Fractal geometry and dynamics
| Where | University of Jyväskylä |
|---|---|
| When | Spring 2026, Period 3 |
| Language | English |
| Updated | January 28, 2026 |
| Contact | alex@rutar.org |
Here is a link to the lecture notes, last updated on Jan 28, 2026.
Here is some supplementary material:
Exercise sheets
- Exercise 1 and Solutions
- Exercise 2 and Solutions
- Exercise 3 due 12:15pm Thursday January 29
- Exercise 4 due 12:15pm Thursday February 5
Final project information
The final project will be due on Sunday, February 22 at 23:59. Here is the current list of projects. I will add more in the next few days if I think of new ones. The pre-requisite material for these projects should mostly be covered by the midpoint of the course.
- Dimensions of self-affine carpets
- Separation conditions for self-similar sets
- Inhomogeneous self-similar sets
- Assouad dimension and weak tangents (reserved for Mikko Liimatainen)
- Multifractal analysis for self-similar measures
- Topological dynamical systems
For detail on the projects, as well as information about the evaluation, see the information sheet.
Important course information
Additional course material
- Fractals in probability and analysis by Chris Bishop and Yuval Peres.
- Techniques in Fractal Geometry by Kenneth Falconer.
- Self-similar and self-affine sets and measures by Balázs Bárány, Károly Simon, and Boris Solomyak. Email me if you want a PDF copy.
- Mike Hochman’s ergodic theory notes.
- Marcelo Viana’s notes on Rokhlin disintegration.
- Gerald Folland’s real analysis textbook.
Evaluation
This course is scored out of 100, but ultimately the evaluation is pass-fail.
To pass, you require at least 60 points, with a minimum of 25 points on the final project. The finer breakdown is as follows:
- written exercise solutions (40 points; your scores on the assignments will be normalized if the sum is not 40)
- presentation of the exercises during exercise classes (10 points; 2 per exercise class with a free pass if you miss one)
- an independent final project with a written (5-10 pages, 25 points) and oral component (45-60 minutes, 25 points).
There will be around 5 available bonus points in the exercises which apply to your overall grade.
Schedule
- Lectures take place from 12:15 to 14:00 on Thursdays and Fridays, starting on January 8 and ending on February 20.
- Exercise classes take place from 14:15 to 16:00 on Thursdays, starting on January 16 and ending on February 20.
- There will be 1 exercise sheet due per week, on January 15, January 22, January 29, Feburary 5, February 12, and February 19.
- There will be a final project, due on February 22, with presentations between February 22 and February 27.
- There will be no exam and the course will end on February 27.
Course details
Description
The goal of this course is to provide an introduction to fractal geometry from the perspective of dynamical systems theory. A particular emphasis is given on objects with some form of invariance under re-scaling, such as self-similarity.
A key intermediate goal is to introduce dynamical systems theory, which (for us) is a useful frame of reference for understanding invariance. This is a ubiquitous tool in analysis, PDEs, geometry, mathematical physics, engineering, …
Some topics which will be covered:
- Iterated function systems
- Dimension of sets and measures
- Measure preserving systems
- Ergodic theorems
- Conditional measures, Rokhlin disintegration
- Exact dimensionality of self-similar measures
Related courses:
Pre-requisites
The following pre-requisites are essential.
- Measure and integration 1 & 2
- Introductory probability theory