Related papers: Hyperbolicity and recurrence in dynamical systems:…
We discuss several topics related to the notion of strong hyperbolicity which are of interest in general relativity. After introducing the concept and showing its relevance we provide some covariant definitions of strong hyperbolicity. We…
Topics concerning metric dimension related invariants in graphs are nowadays intensively studied. This compendium of combinatorial and computational results on this topic is an attempt of surveying those contributions that are of the…
I investigate the problem of finding a statistical description of a complex many-body system whose invariant measure cannot be constructed stemming from classical thermodynamics ensembles. By taking solitons as a reference system and by…
The paper addresses for the first time some basic questions in the theory of finite time dynamics and finite time predictions for slowly mixing non-uniformly hyperbolic dynamical systems. It is concerned with transport in phase spaces of…
We experimentally explore solutions to a model Hamiltonian dynamical system derived in Colliander et al., 2012, to study frequency cascades in the cubic defocusing nonlinear Schr\"odinger equation on the torus. Our results include a…
A dynamical systems approach to turbulence envisions the flow as a trajectory through a high-dimensional state space transiently visiting the neighbourhoods of unstable simple invariant solutions (E. Hopf, Commun. Appl. Maths 1, 303, 1948).…
A majority of methods from dynamical systems analysis, especially those in applied settings, rely on Poincar\'e's geometric picture that focuses on "dynamics of states". While this picture has fueled our field for a century, it has shown…
We construct a smooth nontrivial mixed partially hyperbolic system and explicitly identify its skeleton. This example shares characteristics with the classical examples. Moreover, the support of each physical measure contains three fixed…
In a first part the scope of classical thermodynamics and statistical mechanics is discussed in the broader context of formal dynamical systems, including computer programmes. In this context classical thermodynamics appears as a particular…
Given a Radon probability measure $\mu$ supported in $\mathbb{R}^d$, we are interested in those points $x$ around which the measure is concentrated infinitely many times on thin annuli centered at $x$. Depending on the lower and upper…
Topological dynamics constitutes the study of asymptotic properties of orbits under flows or maps on the Hausdorff phase space. Hyperbolic dynamics is the study of differentiable flows or maps that are usually characterized by the presence…
The theory of physical dimensions and units in physics is outlined. This includes a discussion of the universal applicability and superiority of quantity equations. The International System of Units (SI) is one example thereof. By analyzing…
We study how dynamical quantities such as Lyapunov exponents, metric entropy, topological pressure, recurrence rates, and dimension-like characteristics change under a time reparameterization of a dynamical system. These quantities are…
Multifractal analysis studies level sets of asymptotically defined quantities in a topological dynamical system. We consider the topological pressure function on such level sets, relating it both to the pressure on the entire phase space…
The continuum equations of fluid mechanics are rederived with the intention of keeping certain mechanical and thermodynamic concepts separate. A new "mechanical" mass density is created to be used in computing inertial quantities, whereas…
This essay advocates the view that any problem that has a meaningful empirical content, can be formulated in constructive, more definitely, finite terms. We consider combinatorial models of dynamical systems and approaches to statistical…
We provide a consistent statistical-mechanical treatment for describing the thermodynamics and the structure of fluids embedded in the hyperbolic plane. In particular, we derive a generalization of the virial equation relating the bulk…
We present a novel numerical method aimed to characterize global behaviour, in particular chaotic diffusion, in dynamical systems. It is based on an analysis of the Poincar\'e recurrence statistics on massive grids of initial data or values…
Recurrence is a fundamental characteristic of dynamical systems with complicated behavior. Understanding the inner structure of recurrence is challenging, especially if the system has many degrees of freedom and is subject to noise. We…
This survey paper is aimed to describe a relatively new branch of symbolic dynamics which we call Arithmetic Dynamics. It deals with explicit arithmetic expansions of reals and vectors that have a "dynamical" sense. This means precisely…