Topological Dynamics and Topological Data Analysis

Preface
Contents
An Overview of Unimodal Inverse Limit Spaces
1 Introduction
2 Definitions and Notation
2.1 Chainability
2.2 Symbolic Dynamics
3 End points and Folding Points
4 Composants
5 Ingram Conjecture
References
Dimension Theory of Some Non-Markovian Repellers Part I: A Gentle Introduction
1 Introduction
2 Barnsley's Skew Product Maps
3 The Hausdorff and Box Dimensions
3.1 Fractal Dimensions of Sets
3.2 Hausdorff Dimension of Measures
4 Self-similar Sets
4.1 Self-similar Measures
5 Dimension of the Self-conformal Sets and Measures When OSC Holds
5.1 Hausdorff Dimension of Self-conformal Sets When OSC Is Assumed
5.2 Hausdorff Dimension of an Invariant Measure and Lyapunov Exponents
6 The Hausdorff Dimension of Self-affine Sets
6.1 Singular Value Function, Affinity Dimension, Falconer's Theorem
7 Ergodic Measures for a Self-affine IFS
7.1 Self-affine Measures
8 Ergodic Measures for Barnsley's Skew Product Maps
9 Hofbauer's Pressure
9.1 Pressure and Markov Sets
9.2 Conformal Measure and Small Cylinders
10 The Dimension of Barnsley's Repellers
10.1 The Basic Definitions
10.2 Diagonal and Essentially Non-diagonal System
10.3 Markov Pressure and Hofbauer Pressure
10.4 The Main Results
References
Dimension Theory of Some Non-Markovian Repellers Part II: Dynamically Defined Function Graphs
1 The Weierstrass and Takagi Functions
2 Dynamically Defined Function Graphs
3 Markovian Fractal Interpolation Functions
4 Continuous Generalized Takagi Functions
References
Iterated Function Systems—A Topological Approach. Attractors
1 Introduction
2 IFS
3 Basins and Pointwise Basins
4 Point-Fibred and Strongly Fibred Attractors
5 Fast Basins
6 Homoclinic Attractors Versus Fast Basins
References
Zero-Dimensional Covers of Dynamical Systems
1 Introduction
2 Preliminaries
3 Finite-to-one Zero-Dimensional Covers
4 Zero-Dimensional Decompositions of Dynamical Systems
References
Chaotic Continua in Chaotic Dynamical Systems
1 Introduction
2 Preliminaries
3 Free Tracing Property by Free G-chains
4 Positive Topological Entropy
5 Characterizations of Indecomposable Continua and Free Tracing Property
References
Mandelpinski Necklaces in the Parameter Planes of Rational Maps
1 Introduction
2 Elementary Mapping Properties
3 Some Special Cases
4 Rings in Dynamical Plane
5 Rings in Parameter Plane
6 The Special Case n = 2
References
Some Examples of Hypercyclic Operators and Universal Sequences of Operators
1 Introduction
2 Entire Functions
3 Hilbert Spaces of Analytic Functions
References
Some Basic Properties of Hypercyclic Operators
1 Introduction
2 Common Hypercyclic Vectors
3 Hypercyclic Extension
References
The Testing Ground of Weighted Shift Operators for Hypercyclicity
1 Introduction
2 Weak Topologies
3 Orbital Limit Points
4 Hypercyclic Factorization
References
On δ-deformations of Polygonal Dendrites
1 Preliminaries
1.1 Self-similar Sets
1.2 Dendrites
1.3 Contractible Polygonal Systems
1.4 Main Parameters of a Contractible Polygonal System
2 Generalized Polygonal Systems
3 δ-deformations of Contractible Polygonal Systems
4 Parameter Matching Theorem
5 Main Theorem
5.1 Some Assumptions
References
General Position Theorem and Its Applications
1 Introduction
2 Definitions and Notations
3 General Position Theorem
4 Application of General Position Theorem to Self-similar Sets
4.1 Moving Subpieces Apart from Each Other.
4.2 Non-empty Overlaps of Prescribed Type
References
The nth Itarate of a Map with Dense Orbit
1 Introduction
1.1 The General Separation Theorem
References
Periodic Points of N-Dimensional Toral Automorphisms
1 Introduction
2 Basic Results
3 Main Results
4 Summary
References
Julia Sets in Topological Spaces
1 Introduction
2 Basin of Attractors to a Point
3 Fatou and Julia Sets in General Topological Space
4 Locally Compact and T2 Space
5 Julia Sets
6 Conclusion
References
Julia Set of Some Graphs Using Independence Polynomials
1 Introduction
2 Preliminaries
3 Computing the Independence Polynomial
4 Complete Graph
4.1 Relation of Hausdorff Dimension and Energy of J(I(G,z)) of Complete Graph
4.2 Results on Complete Graph
5 Mandelbrot Graph
5.1 Classification of Mandelbrot Graphs
5.2 Julia Set of Reduced Independence Polynomial of Some Graphs
6 Independence Polynomial of Second Degree of Graphs
7 Conclusion
References
An Introduction to the Notion of Natural Pseudo-distance in Topological Data Analysis
1 Introduction
1.1 Related Literature and Historical Notes
2 The Definition of dG
2.1 The Role of dG in Topological Data Analysis
3 Theoretical Results About dG
3.1 Optimal Homeomorphisms
4 A Link Between dG and Persistent Homology
4.1 Persistent Homology
4.2 Group Equivariant Non-expansive Operators
4.3 Persistent Homology as a Tool to Get Lower Bounds for dG
5 An Open Problem
References
A Brief Introduction to Multidimensional Persistent Betti Numbers
1 Introduction
2 PBNs: Definitions and First Properties
2.1 1-Dimensional PBNs
2.2 The Foliation Method
3 Evaluating the Distance Between Multidimensional PBNs
3.1 Underlying Theoretical Results
4 An Algorithm for Approximating Dmatch
4.1 The 2-Dimensional Case
5 Beyond the Multidimensional Matching Distance Dmatch
5.1 The Coherent Matching Distance for 2-Dimensional Persistent Betti Numbers
References
Some New Methods to Build Group Equivariant Non-expansive Operators in TDA
1 Introduction
2 Our Mathematical Model
3 Building New GENEOs by Means of Power Means
3.1 Examples
4 Series of GENEOs
5 Conclusions
References
Topological Stability of the Hippocampal Spatial Map and Synaptic Transience
1 Introduction
2 Overview of the Results
2.1 Decaying Clique Complexes
2.2 Finite Latency Complexes
3 Discussion
References
Intuitionistic Fuzzy Graph Morphological Topology
1 Introduction
2 Preliminaries
3 Morphological Topology
4 Conclusion
References
Some Properties of the Bitopological Space Associated With the 3-Uniform Semigraph of Cycle Graph
1 Introduction
2 3-Uniform Semigraph of a Cycle Graph
3 Bitopological Space
4 Bitopological Space on Cm,1
5 Some Topological Properties of (V,τE',τM')
6 Conclusion
References
Hypergraph Topology
1 Introduction
2 Preliminaries
3 Hypergraph Topological Space
4 Closed Set
5 Continuity in Hypergraph Topology
6 Metric in Hypergraph Topology
7 Homeomorphism in Hypergraphs
7.1 Subdivision
7.2 Smoothing Out
8 Conclusion
References