Seminar in Complex Analytic Dynamics

MATH 392-02

* * * Midterm Exam * * *

Wednesday, Feb. 29, 7:00 - 8:30 pm, Swords 330

The Midterm Exam covers all the material we have discussed in class from the first day up to and including Friday, Feb. 24. This includes Ch. 15 of Devaney's text as well as the first two sections of the Blanchard paper. You should be sure to review homework assignments #1, 2 and 3, and your class notes.

We will review for the exam in class on Monday, Feb. 27. Please come prepared with specific questions. Some sample practice problems are available here.

Note: No calculators are allowed on the exam so be prepared to answer questions without your personal calculator.

The following topics, definitions and theorems are important material for the exam. You may be asked to define some terms precisely as well as state and/or prove important theorems.

1. Dynamical Systems Theory: iteration, nth iterate, types of orbits (fixed point, periodic point, eventually fixed or periodic, asymptotic orbit, dense orbit), classification of fixed or periodic points (attracting, repelling, neutral, super-attracting), chain rule along a period n-cycle, Attracting Fixed Point Theorem, Repelling Fixed Point Theorem (and their generalizations to periodic cycles), basin of attraction

2. Analytic Conjugacy: definition of when f and g are analytically conjugate, homeomorphism, commutative diagram, the Conjugate Fixed Point Theorem (CFPT), showing two functions are conjugate or finding a conjugacy between two functions, using g(z) = 1/f(1/z) to study dynamics near infinity, dynamical properties preserved under conjugacy

3. Dynamics of Linear Functions: L_alpha(z) = alpha z, nth iterate of L_alpha, general dynamical behavior for different values of alpha, rational versus irrational rotation, dense orbits

4. Chaotic Dynamical Systems: definition of, topological transitivity, sensitive dependence on initial conditions (SDIC), butterfly effect, the Australian's theorem (Banks, et. al.), doubling map is chaotic, chaos is preserved under conjugacy (assuming f and g are continuous)

5. The Fatou and Julia Sets: definition of, degree of a rational function, normal families, pointwise versus uniform convergence, a sequence of analytic functions that converges uniformly will converge to an analytic function, F(z^d) and J(z^d), properties of the Fatou set (open, backwards and forwards invariant, contains the attractors and their basins of attraction), properties of the Julia set (closed, backwards and forwards invariant, contains the repellors, contains the ``interesting'' chaotic dynamics, nonempty if degree > 1), equicontinuous family, Arzela-Ascoli theorem

6. Important Concepts from Complex Analysis: polar form, modulus of a complex number and its properties, complex roots, analytic functions, limits, continuity, definition of the derivative, Riemann sphere, the LIPI theorem, poles, essential singularities, Picard's theorem, meromorphic functions, the Cauchy-Goursat theorem, convergent sequence