

A(T k (Π)) ≤ (1 − m(1 − m))k A(Π)

Proof : Apply the top proposition to each coordinate of f (T ).

Introduction Ergodicity Parametrization Unifying the Maps Conclusion

The Pentagon Case A More General Result

What else can we say about different types of GPMs on convex pentagons ?

Proposition

Let T1 and T2 be GPMs on a convex n-gon Π such that f (T1) = (a0, b0,..., an−1, bn−1) and f (T2) = (x, b0,..., an−1, bn−1) where a0 ≤ x. Then A(T2(Π)) ≤ A(T1(Π)).

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 40 / 45 Proof : Apply the top proposition to each coordinate of f (T ).

Introduction Ergodicity Parametrization Unifying the Maps Conclusion

The Pentagon Case A More General Result

What else can we say about different types of GPMs on convex pentagons ?

Proposition

Let T1 and T2 be GPMs on a convex n-gon Π such that f (T1) = (a0, b0,..., an−1, bn−1) and f (T2) = (x, b0,..., an−1, bn−1) where a0 ≤ x. Then A(T2(Π)) ≤ A(T1(Π)).

A(T k (Π)) ≤ (1 − m(1 − m))k A(Π)

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 40 / 45 Introduction Ergodicity Parametrization Unifying the Maps Conclusion

The Pentagon Case A More General Result

What else can we say about different types of GPMs on convex pentagons ?

Proposition

Let T1 and T2 be GPMs on a convex n-gon Π such that f (T1) = (a0, b0,..., an−1, bn−1) and f (T2) = (x, b0,..., an−1, bn−1) where a0 ≤ x. Then A(T2(Π)) ≤ A(T1(Π)).

A(T k (Π)) ≤ (1 − m(1 − m))k A(Π)

Proof : Apply the top proposition to each coordinate of f (T ).

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 40 / 45 We obtained a better bound of the rate of area decrease for the pentagram map. We proved that a restricted class of GPMs applied to a convex pentagon shrinks to a region of zero area. Furthermore, we provided a bound on the rate of area decrease.

Introduction Ergodicity Parametrization Unifying the Maps Conclusion

The Pentagon Case Review of GPM Results

We proved convergence to a point for a special type of GPM applied to regular polygons.

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 41 / 45 We proved that a restricted class of GPMs applied to a convex pentagon shrinks to a region of zero area. Furthermore, we provided a bound on the rate of area decrease.

Introduction Ergodicity Parametrization Unifying the Maps Conclusion

The Pentagon Case Review of GPM Results

We proved convergence to a point for a special type of GPM applied to regular polygons. We obtained a better bound of the rate of area decrease for the pentagram map.

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 41 / 45 Introduction Ergodicity Parametrization Unifying the Maps Conclusion

The Pentagon Case Review of GPM Results

We proved convergence to a point for a special type of GPM applied to regular polygons. We obtained a better bound of the rate of area decrease for the pentagram map. We proved that a restricted class of GPMs applied to a convex pentagon shrinks to a region of zero area. Furthermore, we provided a bound on the rate of area decrease.

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 41 / 45 Investigate different types of GPMs. Study GPMs on polygons with n > 5 vertices.

Introduction Ergodicity Parametrization Unifying the Maps Conclusion

The Pentagon Case Future GPM Directions

Given a polygon Π, ﬁnd a sufﬁcient condition for a GPM to be a convexity-preserving map on Π.

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 42 / 45 Study GPMs on polygons with n > 5 vertices.

Introduction Ergodicity Parametrization Unifying the Maps Conclusion

The Pentagon Case Future GPM Directions

Given a polygon Π, ﬁnd a sufﬁcient condition for a GPM to be a convexity-preserving map on Π. Investigate different types of GPMs.

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 42 / 45 Introduction Ergodicity Parametrization Unifying the Maps Conclusion

The Pentagon Case Future GPM Directions

Given a polygon Π, ﬁnd a sufﬁcient condition for a GPM to be a convexity-preserving map on Π. Investigate different types of GPMs. Study GPMs on polygons with n > 5 vertices.

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 42 / 45 3 Improved the rate of convergence for Area to 4 . Made a program to assist in computation of the pentagram map. Set up methods for simpler geometric proofs for convergence. Worked with matrices to represent convergence. Built upon previously established results by Richard Schwartz. Proved convergence to a point for a restricted class of pentagons.

Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Overall Results

Generalized the iteration procedure.

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 43 / 45 Made a program to assist in computation of the pentagram map. Set up methods for simpler geometric proofs for convergence. Worked with matrices to represent convergence. Built upon previously established results by Richard Schwartz. Proved convergence to a point for a restricted class of pentagons.

Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Overall Results

Generalized the iteration procedure. 3 Improved the rate of convergence for Area to 4 .

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 43 / 45 Set up methods for simpler geometric proofs for convergence. Worked with matrices to represent convergence. Built upon previously established results by Richard Schwartz. Proved convergence to a point for a restricted class of pentagons.

Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Overall Results

Generalized the iteration procedure. 3 Improved the rate of convergence for Area to 4 . Made a program to assist in computation of the pentagram map.

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 43 / 45 Worked with matrices to represent convergence. Built upon previously established results by Richard Schwartz. Proved convergence to a point for a restricted class of pentagons.

Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Overall Results

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 43 / 45 Built upon previously established results by Richard Schwartz. Proved convergence to a point for a restricted class of pentagons.

Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Overall Results

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 43 / 45 Proved convergence to a point for a restricted class of pentagons.

Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Overall Results

Generalized the iteration procedure. 3 Improved the rate of convergence for Area to 4 . Made a program to assist in computation of the pentagram map. Set up methods for simpler geometric proofs for convergence. Worked with matrices to represent convergence. Built upon previously established results by Richard Schwartz.

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 43 / 45 Introduction Ergodicity Parametrization Unifying the Maps Conclusion

Overall Results

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 43 / 45 Introduction Ergodicity Parametrization Unifying the Maps Conclusion

THANK YOU

Dr. Sun Professor Vollmar for his Python expertise Missouri State University for hosting us The NSF : Grant #1559911

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 44 / 45 Introduction Ergodicity Parametrization Unifying the Maps Conclusion

References

Erik Hintikka and Xingping Sun. Convergence of sequences of polygons. Involve, 2016. Ilse Ipsen and Teresa Selee. Ergodicity coefﬁcients deﬁned by vector norms. Society for Industrial and Applied Mathematics, 32(1) :153 – 200, 2011. Dan Ismailescu et al. Area problems involving kasner polygons. ArXhiv : 0910.0452v1, 2009. Richard Schwartz. The pentagram map. Experimental Mathematics, 51 :71–81, 1994.

H. Dinkins, E. Pavlechko, K. Williams MSU The Pentagram Map July 28th, 2016 45 / 45