A projective Fra ss e presentation of the Menger curve - PowerPoint PPT Presentation

A projective Fra¨ ıss´ e presentation of the Menger curve Aristotelis Panagiotopoulos, joint with S.Solecki UIUC 6th Cornell Conference on Analysis, Probability, and Mathematical Physics on Fractals

Table of Contents The Menger curve The Fra¨ ıss´ e method A projective Fra¨ ıss´ e presentation of the Menger curve

The Menger curve M

The Menger curve M Are they homeomorphic?

A Canonical model of M The definition/construction of such model should:

A Canonical model of M The definition/construction of such model should: • reflect the essential combinatorics of the topology of M ;

A Canonical model of M The definition/construction of such model should: • reflect the essential combinatorics of the topology of M ; Theorem (Characterization, Bestvina (see also Anderson)) M is the unique path-connected, locally path-connected, one dimensional, compact space with the disjoint-arcs property.

A Canonical model of M The definition/construction of such model should: • reflect the essential combinatorics of the topology of M ; Theorem (Characterization, Bestvina (see also Anderson)) M is the unique path-connected, locally path-connected, one dimensional, compact space with the disjoint-arcs property. • make various important properties of M easily accessible.

A Canonical model of M The definition/construction of such model should: • reflect the essential combinatorics of the topology of M ; Theorem (Characterization, Bestvina (see also Anderson)) M is the unique path-connected, locally path-connected, one dimensional, compact space with the disjoint-arcs property. • make various important properties of M easily accessible. Theorem (Homogeneity property, Anderson) Let a 1 , . . . , a n and b 1 , . . . , b n be two sequences of pairwise distinct points of M . Then there is a self-homeomomorphism φ ∈ Homeo ( M ) with φ ( a 1 ) = b 1 , . . . , φ ( a n ) = b n .

Table of Contents The Menger curve The Fra¨ ıss´ e method A projective Fra¨ ıss´ e presentation of the Menger curve

Some model theoretic background

Some model theoretic background ( Q , ≤ ) The random graph ( G , R ) The random graph on Graphs p ( N ), where 0 < p < 1.

Some model theoretic background ( Q , ≤ ) The random graph ( G , R ) The random graph on Graphs p ( N ), where 0 < p < 1. K LO K Graphs finite linear orders finite graphs with embeddings with embeddings

Some model theoretic background ( Q , ≤ ) The random graph ( G , R ) The random graph on Graphs p ( N ), where 0 < p < 1. K LO K Graphs finite linear orders finite graphs with embeddings with embeddings Fra¨ ıss´ e observed that if a class of finite structures K satisfies certain properties (we say K is a Fra¨ ıss´ e class) then there is a unique countable structure N that is saturated with configurations coming from K . This structure N is called the Fra¨ ıss´ e limit of K .

Fra¨ ıss´ e theory

Fra¨ ıss´ e theory Start with a class of finite structures and embeddings. K

Fra¨ ıss´ e theory Start with a class of finite structures and embeddings. Close under countable direct limits. K ω K

Fra¨ ıss´ e theory Start with a class of finite structures and embeddings. Close under countable direct limits. K ω K If K is a Fra¨ ıss´ e class,

Fra¨ ıss´ e theory Start with a class of finite structures and embeddings. Close under countable direct limits. K ω K If K is a Fra¨ ıss´ e class, in particular if: B A C

Fra¨ ıss´ e theory Start with a class of finite structures and embeddings. Close under countable direct limits. K ω K If K is a Fra¨ ıss´ e class, in particular if: B D A C

Fra¨ ıss´ e theory Start with a class of finite structures and embeddings. Close under countable direct limits. K ω K If K is a Fra¨ ıss´ e class, in particular if: B D A C We can form a “generic” sequence:

Fra¨ ıss´ e theory Start with a class of finite structures and embeddings. Close under countable direct limits. K ω K If K is a Fra¨ ıss´ e class, in particular if: B D A C We can form a “generic” sequence: . . . . . . A 0 A 1 A m

Fra¨ ıss´ e theory Start with a class of finite structures and embeddings. Close under countable direct limits. K ω K If K is a Fra¨ ıss´ e class, in particular if: B D A C We can form a “generic” sequence: B . . . . . . A 0 A 1 A m

Fra¨ ıss´ e theory Start with a class of finite structures and embeddings. Close under countable direct limits. K ω K If K is a Fra¨ ıss´ e class, in particular if: B D A C We can form a “generic” sequence: B . . . . . . A 0 A 1 A m A n

Fra¨ ıss´ e theory Start with a class of finite structures and embeddings. Close under countable direct limits. K ω K If K is a Fra¨ ıss´ e class, in particular if: B D A C We can form a “generic” sequence: B . . . . . . . . . A 0 A 1 A m A n M

Projective Fra¨ ıss´ e theory (Irwin, Solecki) Start with a class of finite structures and epimorphisms. Close under inverse limits limits of countable length. K ω K If K is a projective Fra¨ ıss´ e class, in particular if: B D A C We can form a “generic” inverse sequence: B . . . . . . . . . A 0 A 1 A m A n M

Table of Contents The Menger curve The Fra¨ ıss´ e method A projective Fra¨ ıss´ e presentation of the Menger curve

❆ ❇ ❆ ❇ ❆ ❆ ❇ ❇ ❇ ❆ ❆ ❆ ❆ The class K of connected (reflexive) graphs

❇ ❆ ❇ ❆ ❆ ❇ ❇ ❇ ❆ ❆ ❆ ❆ The class K of connected (reflexive) graphs A graph ❆ = ( A , R ) is a set A together with a subset R of A × A , with: ◮ R ( a , a ′ ) = ⇒ R ( a ′ , a ), i.e. R is symmetric; ◮ R ( a , a ) always holds, i.e R is reflexive.

❇ ❆ ❆ ❆ ❆ The class K of connected (reflexive) graphs A graph ❆ = ( A , R ) is a set A together with a subset R of A × A , with: ◮ R ( a , a ′ ) = ⇒ R ( a ′ , a ), i.e. R is symmetric; ◮ R ( a , a ) always holds, i.e R is reflexive. An epimorphism f : ❇ → ❆ is a map f : B → A , with: ◮ f is surjective; ◮ R ( b , b ′ ) in ❇ implies R ( f ( b ) , f ( b ′ )) in ❆ , i.e. f is a homomorphism; ◮ if R ( a , a ′ ) in ❆ , then there is b , b ′ in ❇ with a ′ = f ( b ) , a = f ( b ′ ), and R ( b , b ′ ) in ❇ .

The class K of connected (reflexive) graphs A graph ❆ = ( A , R ) is a set A together with a subset R of A × A , with: ◮ R ( a , a ′ ) = ⇒ R ( a ′ , a ), i.e. R is symmetric; ◮ R ( a , a ) always holds, i.e R is reflexive. An epimorphism f : ❇ → ❆ is a map f : B → A , with: ◮ f is surjective; ◮ R ( b , b ′ ) in ❇ implies R ( f ( b ) , f ( b ′ )) in ❆ , i.e. f is a homomorphism; ◮ if R ( a , a ′ ) in ❆ , then there is b , b ′ in ❇ with a ′ = f ( b ) , a = f ( b ′ ), and R ( b , b ′ ) in ❇ . Definition Let K be the collection of all finite, connected graphs together with all connected epimorpisms between them. An epimorphism f : ❇ → ❆ is connected, if for every connected subgraph ❆ 0 of ❆ we have that f − 1 ( ❆ 0 ) is connected.

The class K of connected (reflexive) graphs Theorem (P., Solecki) K is a projective Fra¨ ıss´ e class.

The class K of connected (reflexive) graphs Theorem (P., Solecki) K is a projective Fra¨ ıss´ e class. B D A C

e limit of K The projective Fra¨ ıss´ In particular, K has a projective Fra¨ ıss´ e limit M .

e limit of K The projective Fra¨ ıss´ In particular, K has a projective Fra¨ ıss´ e limit M . . . . . . . A 0 A 1 A m

e limit of K The projective Fra¨ ıss´ In particular, K has a projective Fra¨ ıss´ e limit M . B . . . . . . A 0 A 1 A m

e limit of K The projective Fra¨ ıss´ In particular, K has a projective Fra¨ ıss´ e limit M . B . . . . . . A 0 A 1 A m A n

e limit of K The projective Fra¨ ıss´ In particular, K has a projective Fra¨ ıss´ e limit M . B . . . . . . . . . A 0 A 1 A m A n M

e limit of K The projective Fra¨ ıss´ In particular, K has a projective Fra¨ ıss´ e limit M . B . . . . . . . . . A 0 A 1 A m A n M • M is a 0-dimensional, compact, metrizable space. • The relation R on M is a compact graph relation and each a ∈ M has at most one R -neighbor.

e limit of K The projective Fra¨ ıss´ In particular, K has a projective Fra¨ ıss´ e limit M . B . . . . . . . . . A 0 A 1 A m A n M • M is a 0-dimensional, compact, metrizable space. • The relation R on M is a compact graph relation and each a ∈ M has at most one R -neighbor. We call M the pre-Menger space...

e limit of K The projective Fra¨ ıss´ In particular, K has a projective Fra¨ ıss´ e limit M . B . . . . . . . . . A 0 A 1 A m A n M • M is a 0-dimensional, compact, metrizable space. • The relation R on M is a compact graph relation and each a ∈ M has at most one R -neighbor. We call M the pre-Menger space... Theorem (P., Solecki) The Menger space M is the quotient M / R of M under the equivalence relation R.

❆ ❆ ❇ ❆ ❇ ❆ ❆ ❆ ❆ Homogeneity of M and M