Mathematics

We give a description of the link between topological dynamical systems and their dimension groups. The focus is on minimal systems and, in particular, on substitution shifts. We describe in detail the various classes of systems including Sturmian shifts and interval exchange shifts. This is a preliminary version of a book which will be published by Cambridge University Press. Any comments are of course welcome.

This is the first article in a two-part series containing some results on dimension estimates for C 1 iterated function systems and repellers. In this part, we prove that the upper box-counting dimension of the attractor of any C 1 iterated function system (IFS) on R d is bounded above by its singularity dimension, and the upper packing dimension of any ergodic invariant measure associated with this IFS is bounded above by its Lyapunov dimension. Similar results are obtained for the repellers for C 1 expanding maps on Riemannian manifolds.

We establish analogues for trees of results relating the density of a set E⊂N , the density of its set of popular differences, and the structure of E . To obtain our results, we formalise a correspondence principle of Furstenberg and Weiss which relates combinatorial data on a tree to the dynamics of a Markov process. Our main tools are Kneser-type inverse theorems for sets of return times in measure-preserving systems. In the ergodic setting we use a recent result of the first author with Björklund and Shkredov and a stability-type extension (proved jointly with Shkredov); we also prove a new result for non-ergodic systems.

Modeling and simulation of disease spreading in pedestrian crowds has been recently become a topic of increasing relevance. In this paper, we consider the influence of the crowd motion in a complex dynamical environment on the course of infection of the pedestrians. To model the pedestrian dynamics we consider a kinetic equation for multi-group pedestrian flow based on a social force model coupled with an Eikonal equation. This model is coupled with a non-local SEIS contagion model for disease spread, where besides the description of local contacts also the influence of contact times has been modelled. Hydrodynamic approximations of the coupled system are derived. Finally, simulations of the hydrodynamic model are carried out using a mesh-free particle method. Different numerical test cases are investigated including uni- and bi-directional flow in a passage with and without obstacles.

Compartmental ordinary differential equation (ODE) models are used extensively in mathematical biology. When transit between compartments occurs at a constant rate, the well-known linear chain trick can be used to show that the ODE model is equivalent to an Erlang distributed delay differential equation (DDE). Here, we demonstrate that compartmental models with non-linear transit rates and possibly delayed arguments are also equivalent to a scalar distributed delay differential equation. To illustrate the utility of these equivalences, we calculate the equilibria of the scalar DDE, and compute the characteristic function-- without calculating a determinant. We derive the equivalent scalar DDE for two examples of models in mathematical biology and use the DDE formulation to identify physiological processes that were otherwise hidden by the compartmental structure of the ODE model.

We study the distribution of non-discrete orbits of geometrically finite groups in SO(n,1) acting on R n+1 , and more generally on the quotient of SO(n,1) by a horospherical subgroup. Using equidistribution of horospherical flows, we obtain both asymptotics for the distribution of orbits of general geometrically finite groups, and quantitative statements with additional assumptions.

This paper investigates the periodic points of the Gauss type shifts associated to the even continued fraction (Schweiger) and to the backward continued fraction (Rényi). We show that they coincide exactly with two sets of quadratic irrationals that we call E -reduced, and respectively B -reduced. We prove that these numbers are equidistributed with respect to the (infinite) Lebesgue absolutely continuous invariant measures of the corresponding Gauss shift.

In this survey we describe how the so-called Dold congruence arises in topology, and how it relates to periodic point counting in dynamical systems.

The Swift-Hohenberg equation is ubiquitous in the study of bistable dynamics. In this paper, we study the dynamic transitions of the Swift-Hohenberg equation with a third-order dispersion term in one spacial dimension with a periodic boundary condition. As a control parameter crosses a critical value, the trivial stable equilibrium solution will lose its stability, and undergoes a dynamic transition to a new physical state, described by a local attractor. The main result of this paper is to fully characterize the type and detailed structure of the transition using dynamic transition theory. In particular, employing techniques from center manifold theory, we reduce this infinite dimensional problem to a finite one since the space on which the exchange of stability occurs is finite dimensional. The problem then reduces to analysis of single or double Hopf bifurcations, and we completely classify the possible phase changes depending on the dispersion for every spacial period.

We study the dynamical Borel-Cantelli lemma for recurrence sets in a measure preserving dynamical system (X,μ,T) with a compatible metric d . We prove that, under some regularity conditions, the μ -measure of the following set R(\psi)= \{x\in X : d(T^n x, x) < \psi(n)\ \text{for infinitely many}\ n\in\N \} obeys a zero-full law according to the convergence or divergence of a certain series, where $\psi:\N\to\R^+$. Some of the applications of our main theorem include the continued fractions dynamical systems, the beta dynamical systems, and the homogeneous self-similar sets.