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A projective Fra ss 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
Aristotelis Panagiotopoulos, joint with S.Solecki
UIUC
6th Cornell Conference on Analysis, Probability, and Mathematical Physics on Fractals
The Menger curve The Fra¨ ıss´ e method A projective Fra¨ ıss´ e presentation of the Menger curve
Are they homeomorphic?
The definition/construction of such model should:
The definition/construction of such model should:
The definition/construction of such model should:
Theorem (Characterization, Bestvina (see also Anderson))
M is the unique path-connected, locally path-connected, one dimensional, compact space with the disjoint-arcs property.
The definition/construction of such model should:
Theorem (Characterization, Bestvina (see also Anderson))
M is the unique path-connected, locally path-connected, one dimensional, compact space with the disjoint-arcs property.
The definition/construction of such model should:
Theorem (Characterization, Bestvina (see also Anderson))
M is the unique path-connected, locally path-connected, one dimensional, compact space with the disjoint-arcs property.
Theorem (Homogeneity property, Anderson)
Let a1, . . . , an and b1, . . . , bn be two sequences of pairwise distinct points of M. Then there is a self-homeomomorphism φ ∈ Homeo(M) with φ(a1) = b1, . . . , φ(an) = bn.
The Menger curve The Fra¨ ıss´ e method A projective Fra¨ ıss´ e presentation of the Menger curve
(Q, ≤) The random graph (G, R) The random graph on Graphsp(N), where 0 < p < 1.
(Q, ≤) The random graph (G, R) The random graph on Graphsp(N), where 0 < p < 1. KLO KGraphs finite linear orders finite graphs with embeddings with embeddings
(Q, ≤) The random graph (G, R) The random graph on Graphsp(N), where 0 < p < 1. KLO KGraphs 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.
Start with a class of finite structures and embeddings. K
Start with a class of finite structures and embeddings. Close under countable direct limits. K Kω
Start with a class of finite structures and embeddings. Close under countable direct limits. K Kω If K is a Fra¨ ıss´ e class,
Start with a class of finite structures and embeddings. Close under countable direct limits. K Kω If K is a Fra¨ ıss´ e class, in particular if: A C B
Start with a class of finite structures and embeddings. Close under countable direct limits. K Kω If K is a Fra¨ ıss´ e class, in particular if: A C B D
Start with a class of finite structures and embeddings. Close under countable direct limits. K Kω If K is a Fra¨ ıss´ e class, in particular if: A C B D We can form a “generic” sequence:
Start with a class of finite structures and embeddings. Close under countable direct limits. K Kω If K is a Fra¨ ıss´ e class, in particular if: A C B D We can form a “generic” sequence: A0 A1 . . . Am . . .
Start with a class of finite structures and embeddings. Close under countable direct limits. K Kω If K is a Fra¨ ıss´ e class, in particular if: A C B D We can form a “generic” sequence: A0 A1 . . . Am . . . B
Start with a class of finite structures and embeddings. Close under countable direct limits. K Kω If K is a Fra¨ ıss´ e class, in particular if: A C B D We can form a “generic” sequence: A0 A1 . . . Am . . . B An
Start with a class of finite structures and embeddings. Close under countable direct limits. K Kω If K is a Fra¨ ıss´ e class, in particular if: A C B D We can form a “generic” sequence: A0 A1 . . . Am . . . B An . . . M
Start with a class of finite structures and epimorphisms. Close under inverse limits limits of countable length. K Kω If K is a projective Fra¨ ıss´ e class, in particular if: A C B D We can form a “generic” inverse sequence: A0 A1 . . . Am . . . B An . . . M
The Menger curve The Fra¨ ıss´ e method A projective Fra¨ ıss´ e presentation of the Menger curve
❆ ❇ ❆ ❇ ❆ ❆ ❇ ❇ ❇ ❆ ❆ ❆ ❆
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.
❇ ❆ ❇ ❆ ❆ ❇ ❇ ❇ ❆ ❆ ❆ ❆
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 ❇. ❇ ❆ ❆ ❆ ❆
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.
Theorem (P., Solecki)
K is a projective Fra¨ ıss´ e class.
Theorem (P., Solecki)
K is a projective Fra¨ ıss´ e class. A C B D
In particular, K has a projective Fra¨ ıss´ e limit M.
In particular, K has a projective Fra¨ ıss´ e limit M. A0 A1 . . . Am . . .
In particular, K has a projective Fra¨ ıss´ e limit M. A0 A1 . . . Am . . . B
In particular, K has a projective Fra¨ ıss´ e limit M. A0 A1 . . . Am . . . B An
In particular, K has a projective Fra¨ ıss´ e limit M. A0 A1 . . . Am . . . B An . . . M
In particular, K has a projective Fra¨ ıss´ e limit M. A0 A1 . . . Am . . . B An . . . M
a ∈ M has at most one R-neighbor.
In particular, K has a projective Fra¨ ıss´ e limit M. A0 A1 . . . Am . . . B An . . . M
a ∈ M has at most one R-neighbor. We call M the pre-Menger space...
In particular, K has a projective Fra¨ ıss´ e limit M. A0 A1 . . . Am . . . B An . . . M
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.
❆ ❆ ❇ ❆ ❇ ❆ ❆ ❆ ❆
From standard Fra¨ ıss´ e theory we get the following projective homogeneity property for M.
Theorem
Let f , g be two continuous connected epimorphism from M to some element ❆ of K. Then there is φ ∈ Aut(M), with f = g ◦ φ. ❆ ❇ ❆ ❇ ❆ ❆ ❆ ❆
From standard Fra¨ ıss´ e theory we get the following projective homogeneity property for M.
Theorem
Let f , g be two continuous connected epimorphism from M to some element ❆ of K. Then there is φ ∈ Aut(M), with f = g ◦ φ. We recover the classical homogeneity result of Anderson for M using the projective homogeneity of M and combinatorial properties of graphs in K such as:
◮ ❆, ❇ ∈ K implies ❆ × ❇ ∈ K; ◮ subdividing an edge e of ❆ ∈ K results to a graph ❆e ∈ K; ◮ introducing a “twin vertex” v′ of some graph ❆ ∈ K results to
a graph ❆v ∈ K;
◮ It is much easier to work with a zero-dimension structure such
as the pre-Menger space M than directly with M. At the same time M collects all relevant combinatorics regarding the dynamics of M;
◮ The Fra¨
ıss´ e approach is unifying. There are many other structures in model theory, topology and analysis which share similar properties to M and M.
◮ The fact that M is a projective Fra¨
ıss´ e limit of the particular class K of connected graphs renders M a very important space for understanding the category of 1-dimensional compact metrizable spaces.