Ergodicity of dynamical systems on 2-adic spheres

Theory of dynamical systems in fields of p-adic numbers is an important part of algebraic and arithmetic dynamics. The study of p-adic dynamical systems is motivated by their applications in various areas of mathematics, e.g., in physics, genetics, biology, cognitive science, neurophysiology, computer science, cryptology, etc.In particular, p-adic dynamical systems found applications in cryptography, which stimulated the interest to nonsmooth dynamical maps. An important class of (in general) nonsmooth maps is given by 1-Lipschitz functions.In this thesis we restrict our study to the class of 1-Lipschitz functions and describe measure-preserving (for the Haar measure on the ring of p-adic integers) and ergodic functions.The main mathematical tool used in this work is the representation of the function by the van der Put series which is actively used in p-adic analysis. The van der Put basis differs fundamentally from previously used ones (for example, the monomial and Mahler basis) which are related to the algebraic structure of p-adic fields. The basic point in the construction of van der Put basis is the continuity of the characteristic function of a p-adic ball.Also we use an algebraic structure (permutations) induced by coordinate functions with partially frozen variables.In this thesis, we present a description of 1-Lipschitz measure-preserving and ergodic functions for arbitrary prime p.

p-adic dynamical systems of (2,2)-rational functions with unique fixed point

The p-adic numbers are a counterintuitive arithmetic system and were firstly introduced circa end of the nineteenth century. In conjunction with the introduction of these numbers, many mathematicians and physicists started to develop new scientific tools using their available, useful, and applicable properties. Several effects of these researches have emerged in mathematics and physics such as p-adic analysis, string theory, p-adic quantum mechanics, quantum field theory, representation theory, algebraic geometry, complex systems, dynamical systems, and genetic codes. One of the important tools of the mentioned advancements is the p-adic integrals. Intense research activities in such an area like p-adic integrals are principally motivated by their significance in p-adic analysis. Recently, p-adic integrals and its diverse extensions have been studied and investigated in detail by many mathematicians. This chapter considers and investigates multifarious extensions of the p-adic integrals elaborately. q-Analogues with diverse extensions of p-adic integrals are also considered such as the weighted p-adic q-integral on \( \mathbb {Z} _{p}\). The two types of the weighted q-Boole polynomials and numbers are introduced and investigated in detail. As several special polynomials and numbers can be derived from the p-adic integrals, some generalized and classical q-polynomials and numbers are obtained from the aforesaid extensions of p-adic integrals. Finally, the importance of these extensions is analyzed.

In this short note, we will study the existence of a vector space of continuous functions f:{mathbb {Z}}_prightarrow mathbb Q_p, where {mathbb {Z}}_p and {mathbb {Q}}_p are, respectively, the ring of p-adic integers and the field of p-adic numbers, such that each nonzero function does not satisfy the Luzin (N) property and the dimension of the vector space is the continuum.

In this paper we study the connection between the use of evaluative language and the building of both personal and social identities, from the perspective of Dynamical System Theory. We primarily discuss two issues: 1) The use of evaluation (in the sense given to the term by Alba-Juez and Thompson (forthcoming)) as a means to the construction of both individual and group identities, thus exploring how the connection between linguistic choices and social identities is shaped by interactional needs for stancetaking. In order to illustrate this connection, we examine examples of the use of evaluative language in a web social network, and we analyze some of the discourse elements showing ways of positioning that act as catalysts for the emergence of a multifactorial dynamic system of identities. 2) The consideration of Dynamical System Theory (DST) as a theoretical framework for the modeling of language and identity. Although originally a mathematical theory, DST has been adopted by cognitive science as a valid framework for the study of cognitive phenomena, on the grounds that natural cognition is a dynamical phenomenon. Within the realm of (socio) linguistics and pragmatics, this study is to a certain degree in line with some recent studies such as Gibbs (2010), Geeraerts, Kristiansen and Peirsman (2010), or Moreno Fernández (2012). Thus, we herein focus on how linguistic evaluation intervenes in the intricate dynamical system of identity, and even though we do not engage in complex mathematical disquisition, we argue that the idea and philosophical foundation underlying DST can lead us towards the ‘integration’ of the complex equation of identity construction, and that consequently the field has great potential for further research.

Although there is a substantial philosophical literature on dynamical systems theory in the cognitive sciences, the same is not the case for neuroscience. This paper attempts to motivate increased discussion via a set of overlapping issues. The first aim is primarily historical and is to demonstrate that dynamical systems theory is currently experiencing a renaissance in neuroscience. Although dynamical concepts and methods are becoming increasingly popular in contemporary neuroscience, the general approach should not be viewed as something entirely new to neuroscience. Instead, it is more appropriate to view the current developments as making central again approaches that facilitated some of neuroscience’s most significant early achievements, namely, the Hodgkin–Huxley and FitzHugh–Nagumo models. The second aim is primarily critical and defends a version of the “dynamical hypothesis” in neuroscience. Whereas the original version centered on defending a noncomputational and nonrepresentational account of cognition, the version I have in mind is broader and includes both cognition and the neural systems that realize it as well. In view of that, I discuss research on motor control as a paradigmatic example demonstrating that the concepts and methods of dynamical systems theory are increasingly and successfully being applied to neural systems in contemporary neuroscience. More significantly, such applications are motivating a stronger metaphysical claim, that is, understanding neural systems as being dynamical systems, which includes not requiring appeal to representations to explain or understand those phenomena. Taken together, the historical claim and the critical claim demonstrate that the dynamical hypothesis is undergoing a renaissance in contemporary neuroscience.

(2008). On Ordering the Natural Numbers or The Sharkovski Theorem. The American Mathematical Monthly: Vol. 115, No. 2, pp. 159-165.

We study the behaviour of the iterates of the Chebyshev polynomials of the first kind in p-adic fields. In particular, we determine in the field of complex p-adic numbers for p > 2, the periodic points of the p-th Chebyshev polynomial of the first kind. These periodic points are attractive points. We describe their basin of attraction. The classification of finite field extensions of the field of p-adic numbers ℚ p , enables one to locate precisely, for any integer ν ≥ 1, the ν-periodic points of T p : they are simple and the nonzero ones lie in the unit circle of the unramified extension of ℚ p , (p > 2) of degree ν. This generalizes a result, stated by M. Zuber in his PhD thesis, giving the fixed points of T p in the field ℚ p , (p > 2). As often happens, we consider separately the case p = 2. Also, if the integer n ≥ 2 is not divisible by p, then any fixed point w of T n is indifferent in the field of p-adic complex numbers and we give for p ≥ 3, the p-adic Siegel disc around w.

This chapter defines dynamic systems, its relevance to sports performance analysis, and explains the inter-personal coordination and perturbations during sport competition. It is concerned with outlining a dominant motor control theory that can aid the explanation and interpretation of performance analysis data dynamic systems theory. The chapter has three central principles of dynamic systems theory: synergies, self-organisation and constraints. It is built upon the existing research using dynamic systems theory to study attacker-defender interactions in rugby union, it could be argued that the practical implications were limited. Future research using dynamic systems theory will be a theoretical framework for analysing inter-personal coordination, ensuring the analysis can be more effectively linked to the attacking/defending principles of play, and be useful and informative for sports coach and performance analysts. There are clear links between the theory and the principles of play of the sport, i.e. the attacking principles of soccer: width, mobility, support, facilitating the exploration of the available playing space.

This work delves into the theory of dynamic systems, focusing on the analysis of entropy in both classical and topological contexts. Beginning with an exposition of fundamental concepts in dynamical systems theory, particular attention is given to topological dynamical systems (TDS). The discussion progresses to explore discrete topological entropy and its significance within dynamical systems, culminating in the introduction of topological entropy pressure as a nuanced form of this concept. The study then investigates various applications of topological entropy within dynamic systems, emphasizing its utility in understanding chaotic systems and its role in ergodic theory. A novel theory, termed Topological Ergodic Entropy Theory (TEET), is presented, offering a fresh perspective on the analysis of ergodic dynamical systems. Furthermore, the work introduces the Ergodic Theory of Turbulent Flow (ETTF), which probes the interplay between topological entropy and the ergodic properties of dynamic systems governed by the Navier-Stokes equations. Through these explorations, the findings contribute significantly to the comprehension of the intricate nature of dynamical systems and their diverse applications across mathematics and physics. By scrutinizing topological entropy and its implications in dynamical systems, this research offers novel insights into the chaotic and stochastic behaviors exhibited by these systems. Additionally, the introduction of pioneering theories like ETTF opens up new avenues for understanding and modeling turbulent phenomena, thereby enriching our understanding of complex dynamical processes.

Most dynamical systems in real-life applications are usually nonlinear and nonstationary, such as biomechanical systems, neuroscience, healthcare, economics, and engineering. Nowadays, the availability of data makes it possible to investigate and discover the hidden dynamics of target systems. However, numerous concerns arise in modeling and analysis of such systems, including complex system structure, data-inherent uncertainty in observations, poor interpretability, etc. To this end, we will investigate dynamical behaviors by proposing data-driven machine learning and dynamical system techniques. Specifically, we present various approaches for extracting characteristics or patterns that have good interpretability in various complex dynamical system applications.In the first study, we consider a pattern recognition problem in biomechanical system using multivariate human kinematic data and propose data-driven analytical methods to model human movement. We first perform recurrence quantification analysis on the phase space of the kinematic system to visualize and evaluate the recurrence behavior of the folded phase space trajectory. However, this method could miss out the relationship between variables. We therefore introduce a new modeling approach based on the self-expressive assumption to discover the interactions between variables and characterize the patterns of human motion. Our model measures correlations between variables by underlying linear subspaces, which can be further extended to nonlinear subspaces. Based on these correlations, our model improves the classification accuracy by 20% - 30% over other standard state-of-art correlation measurements and clustering methods. In the second study, we consider a specific neuroscience problem, seizure detection, to recognize abnormal brain states by learning from highly nonlinear electroencephalography. Based on dynamical systems theory, Koopman operator theory, and dynamic mode decomposition, we project the complex brain dynamics into an infinite-dimensional linear space, where the intrinsic linear dynamic shares the same dynamical behavior as the original nonlinear dynamical system. We propose a new system measurement to unify the projections in different Koopman spaces. Our measurement indicates the stability of the dynamical system, which represents the system state trajectory with respect to the corresponding attractor. We demonstrate the effectiveness of our approach with a seizure onset detection task. In this case study, our model gives high accuracy with a low false-positive rate. In the last study, we consider dynamical systems with multiple modalities. We design a Koopman-based deep learning network to identify and simplify the intrinsic nonlinear dynamics from different sources/observations of a multi-modal dynamical system. Since these different modalities describe the same system, they jointly share the same intrinsic dynamics. We develop an autoencoder framework with custom loss functions to find proper project functions/operators and to obtain the intrinsic Koopman coordinates. Our framework can reconstruct the original system states based on the joint dynamics. Once the intrinsic linear dynamics is learned, future system states can be predicted by the evolution in the Koopman space. This approach allows us to uncover and linearize the hidden nonlinear dynamics, which enables linear dynamical tools, for example, system control, spectrum analysis, to analyze complex dynamical systems.--Author's abstract

Introduction: Children with cerebral palsy may receive various rehabilitative intervention programs aimed at facilitating their development and improving their functional independence in such areas as movement, self-care, playing, leisure and school activities. Occupational therapists working with children with Cerebral Palsy (CP) employ a variety of therapeutic approaches among which neurodevelopmental treatment approach has been traditionally the most common. Considering the expansion of family-centered therapy approaches, the importance of obligation to the International Classification of Functioning, Disabilities and Health principles (ICF, WHO, 2001), limitations of current therapy approaches and recent advances from dynamic systems theory, there is an urgent need to develop a new approach in occupational therapy at the present time. The aims of this study were to present a brief description of CP with an emphasis on the role of occupational therapy in it and to explore the benefits of family-centered approach, dynamic systems theory and family-centered functional therapy in the rehabilitation process of children with cerebral palsy. Materials and Methods: In this study, a comprehensive review of the literature published from 1966 to 2011 was conducted via PubMed, Scopus, ProQuest and Google Scholar data bases. The keywords used in search were “cerebral palsy”, “functional”, “therapy”, “family-centered”. All articles were initially evaluated according to predefined inclusion and exclusion criteria and then selected article were reviewed. Results : Of 128 articles found, 22 articles met the inclusion and exclusion criteria and were considered as appropriate for this study. Conclusion: Family-centered functional therapy, which combines theoretical principles of occupational therapy, family centered approach and dynamic systems theory of motor development, is an optimal response to developmental needs of children with cerebral palsy. However, the efficacy of this approach has not been proven empirically. Keywords: Cerebral palsy, Family centered functional therapy, Dynamic systems theory

This introduction to recent work in p-adic analysis and number theory will make accessible to a relatively general audience the efforts of a number of mathematicians over the last five years. After reviewing the basics (the construction of p-adic numbers and the p-adic analog of the complex number field, power series and Newton polygons), the author develops the properties of p-adic Dirichlet L-series using p-adic measures and integration. p-adic gamma functions are introduced, and their relationship to L-series is explored. Analogies with the corresponding complex analytic case are stressed. Then a formula for Gauss sums in terms of the p-adic gamma function is proved using the cohomology of Fermat and Artin-Schreier curves. Graduate students and research workers in number theory, algebraic geometry and parts of algebra and analysis will welcome this account of current research.

P-adic numbers p-adic interpolation of the reimann zeta-function p-adic power series rationality of the zeta-function of a set of equations over a finite field (Part contents).

Abstract. Numerical climate simulations produce vast amounts of high-resolution data. This poses new challenges to the palaeoclimate community – and indeed to the broader climate community – in how to efficiently process and interpret model output. The palaeoclimate community also faces the additional challenge of having to characterise and compare a much broader range of climates than encountered in other subfields of climate science. Here we propose an analysis framework, grounded in dynamical systems theory, which may contribute to overcoming these challenges. The framework enables the characterisation of the dynamics of a given climate through a small number of metrics. These may be applied to individual climate variables or to several variables at once, and they can diagnose properties such as persistence, active number of degrees of freedom and coupling. Crucially, the metrics provide information on instantaneous states of the chosen variable(s). To illustrate the framework's applicability, we analyse three numerical simulations of mid-Holocene climates over North Africa under different boundary conditions. We find that the three simulations produce climate systems with different dynamical properties, such as persistence of the spatial precipitation patterns and coupling between precipitation and large-scale sea level pressure patterns, which are reflected in the dynamical systems metrics. We conclude that the dynamical systems framework holds significant potential for analysing palaeoclimate simulations. At the same time, an appraisal of the framework's limitations suggests that it should be viewed as a complement to more conventional analyses, rather than as a wholesale substitute.