Related papers: Dynamics in large scale geometry

The development of a metric for structural data is a long-term problem in pattern recognition and machine learning. In this paper, we develop a general metric for comparing nonlinear dynamical systems that is defined with Perron-Frobenius…

We numerically investigate the orbital dynamics of a two-dimensional galactic model, emphasizing the influence of stable and unstable manifolds on the evolution of orbits. In our analysis we use evaluations of the system's Lagrangian…

General Relativity can be reformulated as a geometrodynamical theory, called Shape Dynamics, that is not based on spacetime (in particular refoliation) symmetry but on spatial diffeomorphism and local spatial conformal symmetry. This leads…

Do the spacetime manifolds which feature in our best scientific theories reflect anything metaphysically weighty in the world (e.g., any fundamental substances or relations)? Should we extend our notions of space and time beyond the…

We prove an analogue of the Manin-Mumford conjecture for polynomial dynamical systems over number fields. In our setting the role of torsion points is taken by the small orbit of a point $\alpha$. The small orbit of a point was introduced…

We consider the Lorenz equations, a system of three dimensional ordinary differential equations modeling atmospheric convection. These equations are chaotic and hard to study even numerically, and so a simpler "geometric model" has been…

Geodesic orbit spaces (or g.o. spaces) are defined as those homogeneous Riemannian spaces $(M=G/H,g)$ whose geodesics are orbits of one-parameter subgroups of $G$. The corresponding metric $g$ is called a geodesic orbit metric. We study the…

These are lecture notes from a course in arithmetic dynamics given in Grenoble in June 2017. The main purpose of this text is to explain how arithmetic equidistribution theory can be used in the dynamics of rational maps on P^1. We first…

We study the space geometry of a rotating disk both from a theoretical and operational approach, in particular we give a precise definition of the space of the disk, which is not clearly defined in the literature. To this end we define an…

Newtonian dynamics is derived from prior information codified into an appropriate statistical model. The basic assumption is that there is an irreducible uncertainty in the location of particles so that the state of a particle is defined by…

We provide a construction of Roe (C*-)algebras of general coarse spaces in terms of coarse geometric modules. This extends the classical theory of Roe algebras of metric spaces and gives a unified framework to deal with either uniform or…

Motivated by various results on homogeneous geodesics of Riemannian spaces, we study homogeneous trajectories, i.e. trajectories which are orbits of a one-parameter symmetry group, of Lagrangian and Hamiltonian systems. We present criteria…

It is known that some equations of differential geometry are derived from variational principle in form of Euler-Lagrange equations. The equations of geodesic flow in Riemannian geometry is an example. Conversely, having Lagrangian…

In the statistical description of dynamical systems, an indication of the irreversibility of a given state change is given geometrically by means of a (pre-)ordering of state pairs. Reversible state changes of classical and quantum systems…

The nonlinear coupling between spacetime geometry and matter in the early Universe remains a frontier in theoretical cosmology. By introducing a novel gravitomagnetic-hydrodynamic framework, we reveal a fundamental analogy between…

This paper is an interval dynamics counterpart of three theories founded earlier by the authors, S. Smirnov and others in the setting of the iteration of rational maps on the Riemann sphere: the equivalence of several notions of non-uniform…

We analyze the quantum dynamics of the Friedmann-Robertson-Walker Universe in the context of a Generalized Uncertainty Principle. Since the isotropic Universe dynamics resembles that of a one-dimensional particle, we quantize it with the…

There are many classical results, related to the Denjoy--Wolff Theorem, concerning the relationship between orbits of interior points and orbits of boundary points under iterates of holomorphic self-maps of the unit disc. Here, for the…

Coarse geometry, the branch of topology that studies the global properties of spaces, was originally developed for metric spaces and then Roe introduced coarse structures as a large-scale counterpart of uniformities. In the literature,…

According to the algebraic approach to spacetime, a thoroughgoing dynamicism, physical fields exist without an underlying manifold. This view is usually implemented by postulating an algebraic structure (e.g., commutative ring) of…