Asymptotic counting in dynamical systems
Alan Yan 1
Mentor: Sergiy Merenkov 2
1West-Windsor Plainsboro High School North
2CCNY-CUNY
April 19-20, 2018MIT PRIMES Conference
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Introduction
Question
Given a counting function which quantifies some measurable property of ageometric object, what are the asymptotics of such a function?
Expectation: ∼ cxd , d is “dimension”
Two Main Examples
Fatou Components of Rational MapsLimit Sets of Schottky Groups
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Dimension
Definition (Box-Counting Dimension)
Suppose that N(ε) is the number of boxes of side length ε required tocover a set S . Then the dimension of S is defined as
limε→0
logN(ε)
log(1/ε).
unique d such that N(1/x) ∼ cxd as x →∞.
motivation for power law
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Dimension Example: East Coast of Britain
Figure: East Coast of Britain, d ≈ 1.21
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Dimension Example: Cantor Set
Figure: Cantor Set, d = log3 2
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Julia Set and Fatou Components
Definition
The filled-in Julia set of a complex function f is defined as
K (f ) = {z ∈ C : f k(z) 9∞}.
Definition
The Julia Set of f is defined as the boundary of the filled-in Julia set, i.e.
J(F ) = ∂K (F ).
Definition
A Fatou component of f is a connected component of K (f ).
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Examples: Julia set of f (z) = z2 + c
Figure: c = −0.74543 + 0.11301i
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Examples: Julia set of f (z) = z2 + c
Figure: The Basilica (c = −1)
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Main Theorem on Fatou Components
Definition
Given a set X ⊂ C, define the diameter of X as
diam(X ) = sup{|x − y | : x , y ∈ X}.
The Counting Function
For every function f : C→ C, we associate a counting functionNf : R>0 → R≥0 where Nf (x) is the number of Fatou components of fwhose diameter is at least 1/x .
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Main Result on Fatou Components Cont.
Conjecture
Suppose f (z) = z2 + c has an infinite number of Fatou components. Then,
Nf (x) ∼ cf xd
where d is the dimension of J(f ) and cf > 0 is a constant.
Numerically verified the theorem for f (z) = z2 − 1.
Proved in paper by M. Pollicott, M. Urbanski
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Algorithm: Diameters of the Basilica
Escaping Criterion
Let f (z) = z2 + c be a complex quadratic function. Let R =1+√
1+4|c|2 . If
for some n > 0 we have |f n(z0)| > R, then z0 6∈ K (f ).
1 Construction
2 Distinguish Components
3 Computation of Diameter
4 Problem with (2): Bridges
5
√A Counting Function
Figure: Bridge
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Hyperbolic Geometry: The Poincare Disk Model
Definition
The Poincare disk is the unit disk D equipped with new notions of lines,distance, and angles.
Geodesics are orthogonal circles!
To calculate distance, we canuse the formula
d(A,B) = ln|AQ| · |BP||AP| · |BQ|
Figure: Geodesics
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Angles in D
Figure: Angle between Geodesics
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Isometries and Schottky Groups
Definition
An isometry is a map that preserves distances.
Isometry Groups
G =
{h(z) =
αz + β
βz + α: |α|2 − |β|2 = 1, α, β ∈ D
}.
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Isometries and Schottky Groups (Contd)
Definition
If g is an isometry which does fix z0, then Dz0(g) = D(g) represents theclosed half-plane in D bounded by the perpendicular bisector of thehyperbolic segment [z0, g(z0)] containing g(z0).
Definition
A Schottky group of rank 2 is a subgroup of G generated by twoisometries g1, g2 that satisfies
(D(g1) ∪ D(g−11 )) ∩ (D(g2) ∪ D(g−12 ) = ∅
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Example: Schottky Groups
Figure: Induced Half Planes of the Generators
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The Limit Set of a Schottky Group
Definition
The limit set of L(Γ) of a isometry group Γ is defined by
L(Γ) = Γz ∩ ∂D
Figure: L(S(g1, g2))
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Conjecture on the Limit Sets S(g1, g2)
Definition (Counting Function)
For a Schottky group S(g1, g2), define N(x , p) : R>0 × D→ R≥0 to bethe number of intervals I ∈ ∂D\L(S(g1, g2)) such that the angle betweenthe geodesics from the endpoints of I to p is at least 1/x .
Conjecture
For every point p ∈ D and nontrivial Schottky group S(g1, g2),
N(x , p) ∼ cxd
where d is the dimension of L(S(g1, g2)). The constant c depends on thepoint p.
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Future Research
Fatou Components
Generalize the theorem for different types of rational mapsProve a similar theorem replacing diameters by
√A.
Limits Sets of Schottky Groups
Efficient Algorithm for limit sets of Schottky groupsverify and prove the conjecture on Schottky groupsRelationship between c and p
Discover new counting functions which are asymptotic to cxd .
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Acknowledgements
MIT-PRIMES Program
Prof. Sergiy Merenkov
Dr. Tanya Khovanova
My parents
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References
1 Dal’bo-Milonet, F. (2011). Geodesic and horocyclic trajectories.London: Springer.
2 M. Pollicott, M. Urbanski, Asymptotic Counting in ConformalDynamical Systems (2017)
3 A. Kontorovich, H. Oh, Apollonian Circle Packings and ClosedHorospheres on Hyperbolic 3-Manifolds
4 Carleson, L., Gamelin, T. (1993). Complex dynamics. New York:Springer-Verlag.
5 Falconer, K. (2003). Fractal geometry : mathematical foundationsand applications. Chichester: Wiley.
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Image Sources
1 Alexis Monnerot-Dumaine / CC-BY-SA-3.0 (1)
2 Adam Majewski / CC-BY-SA-3.0 (3)
3 Dal’bo-Milonet, F. (2011). Geodesic and horocyclic trajectories.London: Springer. (8, 9)
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