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Flows on surfaces are one of the most fundamental and classical objects in dynamical systems, and are studied from various areas (e.g. integrable systems, differential equations, fluid mechanics). Though hyperbolic flows and recurrent flows…
We provide an example of how the complex dynamics of a recently introduced model can be understood via a detailed analysis of its associated Riemann surface. Thanks to this geometric description an explicit formula for the period of the…
Many measurements on soft condensed matter (e.g., biological and materials) systems track low-dimensional observables projected from the full system phase space as a function of time. Examples are dynamic structure factors, spectroscopic…
We introduce and study two properties of dynamical systems: topologically transitive and topologically mixing under the set-valued setting. We prove some implications of these two topological properties for set-valued functions and…
Different notions of entropy play a fundamental role in the classical theory of dynamical systems. Unlike many other concepts used to analyze autonomous dynamics, both measure-theoretic and topological entropy can be extended quite…
A set $A \subseteq \mathbb{N}$ is a set of pointwise recurrence if for all minimal dynamical systems $(X, T)$, all $x \in X$, and all open neighborhoods $U \subseteq X$ of $x$, there exists a time $n \in A$ such that $T^n x \in U$. The set…
We consider actions of a tileable amenable group $\Gamma$ on a topological space $X$. For a continuous function on $X$, we define the entropy of the number of homologically detectable critical point of the average of that function over…
This paper considers the problem of characterizing the simplest discrete point sets that are aperiodic, using invariants based on topological dynamics. A Delone set whose patch-counting function N(T), for radius T, is finite for all T is…
We consider metrizable ergodic topological dynamical systems over locally compact, $\sigma$-compact abelian groups. We study pure point spectrum via suitable notions of almost periodicity for the points of the dynamical system. More…
Dynamical systems can display a plethora of ergodic and ergodicity breaking behaviors, ranging from simple periodicity to ergodicity and chaos. Here we report an unusual type of non-ergodic behavior in a many-body discrete-time dynamical…
The interrelation of dynamic processes active on separated time-scales in glasses and viscous liquids is investigated using a model displaying two time-scale bifurcations both between fast and secondary relaxation and between secondary and…
We study the set of irregular points for topologically mixing subshifts of finite type. It is well known that despite the irregular set having zero measure for every invariant measure, it has full topological entropy and full Hausdorff…
Let $(X,d)$ be a compact metric space and $(X,\mathcal{A},\mu,T)$ a measure preserving dynamical system. Furthermore, given a real, positive function $\psi$, let $W(T, \psi)$ and $ R(T,\psi) $ respectively denote the shrinking target set…
We introduce a new type of shift dynamics as an extended model of symbolic dynamics, and investigate the characteristics of shift spaces from the viewpoints of both dynamics and computation. This shift dynamics is called a functional shift…
Let $X$ be a zero-dimensional locally compact Hausdorff space not necessarily metric and $G$ a compactly generated topological group not necessarily abelian or countable. We define recurrence at a point for any continuous action of $G$ on…
The theoretical and numerical understanding of the key concept of topological entropy is an important problem in dynamical systems. Most studies have been carried out on maps (discrete-time systems). We analyse a scenario of global changes…
In this paper we present a novel methodology based on a topological entropy, the so-called persistent entropy, for addressing the comparison between discrete piecewise linear functions. The comparison is certified by the stability theorem…
Compact objects evolving in an astrophysical environment experience a gravitational drag force known as dynamical friction. We present a multipole-frequency decomposition to evaluate the orbit-averaged energy and angular momentum…
We consider the collection of uniformly discrete point sets in Euclidean space equipped with the vague topology. For a point set in this collection, we characterise minimality of an associated dynamical system by almost repetitivity of the…
Aspects of the dynamical glass transition are considered within a mean field spin glass model. At the dynamical transition the the system condenses in a state of lower entropy. The difference, the information entropy or complexity, is…