1. Julia sets: Julia sets appear as limit sets of repelling periodic points under iterations of complex-valued maps. They can take varied, mesmerizing forms; in the introductory lectures, we saw some examples that were linked to the iteration of $f(z)=z^2+c$ The Julia set has an intimate connection with another fractal called the Mandelbrot set.
    Reference: Chapter 2 “Julia Sets and Their Computergraphical Generation” of “The Beauty of Fractals”. The presentation could include some of the figures at the end of that chapter. Ideally, this presentation would also include some aspects treated by Douady in the same book (“Julia Sets and the Mandelbrot Set”).
  2. Fractals and multifractals: The Cantor set is a particular example of a fractal. A fractal is an object that has a noninteger dimension; for instance, the Cantor set is “too big” to have dimension zero (it is uncountable) but “too small” to have dimension one (it has length 0). Self-similar sets defined by recursive rules are notable examples. A multifractal is a sort of union of fractal sets, possibly of different dimensions. To study them, one has to introduce a generalized notion of dimension, borrowing some ideas from probability theory/statistics.
    Reference: Tél’s “Fractals, Multifractals, and Thermodynamics” discusses the notion of fractal and multifractal (sections I and II). If you’re curious about thermodynamics, the presentation could also cover section III.
  3. The ergodic theorem and measure-theoretic entropy: It was realized early by people like Boltzmann and later Birkhoff that for certain maps on probability spaces, that nowadays we call ergodic, time averages of the form $\frac 1n \sum_{i=0}^n f^k(x_0)$ converge to a “spatial” expected value $\mathbb E(f)$. There are many nonequivalent ways in which one can make mathematical sense of this intuition. Some version of this ergodic theorem can be used then to prove the convergence of the “maximal information given by a sequence of experiments evolving under $f$ per unit of time”. This quantity, known as the entropy of $f$, turns out to be an important conjugacy invariant (two conjugate maps have the same entropy) and provides great insight in dynamical systems.
    Reference: Halmos, “Recent progress in ergodic theory” (very well written). This could be complemented with some additional historical remarks in the first sections of Katok’s “Fifty years of entropy in dynamics”.
  4. Feigenbaum universality: It was realized by Feigenbaum that all unimodal maps (maps with a single maximum/minimum which is locally convex) such as the logistic map or the quadratic maps undergo period doubling bifurcations in a similar way. Specifically, if the family $f_c$ undergoes the $n$-th period doubling at $c_n$, then \(\lim_{n\to \infty} \frac{c_n-c_{n-1}}{c_{n-1}-c_{n-2}} = 4.66920...\)
    a number that is nowadays known as Feigenbaum’s constant.
    Possible reading: preferably Feigenbaum’s original article “Quantitative universality for a class of nonlinear transformations”, although there’s some relevant information in Alligood et al, particularly chapter 12. A more advanced reference, treating the higher dimensional case, is the article “Feigenbaum universality and the thermodynamic formalism” by Sinai, Khanin and Vul.
  5. Control theory: Stabilization of the dynamics around an unstable periodic orbit by making small time-dependent perturbations.
    Reference: “Controlling chaos” by Ott et al; a modification of these ideas was introduced here to control arrhythmias (“Controlling cardiac chaos</a>”).
  6. Cellular automata: A one-dimensional cellular automaton, or “block map”, sends a sequence $x$ to another sequence $y$ in such a way that each symbol $y_i$ of the target sequence only depends on a finite block of n consecutive symbols, this is $y_i = f(x_ix_{i+1}\cdots x_{i+n-1})$, where $n$ is fixed. Cellular automata are systems defined by simple rules that can present a very rich behavior. The purpose of this presentation would be to introduce automata and to present the proof of the Curtis–Hedlund–Lyndon theorem, which characterizes cellular automata as continuous equivariant maps between shifts.
    Reference: The theorem is in the first three sections of Hedlund’s “Endomorphisms and automorphisms of the shift dynamical system”; if time allows, other theorems in that article could be discussed.
  7. Existence of semiconjugacies: **Although we worked with (semi)conjugacies, we did not discuss how to find them, approximate them, or show that they exist. These subjects are treated in Chapter 12 of Banks et al, “Chaos: A Mathematical Introduction”. A more result can be found in “A constructive proof of the existence of a semi-conjugacy for a one-dimensional map” by Ou and Palmer
  8. Lorenz attractor: The main object of this presentation would be Tucker’s theorem: the Lorenz system of equations supports a robust “strange attractor”. The Lorenz system was introduced as a toy model of nonlinear atmospheric dynamics that exhibits sensitivity to initial conditions.
    Reference: the review article “What’s New on Lorenz Strange Attractors?” by Viana
  9. Relations with matrix theory: Some subshifts of “finite type” are defined by a matrix $A$ of possible transitions between a symbol $x_i$ and $x_{i+1}$. One can study the matrix $A$ to get information about the dynamical behavior of the shift.
    Reference: first four sections of Boyle’s “Symbolic dynamics and matrices”.
  10. On the relationship between entropy and chaos: Ref. Falniowski et al., “Two result on entropy, chaos and independence in symbolic dynamics”