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Jaerisch Lab
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Research that explores the connection between chaotic dynamical systems and fractals through ergodic theory.
AboutAbout the lab
Exploring the fractals behind chaos.
Prof. Jaerisch studies dynamical systems, centering on ergodic theory and fractal geometry.
The initial values of orbits that behave chaotically often form a fractal set, such as a Julia set.
For conformal dynamical systems, Prof. Jaerisch investigates the relationship between their properties as dynamical systems and their properties as fractals, using thermodynamic formalism as a tool.
In the message to students, the professor says that fractal geometry is a new field, so students can take on new challenges from a fairly early stage.
Key terms
- dynamical system
- A mathematical model describing how points in a space move over time according to a fixed rule
- fractal
- A set with a very irregular and complex structure, such as a Julia set
- ergodic theory
- A field that studies the long-term average behavior of dynamical systems using the ideas of probability (measure)
- Kleinian group
- A group made up of discrete (isolated) transformations of hyperbolic space
ResearchResearch themes
The lab studies conformal dynamical systems from both the ergodic theory side and the fractal geometry side. It is also looking at generalizing thermodynamic formalism, which originates in statistical physics, to a broader class of dynamical systems.
*Illustrative imageConformal Dynamical Systems and Fractal Geometry
The lab focuses on the relationship between the dynamical properties and the fractal-geometric properties of conformal semigroup actions. The main examples are actions of semigroups of rational functions on the Riemann sphere, actions of Kleinian groups on hyperbolic space, and conformal iterated function systems. The lab also studies random dynamical systems and the regularity of fractal functions such as generalized Takagi functions.
*Illustrative imageGeneralizing Thermodynamic Formalism
Thermodynamic formalism, developed in the 1970s by Sinai, Ruelle, Bowen, and others, is a powerful tool for studying the relationship between geometry and dynamical systems. The lab is working on generalizing the theory, which is well established for uniformly hyperbolic dynamical systems, to a broader class of systems with weaker types of hyperbolicity.
MembersFaculty
- Associate ProfessorJohannes Jaerisch
VoicesStudent voices
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Report an errorAbout the images
The top and research theme images are illustrative and were not taken at the lab. They are cropped from Wikimedia Commons images. Cropped CC BY-SA images are provided under the same CC BY-SA license as the original (version shown on each line).
- Top image: “Julia set (highres 01)” Solkoll / Public domain (cropped)
- Conformal Dynamical Systems and Fractal Geometry: “Mandelbrot Set Image 113” Aokoroko / CC0 (cropped)
- Generalizing Thermodynamic Formalism: “Mandel zoom 00 mandelbrot set” Created by Wolfgang Beyer with the program Ultra Fractal 3. / CC BY-SA 3.0 (cropped)
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