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 x0 under f is Of (x0) = {x0, f (x0), f (f (x0)), · · · }. f (x) = x2

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 x0 under f is Of (x0) = {x0, f (x0), f (f (x0)), · · · }. f (x) = x2 Question (Central question) Let f , g ∈ C[x]. Are there points x0 with finite multi-orbits, Of ,g(x0) = {x0, f (x0), g(x0), f (g(x0)), · · · }?

Talia Blum, Colby Kelln, Henry Talbott 2 / 20 February 1, 2020 2 / 20

Beyond classical dynamics

Classical dynamics f (x) = x2 Generalized dynamics f (x) = x2 g(x) = x2 − 2

Talia Blum, Colby Kelln, Henry Talbott 3 / 20 February 1, 2020 3 / 20

Realization spaces of portraits

Portrait f (x0) = x2 . . . f (x3) = x3 such that f has degree 2 Realization space (x0, x1, x2, x3) = (0, 1, t, t2 − t + 1) f (x) =

1 t2−t x2 − 1 t2−t x

Talia Blum, Colby Kelln, Henry Talbott 4 / 20 February 1, 2020 4 / 20

Realization spaces of portraits

Portrait f (x0) = x2 g(x0) = x0 . . . . . . f (x3) = x3 g(x3) = x1 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

206 560 1 14 Two cubics acting on six points Dimension #(Portraits)1

• 1

52,238 1,251,585 1 1,009 2 16

1for 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) (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) (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 2d 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

206 560 1 14 Two quadratics acting on five points Dimension #(Portraits)

• 1

16590 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 NSF2, NSA NCUWM Organizers

2Grant No. DMS-1439786 Talia Blum, Colby Kelln, Henry Talbott 20 / 20 February 1, 2020 20 / 20