Related papers: Dynamics in dimension zero. A survey
In the paper we study aperiodic substitutional dynamical systems arisen from non-primitive substitutions. We prove that the Vershik homeomorphism $\phi$ of a stationary ordered Bratteli diagram is homeomorphic to an aperiodic substitutional…
Leveraging recent work on data-driven methods for constructing a finite state space Markov process from dynamical systems, we address two problems for obtaining further reduced statistical representations. The first problem is to extract…
In another article we associated a dynamical system to a non-properly ordered Bratteli diagram. In this article we describe how to compute the $K-$group $K_0$ of the dynamical system in terms of the Bratteli diagram. In the case of properly…
We consider Bratteli diagrams of finite rank (not necessarily simple) and ergodic invariant measures with respect to the cofinal equivalence relation on their path spaces. It is shown that every ergodic invariant measure (finite or…
We study stationary ordered Bratteli diagrams and give necessary and sufficient conditions for these orders to generate a continuous Vershik map. We apply this to finding adic representations for one sided substitution subshifts. We give an…
We study dynamical systems using measures taking values in a non-Archimedean field. The underlying space for such measure is a zero-dimensional topological space. In this paper we elaborate on the natural translation of several notions,…
A scheme is presented to extract detailed dynamical signatures from successive measurements of complex systems. Relative entropy based time series tools are used to quantify the gain in predictive power of increasing past knowledge. By…
We provide an explicit S-adic representation of rank one subshifts with bounded spacers and call the subshifts obtained in this way ''Ferenczi subshifts''. We aim to show that this approach is very convenient to study the dynamical behavior…
We present a series of statements about scaling entropy, a metric invariant of dynamical systems proposed by A. M. Vershik in the late 90's. We show that it is a metric invariant indeed and calculate it in several special cases.
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…
Recalling the construction of a flat surface from a Bratteli diagram, this paper considers the dynamics of the shift map on the space of all bi-infinite Bratteli diagrams as the renormalizing dynamics on a moduli space of flat surfaces of…
We approximate stochastic processes in finite dimension by dynamical systems. We provide trajectorial estimates which are uniform with respect to the initial condition for a well chosen distance. This relies on some non-expansivity property…
For any adic transformation $T$ defined on the path space $X$ of an ordered Bratteli diagram, endowed with a Markov measure $\mu$, we construct an explicit dimension space (which corresponds to a matrix values random walk on $\mathbb{Z}$)…
Dynamical systems theory has long provided a foundation for understanding evolving phenomena across scientific domains. Yet, the application of this theory to complex real-world systems remains challenging due to issues in mathematical…
We characterise the evolution of a dynamical system by combining two well-known complex systems' tools, namely, symbolic ordinal analysis and networks. From the ordinal representation of a time-series we construct a network in which every…
The notion of metric entropy dimension is introduced to measure the complexity of entropy zero dynamical systems. For measure preserving systems, we define entropy dimension via the dimension of entropy generating sequences. This…
Two low-dimensional magnetohydrodynamic models containing three velocity and three magnetic modes are described. One of them (nonhelical model) has zero kinetic and current helicity, while the other model (helical) has nonzero kinetic and…
Iterations of odd piecewise continuous maps with two discontinuities, i.e., symmetric discontinuous bimodal maps, are studied. Symbolic dynamics is introduced. The tools of kneading theory are used to study the homology of the discrete…
Dynamic evolution behaviors of dimension-varying control systems often appear in the genetic regulatory network and the vehicle clutch system etc. An interesting and significant study on dimension-varying control systems is how to realize…
A cross-dimensional Euclidian space ($\Omega$) is proposed for the state space of dimension varying (control) systems. It is shown that the topological structure of $\Omega$ is consistent with all $\mathbb{R}^n$, which are the state spaces…