Pasting polynomials together Sarah C. Koch University of - PowerPoint PPT Presentation

Sep 05, 2022 •107 likes •437 views

Pasting polynomials together Sarah C. Koch University of Michigan The Basilica San Marco Cathedral Venice, Italy The Rabbit p ( z ) = z 2 +( 0 . 1226+0 . 7449 i ) The Corabbit p ( z ) = z 2 +( 0 . 1226 0 . 7449 i ) A

Pasting polynomials together Sarah C. Koch University of Michigan
The Basilica San Marco Cathedral Venice, Italy
The Rabbit − p ( z ) = z 2 +( − 0 . 1226+0 . 7449 i )
The Corabbit − p ( z ) = z 2 +( − 0 . 1226 − 0 . 7449 i )
A dendrite − p ( z ) = z 2 + i
Kokopelli p ( z ) = z 2 � 0 . 156 + 1 . 302 i
Cokokopelli p ( z ) = z 2 � 0 . 156 � 1 . 302 i
p ( z ) = z 2 + (0 . 5 + i ) A Cantor set
Keeping track of shapes z 2 z 2 − 1 basilica − z 2 + ( − 0 . 1 + 0 . 75 i ) rabbit − z 2 + ( − 0 . 1 + 0 . 75 i ) corabbit − z 2 − 0 . 156 + 1 . 302 i kokopelli − z 2 + i dendrite
Parameter space: coloring scheme? c plane
z 2 z 2 − 1 basilica − z 2 + ( − 0 . 1 + 0 . 75 i ) rabbit − z 2 + ( − 0 . 1 + 0 . 75 i ) corabbit − z 2 − 0 . 156 + 1 . 302 i kokopelli − z 2 + i dendrite The Mandelbrot Set
the basilica the rabbit
The mating of the basilica and the rabbit √ F ( z ) = 2 z 2 + 1 − 3 2 z 2 − 2 C ⇥ n
hmmm... let’s see that again.
Which quadratic polynomials can be mated?
Theorem. (Tan Lei, Rees, Shishikura) Let p : z 7! z 2 + c 1 and q : z 7! z 2 + c 2 be postcritically finite. Then p and q can be mated if and only if c 1 and c 2 do not belong to conjugate limbs of the Mandelbrot Set.
A shared mating
Arnaud Cheritat polynomial matings: https://www.math.univ- toulouse.fr/~cheritat/MatMovies/
Fractal Stream Software: Dynamics Explorer Mandel Books: An introduction to chaotic dynamical Dynamics in one complex variable systems by John Milnor by Robert Devaney complex analysis, topology, Classes: differential geometry, algebraic topology
Thank you!

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