| Instructor: | Semyon Dyatlov |
| Class hours: | TR 1–2:30 in 2-142 |
| Office hours: | Tue 11–noon in 2-377 and by appointment on Zoom |
| Grading: | Based on problem sets due once every 2 weeks. You may use TeX or you may write your solutions by hand; in the latter case please make sure they are easy to read! You may consult anyone and anything but you need to absorb and write out the solutions yourself. Anything close to copying is not permitted. Your scores will be posted on Canvas and the papers will be returned in class or during office hours. You might find Pset Partners useful for forming study groups. |
| Description: |
This is an introduction to the mathematical theory of chaotic dynamical systems.
A basic example of a question it answers is the following: why do billiard trajectories inside a disk or a rectangle stay neatly together even after very long time, but inside a more complicated domain they spread out very fast? (For some motivating movies, click here.) We won't actually be able to handle billiards in this course (boundaries are too tough) and will instead focus on the boundaryless case, where the fundamental examples are geodesic flows on negatively curved manifolds. Here are some topics I am planning to cover:
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| Prerequisites: | Basic knowledge of analysis (18.100) is required. Manifolds (18.101) will be needed too – not an official prerequisite but you will have to know their basics once we get to geodesic flows. It also would be helpful to be familiar with functional analysis (18.102), Lebesgue integration (18.102 or 18.125), and a bit of Riemannian geometry (the beginning of 18.965) but I will review some of these briefly in the course. Distributions (18.155) might be useful towards the end of the course if I decide to cover e.g. the Atiyah–Bott–Guillemin trace formula, but this decision will be made based on the participants' interests. |
| Materials: | No official textbook but here are some sources I will use in my preparation:
[KH] Katok–Hasselblatt: a monumental volume on dynamical systems |
| Tue | May 10 |
Selberg zeta function and its holomorphic continuation via determinants and transfer operators.
Prime geodesic theorem and its relation to the zeta function.
More recent results on zeta functions. (No proofs.)
Lecture notes §14 References: [Bor, §§15.3,14.5–14.6] |
| Thu | May 5 | Transfer operators. Ruelle–Perron–Frobenius Theorem. Existence of the Patterson–Sullivan measure. The various meanings of δ. (See §13.) |
| Tue | May 3 |
Geodesic flow, the trapped set and its relation to the limit set. (See §12.)
Patterson–Sullivan measure.
Lecture notes §13 Slides from a minicourse given at EPFL with some useful pictures References: [Bor, §§14.1–14.3,15.2], [Bow, §§1.A–1.B] |
| Thu | Apr 28 |
Schottky representation. Limit set. (See §12.)
Pictures used in lecture: a 3-funnel surface 1, 2, 3, 4, a hyperbolic cylinder 1, 2 Problem set 6, due May 10 |
| Tue | Apr 26 |
Convex co-compact hyperbolic surfaces: definition, examples.
Lecture notes §12 Reference: [Bor, §§2.1–2.4, 15.1] |
| Thu | Apr 21 |
Variational principle for entropy: end of the proof.
Symbolic dynamics and the Markov partition for a hyperbolic toral automorphism.
(See §11.) Figures used in the lecture: 1, 2, 3, 4 Reference: [BS, §5.12] |
| Tue | Apr 19 |
Variational principle for entropy.
Lecture notes §11 Reference: [KH, §4.5] |
| Thu | Apr 14 | Example of measure-theoretic entropy: hyperbolic toral automorphisms. Generating partitions; a generating partition is enough to compute the entropy of a map. (See §10.) |
| Tue | Apr 12 |
Conditional entropy of partitions. Refinements and joint partitions. Measure-theoretic entropy of maps. Examples: irrational shift on the circle,
expanding map on the circle. (See §10.) The figures used in lecture Problem set 5, due Thu Apr 21 |
| Thu | Apr 7 |
Examples of topological entropy: hyperbolic toral automorphisms, geodesic flows on hyperbolic surfaces.
Entropy and closed orbits (no proof). (See §9.)
Entropy of partitions.
Lecture notes §10 Reference: [KH, §§4.3–4.4] |
| Tue | Apr 5 |
Topological entropy. Examples: irrational shift on the circle, expanding map on the circle. (See §9.)
Reference: [KH, §§3.1.b,3.2] |
| Thu | Mar 31 |
An even more general
case: Stable/Unstable Manifold Theorem for hyperbolic sets. (See §8.)
Topological entropy
Lecture notes §9 Problem set 4, due Thu Apr 7 |
| Tue | Mar 29 | More properties of stable/unstable manifolds for the model case. Upgrading the model case to higher dimensions and reducing the general case to the model case. (See §8.) |
| Thu | Mar 17 | Stable/Unstable Manifold Theorem for a model case of a hyperbolic fixed point. (See §8.) |
| Tue | Mar 15 |
Hyperbolicity of the geodesic flow on compact negatively curved surfaces: construction of stable/unstable cones. (See §7.)
Stable/Unstable Manifold Theorem for hyperbolic fixed points: overview and motivation. Lecture notes §8 Reference: [D, §§2–4] |
| Thu | Mar 10 |
Characterization of hyperbolicity via cones (continued).
Hyperbolicity of the geodesic flow on compact negatively curved surfaces: Jacobi's equations. (See §7.)
Problem set 3, due Thu Mar 17 |
| Tue | Mar 8 |
General hyperbolic maps: continuity of the stable/unstable spaces,
adapted metrics. Characterization of hyperbolicity via cones.
Stability of Anosov maps and flows under perturbations.
Lecture notes §7 References: [D, §§4.2–4.3,5.1], [KH, §6.2.c, Steps 1–2] |
| Thu | Mar 3 |
Mixing of the geodesic flow with respect to the Liouville measure via
Hopf's argument. (See §6.) Reference: [C, Chapter 4] |
| Tue | Mar 1 | Canonical vector fields V, W, V⟂, U+, U- on the unit tangent bundle of a hyperbolic surface. Hyperbolicity of the geodesic flow on a compact hyperbolic surface. Unique ergodicity of horocycle flows (without proof). (See §6.) |
| Thu | Feb 24 |
Some important results about geodesic flows, Anosov maps, and Anosov flows. (See §5.)
Hyperbolic plane: metric, geodesics, the isometry group PSL(2,R).
Hyperbolic surfaces: Teichmüller theory (gluing from pairs of pants),
representation as quotients of the hyperbolic plane. Lecture notes §6 Problem set 2, due Thu Mar 3 |
| Thu | Feb 17 |
Contact forms, Reeb vector field, Liouville measure. Geodesic flows. Geodesic flows as contact flows. Lecture notes §5 |
| Tue | Feb 15 |
Hyperbolic maps. Examples: hyperbolic toral automorphisms, hyperbolic periodic trajectories.
Billiard ball maps for domains in R2 and which period 2 trajectories
are hyperbolic. Hyperbolic flows. Suspensions of hyperbolic maps. Lecture notes §4 References: [D, §§4.1(beginning),4.6(beginning),5.2] |
| Thu | Feb 10 |
Mixing. Examples of mixing and non-mixing systems: irrational shift
on the circle, expanding map on the circle, hyperbolic toral automorphism.
Poincaré Recurrence Theorem. Lecture notes §3 References: [KH, §§4.2.d,4.2.e,4.1.f], [C, Chapter 3] |
| Tue | Feb 8 |
Example: an expanding map on the circle.
The Birkhoff (almost everywhere) ergodic theorem.
(See §2.)
References: [KH, §§4.1.c,4.2.c], [C, Chapter 2] Problem set 1, due Thu Feb 17 |
| Thu | Feb 3 |
Invariant measures
and the Krylov–Bogolyubov Theorem on their existence. Unique ergodicity.
(See §1.) The von Neumann (L2) ergodic theorem.
Equivalent definitions of ergodicity of a measure.
Lecture notes §2 References: [KH, §§4.1.b], [C, Chapter 1] |
| Tue | Feb 1 |
Introduction: map iterations and flows. Example: unique ergodicity of irrational shift. Review of Lebesgue integral (very brief). Weak convergence of probability measures and compactness theorem for them.
Lecture notes §1 References: [KH, §§4.1.a,4.2.a] |
