Mathew Penrose
- Published in print:
- 2003
- Published Online:
- September 2007
- ISBN:
- 9780198506263
- eISBN:
- 9780191707858
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198506263.003.0009
- Subject:
- Mathematics, Probability / Statistics
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) ...
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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
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.
Alexandre Dezotti and Pascale Roesch
Araceli Bonifant, Mikhail Lyubich, and Scott Sutherland (eds)
- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691159294
- eISBN:
- 9781400851317
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691159294.003.0009
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics
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 ...
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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
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.
Tim Maudlin
- Published in print:
- 2014
- Published Online:
- April 2014
- ISBN:
- 9780198701309
- eISBN:
- 9780191771613
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198701309.003.0004
- Subject:
- Philosophy, Logic/Philosophy of Mathematics
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 ...
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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
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.
Tim Maudlin
- Published in print:
- 2014
- Published Online:
- April 2014
- ISBN:
- 9780198701309
- eISBN:
- 9780191771613
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198701309.003.0002
- Subject:
- Philosophy, Logic/Philosophy of Mathematics
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 ...
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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
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.
