scholarly journals Randomness and Topological Invariants in Pentagonal Tiling Spaces

2011 ◽  
Vol 2011 ◽  
pp. 1-23 ◽  
Author(s):  
Juan García Escudero
Keyword(s):  
Dynamical Systems ◽  
Configurational Entropy ◽  
Point Of View ◽  
Topological Invariants ◽  
Aperiodic Tilings ◽  
Tiling Spaces ◽  
Substitution Process ◽  
Cech Cohomology ◽  
Random Substitution ◽  
Inflation Rules

We analyze substitution tiling spaces with fivefold symmetry. In the substitution process, the introduction of randomness can be done by means of two methods which may be combined: composition of inflation rules for a given prototile set and tile rearrangements. The configurational entropy of the random substitution process is computed in the case of prototile subdivision followed by tile rearrangement. When aperiodic tilings are studied from the point of view of dynamical systems, rather than treating a single one, a collection of them is considered. Tiling spaces are defined for deterministic substitutions, which can be seen as the set of tilings that locally look like translates of a given tiling. Čech cohomology groups are the simplest topological invariants of such spaces. The cohomologies of two deterministic pentagonal tiling spaces are studied.

scholarly journals On some kinds of dense orbits in general nonautonomous dynamical systems

2020 ◽  
Vol 7 (1) ◽  
pp. 163-175
Author(s):  
Mehdi Pourbarat
Keyword(s):  
Dynamical Systems ◽  
Initial Conditions ◽  
Point Of View ◽  
Topological Conjugacy ◽  
Topological Transitivity ◽  
Nonautonomous Systems ◽  
Nonautonomous Dynamical Systems ◽  
Sensitive Dependence ◽  
Dense Orbits

AbstractWe study the theory of universality for the nonautonomous dynamical systems from topological point of view related to hypercyclicity. The conditions are provided in a way that Birkhoff transitivity theorem can be extended. In the context of generalized linear nonautonomous systems, we show that either one of the topological transitivity or hypercyclicity give sensitive dependence on initial conditions. Meanwhile, some examples are presented for topological transitivity, hypercyclicity and topological conjugacy.

SURVEY Towards a global view of dynamical systems, for the C1-topology

2011 ◽  
Vol 31 (4) ◽  
pp. 959-993 ◽  
Author(s):  
C. BONATTI
Keyword(s):  
Dynamical Systems ◽  
Compact Manifold ◽  
Global Structure ◽  
Point Of View ◽  
List Type ◽  
Global View ◽  
Local Phenomenon ◽  
Generic Dynamics

AbstractThis paper suggests a program for getting a global view of the dynamics of diffeomorphisms, from the point of view of the C1-topology. More precisely, given any compact manifold M, one splits Diff1(M) into disjoint C1-open regions whose union is C1-dense, and conjectures state that each of these open sets and their complements is characterized by the presence of: •either a robust local phenomenon;•or a global structure forbidding this local phenomenon. Other conjectures state that some of these regions are empty. This set of conjectures draws a global view of the dynamics, putting in evidence the coherence of the numerous recent results on C1-generic dynamics.

scholarly journals Classification of rough transformations of a circle from a modern point of view

2018 ◽  
Vol 20 (4) ◽  
pp. 408-418
Author(s):  
A. E. Kolobyanina ◽  
E. V. Nozdrinova ◽  
O. V. Pochinka
Keyword(s):  
Invariant Manifolds ◽  
Sufficient Conditions ◽  
Periodic Points ◽  
Point Of View ◽  
Topological Classification ◽  
Topological Invariants ◽  
Modern Theory ◽  
Ambient Manifold ◽  
Periodic Data

In this paper the authors use modern methods and approaches to present a solution to the problem of the topological classification of circle’s rough transformations in canonical formulation. In the modern theory of dynamical systems such problems are understood as the complete topological classification: finding topological invariants, proving the completeness of the set of invariants found and constructing a standard representative from a given set of topological invariants. Namely, in the first theorem of this paper the type of periodic data of circle’s rough transformations is established. In the second theorem necessary and sufficient conditions of their conjugacy are proved. These conditions mean coincidence of periodic data and rotation numbers. In the third theorem the admissible set of parameters is implemented by a rough transformation of a circle. While proving the theorems, we assume that the results on the local topological classification of hyperbolic periodic points, as well as the results on the global representation of the ambient manifold as a union of invariant manifolds of periodic points, are known.

CROSSED PRODUCT C*-ALGEBRAS AND ALGEBRAIC TOPOLOGY

1996 ◽  
Vol 08 (04) ◽  
pp. 623-637
Author(s):  
JUDITH A. PACKER
Keyword(s):  
Algebraic Topology ◽  
Equivariant Cohomology ◽  
Automorphism Groups ◽  
Crossed Product ◽  
Fiber Bundles ◽  
Topological Invariants ◽  
C Algebras ◽  
Recent Developments ◽  
Continuous Trace ◽  
Cech Cohomology

We discuss some recent developments that illustrate the interplay between the theory of crossed products of continuous trace C*-algebras and algebraic topology, summarizing results relating topological invariants coming from the theory of fiber bundles to continuous trace C*-algebras and their automorphism groups and the structure of the associated crossed product C*-algebras. This survey article starts from the classical theory of Dixmier, Douady, and Fell, and discusses the more recent work of Echterhoff, Phillips, Raeburn, Rosenberg, and Williams, among others. The topological invariants involved are Čech cohomology, the cohomology of locally compact groups with Borel cochains of C. Moore, and the recently introduced equivariant cohomology theory of Crocker, Kumjian, Raeburn and Williams.

Persistent Homology, Regimes and Climate Data

2021 ◽  
Author(s):  
Kristian Strommen ◽  
Nina Otter ◽  
Matthew Chantry ◽  
Joshua Dorrington
Keyword(s):  
Dynamical Systems ◽  
State Of The Art ◽  
Persistent Homology ◽  
Topological Invariants ◽  
Climate Data ◽  
Future Directions ◽  
Atlantic Sector ◽  
Linear Dynamical Systems ◽  
Markovian Dynamics ◽  
Shed Light

<p>The concept of weather or climate 'regimes' have been studied since the 70s, to a large extent because of the possibility they offer of truncating complicated dynamics to vastly simpler, Markovian, dynamics. Despite their attraction, detecting them in data is often problematic, and a unified definition remains nebulous. We argue that the crucial common feature across different dynamical systems with regimes is the non-trivial topology of the underlying phase space. Such non-trivial topology can be detected in a robust and explicit manner using persistent homology, a powerful new tool to compute topological invariants in arbitrary datasets. We show some state of the art examples of the application of persistent homology to various non-linear dynamical systems, including real-world climate data, and show how these techniques can shed light on questions such as how many regimes there really are in e.g. the Euro-Atlantic sector. Future directions are also discussed.</p>

Entropy of Nanostructures

2017 ◽  
pp. 600-614
Author(s):  
Ottorino Ori ◽  
Franco Cataldo ◽  
Mihai V. Putz
Keyword(s):  
Mechanical Properties ◽  
Free Energy ◽  
Gibbs Free Energy ◽  
Defect Formation ◽  
Vacancy Concentration ◽  
Structural Evolution ◽  
Point Of View ◽  
Structural Defects ◽  
Formation Mechanisms ◽  
Topological Invariants

Recent advances in graphene studies deal with the influence of structural defects on graphene chemical, electrical, magnetic and mechanical properties. Here the complex mechanisms leading to the formation of clusters of vacancies in 2D honeycomb HD lattices are described by a pure topological point of view, aiming to correlate the variation of specific topological invariants, sensible to vacancy concentration, to the structural evolution of the defective networks driven by the topo-thermodynamical Gibbs free energy. Interesting predictions on defect formation mechanisms add details on the topological mechanisms featured by the graphenic structures with defects. Future roles of bondonic particles in defective HD materials are also envisaged.

scholarly journals Coherent structures and chaotic advection in three dimensions

2010 ◽  
Vol 654 ◽  
pp. 1-4 ◽  
Author(s):  
STEPHEN WIGGINS
Keyword(s):  
Dynamical Systems ◽  
Fluid Mechanics ◽  
Coherent Structures ◽  
Three Dimensional ◽  
Point Of View ◽  
Three Dimensions ◽  
Chaotic Advection ◽  
Periodic Flow ◽  
Two Dimensional ◽  
Time Periodic

In the 1980s the incorporation of ideas from dynamical systems theory into theoretical fluid mechanics, reinforced by elegant experiments, fundamentally changed the way in which we view and analyse Lagrangian transport. The majority of work along these lines was restricted to two-dimensional flows and the generalization of the dynamical systems point of view to fully three-dimensional flows has seen less progress. This situation may now change with the work of Pouransari et al. (J. Fluid Mech., this issue, vol. 654, 2010, pp. 5–34) who study transport in a three-dimensional time-periodic flow and show that completely new types of dynamical systems structures and consequently, coherent structures, form a geometrical template governing transport.

scholarly journals Closed braids and knot holders associated to some laser dynamical systems: A pump-modulated Nd-doped Â…ber laser

2017 ◽  
Vol 12 (12) ◽  
pp. 6894-6900
Author(s):  
Elsaued Elrifai
Keyword(s):  
Dynamical Systems ◽  
Fiber Laser ◽  
Crossing Number ◽  
Topological Invariants ◽  
Control Parameters ◽  
Linking Number ◽  
Fibered Knots ◽  
Braid Theory ◽  
Knots And Links

In this work the arising knots and links for the pump-modulated Nd-doped fiber laser is investigated. For the associated templates, some of their topological invariants, such as braid linking matrix, braid words, crossing number and linking number, are studied. It is recognized that the derived topological invariants are quietly dependent on the control parameters Using the tools of the braid theory, it is shown that pump-modulated Nd-doped fiber laser knots and links are positive fibered knots and links.

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