Unlikely Intersections and Portraits of Dynamical Semigroups Talia - PowerPoint PPT Presentation

Unlikely Intersections and Portraits of Dynamical Semigroups Talia Blum Colby Kelln Henry Talbott February 1, 2020 Nebraska Conference for Undergraduate Women in Math Talia Blum, Colby Kelln, Henry Talbott 1 / 20 February 1, 2020 1 / 20

Classical dynamics Definition A dynamical system is a rational function f ∈ C ( x ) together with a set S ⊆ C such that f : S → S. Definition The orbit of a point x 0 under f is O f ( x 0 ) = { x 0 , f ( x 0 ) , f ( f ( x 0 )) , · · · } . f ( x ) = x 2 Talia Blum, Colby Kelln, Henry Talbott 2 / 20 February 1, 2020 2 / 20

Classical dynamics Definition A dynamical system is a rational function f ∈ C ( x ) together with a set S ⊆ C such that f : S → S. Definition The orbit of a point x 0 under f is O f ( x 0 ) = { x 0 , f ( x 0 ) , f ( f ( x 0 )) , · · · } . f ( x ) = x 2 Question (Central question) Let f , g ∈ C [ x ] . Are there points x 0 with finite multi-orbits, O f , g ( x 0 ) = { x 0 , f ( x 0 ) , g ( x 0 ) , f ( g ( x 0 )) , · · · } ? Talia Blum, Colby Kelln, Henry Talbott 2 / 20 February 1, 2020 2 / 20

Beyond classical dynamics Classical dynamics Generalized dynamics f ( x ) = x 2 f ( x ) = x 2 g ( x ) = x 2 − 2 Talia Blum, Colby Kelln, Henry Talbott 3 / 20 February 1, 2020 3 / 20

Realization spaces of portraits Portrait f ( x 0 ) = x 2 . . . f ( x 3 ) = x 3 such that f has degree 2 Realization space ( x 0 , x 1 , x 2 , x 3 ) = (0 , 1 , t , t 2 − t + 1) t 2 − t x 2 − 1 1 f ( x ) = t 2 − t x Talia Blum, Colby Kelln, Henry Talbott 4 / 20 February 1, 2020 4 / 20

Realization spaces of portraits Portrait f ( x 0 ) = x 2 g ( x 0 ) = x 0 . . . . . . f ( x 3 ) = x 3 g ( x 3 ) = x 1 such that f , g have degree 2 Intersection possibilities Talia Blum, Colby Kelln, Henry Talbott 5 / 20 February 1, 2020 5 / 20

Dimension counting heuristic Expected dimension heuristic For a portrait’s system of equations, #( variables ) − #( equations ) − 2 symmetries of C = expected dimension . Quadratic example + 6 coefficient variables + 4 point variables − 8 equations − 2 symmetries of C = 0 dim realization space expected Talia Blum, Colby Kelln, Henry Talbott 6 / 20 February 1, 2020 6 / 20

Data collection Two quadratics acting on four points Dimension #(Portraits) -1 206 0 560 1 14 Two cubics acting on six points #(Portraits) 1 Dimension -1 52,238 0 1,251,585 1 1,009 2 16 1 for the computed 1,304,848 out of 1,350,742 data points Talia Blum, Colby Kelln, Henry Talbott 7 / 20 February 1, 2020 7 / 20

1-dimensional: 2 quadratics on 4 points Talia Blum, Colby Kelln, Henry Talbott 8 / 20 February 1, 2020 8 / 20

Portraits with only two images Quadratic: Talia Blum, Colby Kelln, Henry Talbott 9 / 20 February 1, 2020 9 / 20

Portraits with only two images Quadratic: Cubic: Talia Blum, Colby Kelln, Henry Talbott 9 / 20 February 1, 2020 9 / 20

Classification by number of images Question Is there a relationship between the number of images and dimension of realization space? Two quadratics acting on four points: Dim #Images of (f,g) -1 (2, 2), (2, 3), (2, 4), (3, 3), (3, 4), (4, 4) 0 (2, 3), (3, 3), (3, 4), (4, 4) 1 (2, 2), (3, 3) Two cubics acting on six points: Dim #Images of (f,g) -1 (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (3, 3), (3, 4), (3, 5), (3, 6), (4, 4), (4, 5), (4, 6), (5, 5), (5, 6), (6, 6) 0 (2, 3), (2, 4), (2, 5), (2, 6), (3, 3), (3, 4), (3, 5), (3, 6), (4, 4), (4, 5), (4, 6), (5, 5), (5, 6), (6, 6) 1 (2, 6), (3, 3), (3, 4), (3, 5), (3, 6), (4, 4), (4, 5), (4, 6), (6, 6) 2 (2, 2) Talia Blum, Colby Kelln, Henry Talbott 10 / 20 February 1, 2020 10 / 20

Classification of two-image portraits Theorem Given a portrait of degree d on 2 d points, if each polynomial has two images, then the realization space has dimension d − 1 or is empty. Talia Blum, Colby Kelln, Henry Talbott 11 / 20 February 1, 2020 11 / 20

More data Two quadratics acting on four points Dimension #(Portraits) -1 206 0 560 1 14 Two quadratics acting on five points Dimension #(Portraits) -1 16590 0 246 1 3 Talia Blum, Colby Kelln, Henry Talbott 12 / 20 February 1, 2020 12 / 20

1-dimensional: 2 quadratics on 5 points Talia Blum, Colby Kelln, Henry Talbott 13 / 20 February 1, 2020 13 / 20

1-dimensional: 2 quadratics on 5 points Talia Blum, Colby Kelln, Henry Talbott 14 / 20 February 1, 2020 14 / 20

Adding points that preserve dimension Talia Blum, Colby Kelln, Henry Talbott 15 / 20 February 1, 2020 15 / 20

Sufficient condition for building large portraits Talia Blum, Colby Kelln, Henry Talbott 16 / 20 February 1, 2020 16 / 20

Large portraits of maximal dimension Theorem Let f ∈ C ( x ) , and let S be a set such that f ( S ) ⊂ S and for y ∈ f ( S ) , f − 1 ( y ) ⊂ S. If there exists a degree 1 rational function ℓ ( x ) such that f ◦ ℓ = f , then ( ℓ ◦ f )( S ) ⊆ S. Talia Blum, Colby Kelln, Henry Talbott 17 / 20 February 1, 2020 17 / 20

Many functions 28 quadratics acting on four points! Talia Blum, Colby Kelln, Henry Talbott 18 / 20 February 1, 2020 18 / 20

Future work If the realization space is... finite: Derive a sharp bound for #(realizations) Examine which number fields realizations belong to positive-dimensional: Assess geometric properties of realization space empty: Find combinatorial properties of portraits that guarantee empty realization space Talia Blum, Colby Kelln, Henry Talbott 19 / 20 February 1, 2020 19 / 20

Acknowledgements Mentors: Trevor Hyde, John Doyle, Max Weinreich Summer@ICERM organizers: John Doyle, Ben Hutz, Bianca Thompson, Adam Towsley ICERM NSF 2 , NSA NCUWM Organizers 2 Grant No. DMS-1439786 Talia Blum, Colby Kelln, Henry Talbott 20 / 20 February 1, 2020 20 / 20