Related papers: A new notion of dimension for dynamical systems an…
The commonly used spatial entropy $h_{r}(\mathcal{U})$ of the multi-dimensional shift space $\mathcal{U}$ is the limit of growth rate of admissible local patterns on finite rectangular sublattices which expands to whole space…
We study systems with periodically oscillating parameters that can give way to complex periodic or non periodic orbits. Performing the long time limit, we can define ergodic averages such as Lyapunov exponents, where a negative maximal…
The dynamics of many important high-dimensional dynamical systems are both chaotic and complex, meaning that strong reducing hypotheses are required to understand the dynamics. The highly influential chaotic hypothesis of Gallavotti and…
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…
In this paper we study the shifts, which are the shift-invariant and topologically closed sets of configurations over a finite alphabet in $\mathbb{Z}^d$. The minimal shifts are those shifts in which all configurations contain exactly the…
Let $x_{j+1}=\phi(x_{j})$, $x_{j}\in\mathbb{R}^{d}$, be a dynamical system with $\phi$ being a diffeomorphism. Although the state vector $x_{j}$ is often unobservable, the dynamics can be recovered from the delay vector…
We generalize the proof of Karamata's Theorem by the method of approximation by polynomials to the operator case. As a consequence, we offer a simple proof of \emph{uniform dual ergodicity} for a very large class of dynamical systems with…
In this chapter, we utilize dynamical systems to analyze several aspects of machine learning algorithms. As an expository contribution we demonstrate how to re-formulate a wide variety of challenges from deep neural networks, (stochastic)…
In this paper we define unstable topological entropy for any subsets (not necessarily compact or invariant) in partially hyperbolic systems as a Carath\'{e}odory dimension characteristic, motivated by the work of Bowen and Pesin etc. We…
The paper develops and studies a very general notion of dichotomy, referred to as "nonuniform $(h,k,\mu,\nu)$-dichotomy". The new notion contains as special cases most versions of dichotomy existing in the literature. The paper then…
A new measure to characterize stability of complex dynamical systems against large perturbation is suggested, the stability threshold (ST). It quantifies the magnitude of the weakest perturbation capable to disrupt the system and switch it…
We investigate sensitivity to cumulative perturbations for a few dynamical system classes of practical interest. A system is said to have bounded sensitivity to cumulative perturbations (bounded sensitivity, for short) if an additive…
We study the parametrized complexity of fundamental relations between multidimensional subshifts, such as equality, conjugacy, inclusion, and embedding, for subshifts of finite type (SFTs) and effective subshifts. We build on previous work…
Our conventional system of physical units is based on local or microscopic {\it dimensional} quantities which are {\it defined}, for convenience or otherwise aesthetic reasons, to be spacetime-independent. A more general choice of units may…
We consider some of the main notions of Gibbs measures on subshifts introduced by different communities, such as dynamical systems, probability, operator algebras, and mathematical physics. For potentials with $d$-summable variation, we…
The use in the action integral of a volume element of the form $\Phi d^{D}x$, where $\Phi$ is a metric-independent measure density, can yield new interesting results in all types of known generally coordinate-invariant theories: (1) 4-D…
This is an expository article, originally written in Japanese, on a dynamical system over a non-archimedean field. The main viewpoint is from complex and non-archimedean potential theories. After quickly introducing the Berkovich projective…
Let $ E $ be a non-empty compact subset of the Riemann sphere and $T$ be a rational map of degree at least two. We study the associated \emph{orbital set}, that is, the backwards orbit of $E$ under $T$, and study the relationship between…
In this paper, the dynamical system analysis has been performed to analyze the dynamical behavior of the Universe in $f(R,L_m,T)$ gravity with a scalar field. A well motivated potential function and the linear form of the functional…
This paper is a preliminary work to address the problem of dynamical systems with parameters varying in time. An idea to predict their behaviour is proposed. These systems are called \emph{transient systems}, and are distinguished from…