## Syllabus

### Two lectures per week, one homework

For bird's eye overview, read the summary of part II of ChaosBook.#### Week 9

- Chapter 17 - Walkabout: Transition graphs
- Topological dynamics encoded by means of transition matrices/Markov graphs. Read all of it.
- Chapter 18 - Counting
- You learn here how to count distinct orbits, and in the process touch upon all the main themes of this book, going the whole distance from diagnosing chaotic dynamics to - while computing the topological entropy from transition matrices/Markov graphs - our first zeta function. The long time dynamics of a discrete-time system is described by (1) generating all orbits by action of the 1-time step transition matrix, and (2) expressing the result as a "generating function". Here we derive the trace formula, the determinant and the topological zeta function. If you do not understand how to derive these, you'll be lost for the rest of the semester, and what fun is that? Read sects. 18.1 - 18.4; 18.6 - 18.7, ChaosBook vers. approx 15.3.
- Homework

#### Week 10

- Chapter 19 - Transporting densities
- A first attempt to move the whole phase space around - natural measure and fancy operators. Skip sects. 19.3 and 19.6.
- Chapter 20 - Averaging
- On the necessity of studying the averages of observables in chaotic dynamics. Formulas for averages are cast in a multiplicative form that motivates the introduction of evolution operators. Discrete-time and generating functions continued and compared. Infinitesimal time evolution related to the infinite time dynamics via a Laplace transform. The stage is set for the classical trace formula. Read sects. 20.1 and 20.2. Skip sect. 20.1.3 "Moments, cumulants"
- Homework

#### Week 11

- Chapter 21 - Trace formulas
- If there is one idea that one should learn about chaotic dynamics, it happens in this chapter: the (global) spectrum of the evolution is dual to the (local) spectrum of periodic orbits. The duality is made precise by means of trace formulas. The course is OVER. Trace formulae are beautiful, and there is nothing more to say. Just some mopping up to do.
- Chapter 22 - Spectral determinants
- We derive the spectral determinants, dynamical zeta functions. While traces and determinants are formally equivalent, determinants are the tool of choice when it comes to computing spectra. Skip sects. 20.5 and 20.6.
- Homework

#### Week 12

- Chapter 23 - Cycle expansions
- Spectral eigenvalues and dynamical averages are computed by expanding spectral determinants into cycle expansions, expansions ordered by the topological lengths of periodic orbits. Skip sects. 23.3.1 "Newton algorithm for determining the evolution operator eigenvalues", and 23.6 "Stability ordering of cycle expansions".
- Chapter 20 - Averaging
- Read sect. 20.1, including sect. 20.1.3 "Moments, cumulants"
- Chapter 24 - Deterministic diffusion
- We derive exact formulas for diffusion constants transport coefficients. All from first principles, without invoking any Boltzmann-Gibbs probabilistic notions. Read sect. 28.1: Foundations of statistical mechanics illuminated by 2-dimensional Lorentz gas.
- Homework

#### Week 13

- Chapter 24 - Deterministic diffusion
- Read sect. 28.2: A class of simple 1-dimensional dynamical systems where all transport coefficients can be evaluated analytically, by hand. Diffusion is a non-monotonic function of the local expansion rate, and it is non-gaussian, with a non-vanishing kurtosis. Perhaps the most fundamental diagnostic of deterministic chaos is the non-differentiable dependence of its transport coefficients on smooth variations of system parameters.
- Chapter xx - Finite groups
- Self-study week part I: Finite groups. Cyclic groups of two and three elements. Symmetries of a triangle, six element group multiplication table. Matrix representations. Regular representations. This is standard material, not written up in ChaosBook, but necessary for the course. We liked Tinkham, chapter 2, or Dresselhaus chapters 1 and 2 (early version available as lecture notes).
- Homework

#### Week 14

- Chapter xx- Irreducible representations
- Self-study week part II: Characters. Orthogornality relations. Character tables. If you have understood character projection operators, we are set. We liked Tinkham, chapter 3, or Dresselhaus chapters 3 and 4 (early version available as lecture notes).
- Chapter 25 - Discrete symmetry factorization
- Symmetries simplify and improve the cycle expansions in a rather beautiful, not entirely obvious way, by factorizing cycle expansions. Read sects. 25.1 and 25.2. For a one-d map with reflection symmetry determinants factorize into symmetric and and antisymmetric ones, and each one receives contributions from all kinds of orbits. In a not entirely obvious way. A triple home run: simpler symbolic dynamics, fewer cycles needed, much better convergence of cycle expansions. Once you master this, going back is unthinkable.
- Homework

#### Week 15

- Chapter 25 - Discrete symmetry factorization
- 3 disk pinball symmetries suffice to illustrate all that is needed to factorized spectral determinants for any system with a discrete symmetry: character. Discrete symmetry tiles the state space, and dynamics can be reduced to dynamics on the fundamental domain, together with a finite matrix that keeps track of the tile the full state space trajectory lands on. We need some group theory (one needs to underatand the projection to irreducible representations) and illustrate how different classes of periodic orbits contribute to different invariant subspaces for the 3-disk pinball. Read sects. 25.2 - 25.6.
- Chapter 26 - Continuous symmetry factorization
- Trace formulas relate short time dynamics (unstable period ic orbits) to long time invariant state space densities (natural measure). A trace formula for a partially hyperbolic (N+ 1)-dimensional compact manifold invariant under a global continuous symmetry is derived. In this extension of “periodic orbit” theory there are no or very few periodic orbits - the relative periodic orbits that the trace formula has support on are almost never eventually periodic
- Homework

#### Week 16

- Chapter 30 - “Turbulence” - a one spatial dimension warmup
- Flows described by PDEs are said to be `infinite dimensional' because if one writes them down as a set of ODEs, one needs infinitely many of them to represent the dynamics of one PDE. The long-time dynamics of many such systems of physical interest is finite-dimensional. Here we cure you of the fear of infinite-dimensional flows.
- Tutorial - Turbulence? A stroll through 61,506 dimensions
- In the world of everyday, moderately turbulent fluids flowing across planes and down pipes, a velvet revolution is taking place. Experiments are as detailed as simulations, there is a zoo of exact numerical solutions that one dared not dream about a decade ago, and portraits of turbulent fluid's state space geometry are unexpectedly elegant. We take you on a tour of this newly breached, hitherto inaccessible territory. Mastery of fluid mechanics is no prerequisite, and perhaps a hindrance: the tutorial is aimed at anyone who had ever wondered how we know a cloud when we see one, if no cloud is ever seen twice? And how do we turn that into mathematics?
- Homework