Tyll Krüger

Local instability of orbits in polygonal and polyhedral billiards

We classify when local instability of orbits of closeby points can occur for billiards in two dimensional polygons, for billiards inside three dimensional polyhedra and for geodesic flows on surfaces of three dimensional polyhedra. We sharpen a theorem of Boldrighini, Keane and Marchetti. We show that polygonal and polyhedral billiards have zero topological entropy. We also prove that billiards in polygons are positive expansive when restricted to the set of non-periodic points. The methods used are elementary geometry and symbolic dynamics.

Periodic billiard orbits are dense in rational polygons

We show that periodic orbits are dense in the phase space for billiards in polygons for which the angle between each pair of sides is a rational multiple of 7r.

The “Cameo Principle” and the Origin of Scale-Free Graphs in Social Networks

We formulate a simple edge generation rule based on an inverse like mass action principle for random graphs over a structured vertex set. We show that under very weak assumptions on this structure one obtains a scale free distribution for the degree. We furthermore introduce and study a “my friends are your friends” local search principle which makes the clustering coefficient large.

Acceleration of bouncing balls in external fields

We introduce two models, the Fermi-Ulam model in an external field and a one dimensional system of bouncing balls in an external field above a periodically oscillating plate. For both models we investigate the possibility of unbounded motion. In a special case the two models are equivalent.

Markov partitions and shadowing for non-uniformly hyperbolic systems with singularities

We show the existence of countable Markov partitions for a large class of non-uniformly hyperbolic systems with singularities including dispersing billiards in any dimension.

What Can One Learn About Self-Organized Criticality from Dynamical Systems Theory?

We develop a dynamical system approach for the Zhang model of self-organized criticality, for which the dynamics can be described either in terms of iterated function systems or as a piecewise hyperbolic dynamical system of skew-product type. In this setting we describe the SOC attractor, and discuss its fractal structure. We show how the Lyapunov exponents, the Haussdorf dimensions, and the system size are related to the probability distribution of the avalanche size via the Ledrappier–Young formula.

Lyapunov exponents and transport in the Zhang model of self-organized criticality.

We discuss the role played by Lyapunov exponents in the dynamics of Zhangs model of self-organized criticality. We show that a large part of the spectrum (the slowest modes) is associated with energy transport in the lattice. In particular, we give bounds on the first negative Lyapunov exponent in terms of the energy flux dissipated at the boundaries per unit of time. We then establish an explicit formula for the transport modes that appear as diffusion modes in a landscape where the metric is given by the density of active sites. We use a finite size scaling ansatz for the Lyapunov spectrum, and relate the scaling exponent to the scaling of quantities such as avalanche size, duration, density of active sites, etc.

Topological and symbolic dynamics for hyperbolic systems with holes

We consider an axiom A diffeomorphism or a Markov map of an interval and the invariant set Omega* of orbits which never falls into a fixed hole. We study various aspects of the symbolic representation of Omega* and of its non-wandering set Omega(nw). Our results are on the cardinality of the set of topologically transitive components of Omega(nw) and their structure. We also prove that Omega* is generically a subshift of finite type in several senses.

Self-organized criticality and thermodynamic formalism

We develop a thermodynamic formalism for a dissipative version of the Zhang model of Self-Organized Criticality, where a parameter allows us to tune the local energy dissipation. By constructing a suitable Markov partition we define Gibbs measures (in the sense of Sinai, Ruelle, and Bowen), partition functions, and topological pressure allowing the analysis of probability distributions of avalanches. We discuss the infinite-size limit in this setting. In particular, we show that a Lee–Yang phenomenon occurs in the conservative case. This suggests new connections to classical critical phenomena.

Complexity, randomness, discretization: some remarks on a program of J. Ford

Abstract We discuss certain aspects of the complexity of periodic orbits of chaotic dynamical systems in a symbolic setting. Our theory is based on a generalization of the notion of random sequences in the sense of Kolmogorov and Martin-Lof. Applying our results to the systems studied by Ford et al. leads to completely different conclusions.