Search Results

PERCOLATIVE INGREDIENTS

Mathew Penrose

in Random Geometric Graphs

This chapter contains some known results on connectivity which are used later on. The notion of unicoherence of a simply-connected set is explained and extended to lattices. Peierls (counting) ... More

This chapter contains some known results on connectivity which are used later on. The notion of unicoherence of a simply-connected set is explained and extended to lattices. Peierls (counting) arguments are described for estimating the number of connected sets in the lattice, and elements of (lattice) percolation theory are described. A multiparameter ergodic theorem is given, and the basic theory of continuum percolation is described. Some of the theory of Poisson point processes are recalled, including the superposition, thinning, and scaling theorems.Less

On (non-)local connectivity of some Julia sets

Alexandre Dezotti and Pascale Roesch

Araceli Bonifant, Mikhail Lyubich, and Scott Sutherland (eds)

in Frontiers in Complex Dynamics: In Celebration of John Milnor's 80th Birthday

This chapter deals with the question of local connectivity of the Julia set of polynomials and rational maps. It discusses when the Julia set of a rational map is considered connected but not locally ... More

This chapter deals with the question of local connectivity of the Julia set of polynomials and rational maps. It discusses when the Julia set of a rational map is considered connected but not locally connected. The question of the local connectivity of the Julia set has been studied extensively for quadratic polynomials, but there is still no complete characterization of when a quadratic polynomial has a connected and locally connected Julia set. This chapter thus proposes some conjectures and develops a model of non-locally connected Julia sets in the case of infinitely renormalizable quadratic polynomials. This model presents the structure of what the post-critical set in that setting should be.Less

Closed Sets, Open Sets (Again), Connected Spaces

Tim Maudlin

in New Foundations for Physical Geometry: The Theory of Linear Structures

Chapter 3 presents the definition of a closed set in the Theory of Linear Structures, which does not correspond to the definition in standard topology. An alternative and parallel definition of an ... More

Chapter 3 presents the definition of a closed set in the Theory of Linear Structures, which does not correspond to the definition in standard topology. An alternative and parallel definition of an open set is produced and shown to be equivalent to that of Chapter 2. The definition of a connected space is given, and contrasted with the standard definition. The interior and boundary points of a set are defined.Less

Topology and Its Shortcomings

Tim Maudlin

in New Foundations for Physical Geometry: The Theory of Linear Structures

Chapter 1 reviews the structure of standard topology as an axiomatic system used to implicitly define the notion of an open set. Standard definitions of continuity, connectedness, the boundary of a ... More

Chapter 1 reviews the structure of standard topology as an axiomatic system used to implicitly define the notion of an open set. Standard definitions of continuity, connectedness, the boundary of a set, etc. are discussed. The chapter considers in particular the problems that standard topology has in characterizing the geometry of discrete spaces and finite-point spaces. There is a detailed discussion of Sierpinski spaces.Less