MTH 555, Fall 2019

Topics and Assignments

Date Topics Assignments

Sep 4

1.2 Dynamical Systems and the Logistic Family

Sep 9 & 11

1.3 Background Material and Notation

2. First Steps in Complex Iteration
2.1 & 2.2 Orbits and Iteration

1.3 Colin Merrick
2.1 & 2.2 Christine Marcotte

Sep 16 & 18

2.3 Iterating the Logistics Maps

2.4 Periodic Points and their Multipliers

2.3 Erika Fiore

2.4 John Jones

Sep 23 & 25

2.5 The Basin of Infinity of a Polynomial and
2.6 Iterating Transcendental Functions

2.7 Affine Conjugacy

Brian Camara

Connor O'Loughlin

Sep 30 & Oct 2

2.8 Equicontinuity

3. Riemann Sphere
3.1 Stereographic Projection/Chordal Distance and
3.2 Spherical Derivative and Meromorphic Functions

2.8 Araceli
Homework I
3.1 & 3.2 Eric Ferias

Oct 7 & 9

3.3 Moebius Transformations

4. Fatou and Julia sets
4.1 Definitions of the Fatou and Julia sets
The Geometry of Julia Sets

3.3 Jonathan Curtis
Homework II
4.1 Araceli

Oct 15 & 16

4.2 The Filled-in Julia set of a Polynomial and Exceptional Set

4.3 The Boundary of the Filled-in Julia Set

4.2 Christine Marcotte

4.3 Brian Camara
Homework III

Oct 21 & 23

5. Components of the Fatou set/Normal Families
5.1 Normal Families and Compact Convergence

5.2 Components of the Fatou Set: Constant Limit Functions

5.1 Erika Fiore
5.2 Eric Ferias

Oct 28 & 30

5.3 Parabolic Points

5.4 Non-Constant Limit Functions and the Classification of Fatou Components

5.3 John Jones
5.4 Connor O'Loughlin
Homework IV

Nov 4 & 6

6. Further Properties of the Julia sets; The Fundamental Theorem
6.1 Essential Singularities

6.2 The Julia Set of an Entire Function

6.1 Jonathan Curtis

Nov 13

6.3 Montel's Theorem and Properties of the Julia Set

6.4 Connecting Normal Families and the Entire Functions: Zalcman's Lemma

Nov 18 & 20

6.5 Density of Repelling Periodic Points

7. Singular Values and the Mandelbrot Set
7.1 Singular Values

Homework V

Nov 25

7.2 The Influence of Singular Values and
7.3 The Mandelbrot Set

7.4 The Boundary of the Mandelbrot Set and
The Exponential Map is Chaotic: An Invitation to Transcendental Dynamics I

Homework VI

Dec 2 & 4

The Exponential Map is Chaotic: An Invitation to Transcendental Dynamics II

The Exponential Map is Chaotic: An Invitation to Transcendental Dynamics III

Dec 9

Review

Date	Topics	Assignments
Sep 4	1.2 Dynamical Systems and the Logistic Family
Sep 9 & 11	1.3 Background Material and Notation 2. First Steps in Complex Iteration 2.1 & 2.2 Orbits and Iteration	1.3 Colin Merrick 2.1 & 2.2 Christine Marcotte
Sep 16 & 18	2.3 Iterating the Logistics Maps 2.4 Periodic Points and their Multipliers	2.3 Erika Fiore 2.4 John Jones
Sep 23 & 25	2.5 The Basin of Infinity of a Polynomial and 2.6 Iterating Transcendental Functions 2.7 Affine Conjugacy	Brian Camara Connor O'Loughlin
Sep 30 & Oct 2	2.8 Equicontinuity 3. Riemann Sphere 3.1 Stereographic Projection/Chordal Distance and 3.2 Spherical Derivative and Meromorphic Functions	2.8 Araceli Homework I 3.1 & 3.2 Eric Ferias
Oct 7 & 9	3.3 Moebius Transformations 4. Fatou and Julia sets 4.1 Definitions of the Fatou and Julia sets The Geometry of Julia Sets	3.3 Jonathan Curtis Homework II 4.1 Araceli
Oct 15 & 16	4.2 The Filled-in Julia set of a Polynomial and Exceptional Set 4.3 The Boundary of the Filled-in Julia Set	4.2 Christine Marcotte 4.3 Brian Camara Homework III
Oct 21 & 23	5. Components of the Fatou set/Normal Families 5.1 Normal Families and Compact Convergence 5.2 Components of the Fatou Set: Constant Limit Functions	5.1 Erika Fiore 5.2 Eric Ferias
Oct 28 & 30	5.3 Parabolic Points 5.4 Non-Constant Limit Functions and the Classification of Fatou Components	5.3 John Jones 5.4 Connor O'Loughlin Homework IV
Nov 4 & 6	6. Further Properties of the Julia sets; The Fundamental Theorem 6.1 Essential Singularities 6.2 The Julia Set of an Entire Function	6.1 Jonathan Curtis
Nov 13	6.3 Montel's Theorem and Properties of the Julia Set 6.4 Connecting Normal Families and the Entire Functions: Zalcman's Lemma
Nov 18 & 20	6.5 Density of Repelling Periodic Points 7. Singular Values and the Mandelbrot Set 7.1 Singular Values	Homework V
Nov 25	7.2 The Influence of Singular Values and 7.3 The Mandelbrot Set 7.4 The Boundary of the Mandelbrot Set and The Exponential Map is Chaotic: An Invitation to Transcendental Dynamics I	Homework VI
Dec 2 & 4	The Exponential Map is Chaotic: An Invitation to Transcendental Dynamics II The Exponential Map is Chaotic: An Invitation to Transcendental Dynamics III
Dec 9	Review