Waves in complex structures

Waves provide a useful tool to understand the fine properties of various structures such as manifolds, fractals or graphs. In our recent work we have been studying waves in complex environments and relate the geometric structure to the properties of the wave.

Our group has been working on attractors to chaotic dynamical systems and describe the oscillations of the waves arising from the natural invariant measures associated to the attractors and how they relate to the chaotic dynamics. This problem is particularly important in Quantum Chaos (Bourgain-Dyatlov, 2017).

We have also been involved in a project led by Etienne Le Masson (University of Bristol), which relates waves on large graphs to waves on manifolds. Here we used ideas of waves on graphs to provide new insight in Quantum Chaos on manifolds (Le Masson-Sahlsten, 2017). Waves on large graphs are particularly important in applications such as social networks. For example, in community detection it can provide more effective tool than random walks (Sahai-Speranzon-Banaszuk, 2012).

**Output**

- Thomas Jordan and Tuomas Sahlsten:
*Fourier transforms of Gibbs measures for the Gauss map*, Math. Ann. (2016) Vol 364 (3): 983-1023 - Tuomas Sahlsten and Connor Stevens:
*Fourier decay in nonlinear dynamics*, in preparation - Etienne Le Masson and Tuomas Sahlsten:
*Quantum ergodicity and Benjamini-Schramm convergence of hyperbolic surfaces*, Duke Math. J. 166 (2017), no. 18, 3425-3460. - Etienne Le Masson and Tuomas Sahlsten:
*Equidistribution of Maass cusp forms in the level aspect*, in preparation

Perception and Dynamics of Structural Causal Models

Output:

- Vadim Kulikov and Tuomas Sahlsten:
*Dynamics of Causality*, in preparation

Methods in nonsmooth dynamics

Typical methods in entropy and dimension theory of chaotic dynamical systems involve smoothness assumptions such as \(C^1\). This can be quite restrictive for e.g. uniformly quasiregular dynamics, which provide a natural higher dimensional analogue of complex dynamics. Here the dynamical systems may not even be differentiable and only weak Sobolev regularity exists. Still questions like entropy and dimension remain as natural problems to study in these systems. The community working around quasiregular maps (e.g. Rickman 1993) have developed wide variety of methods and tools, which we can use to approach problems in dynamics and recent works of Kangasniemi, Okuyama and Pankka (ref 1, 2, 3) have taken major steps in particular towards ergodic theory of invariant measures for uniformly quasiregular dynamics using cohomology theory. In our project we attempt to approach in particular entropy and dimension theory for uniformly quasiregular maps.

Output:

- Ilmari Kangasniemi, Yusuke Okuyama, Pekka Pankka and Tuomas Sahlsten:
*Entropy of uniformly quasiregular mappings*, in preparation