Smyths finite approximations University of Birmingham Chris Good, - PowerPoint PPT Presentation

Smyth’s paper Smyth’s finite approximations University of Birmingham Chris Good, Mate Puljiz, David Parker, Jonathan E. Rowe Sheffield, April 4, 2014

Motivation Smyth’s paper Inverse limit construction The End Universality A space is universal for a certain class of spaces if all the spaces from that class can be realised as the quotients of the initial space. It is a well know fact that ω ω is the universal Polish space (complete, metric, separable) and Cantor set is universal for the separable compact metric spaces. Eg, the binary expansion of a real number { 0 , 1 } ω → [0 , 1] .

Motivation Smyth’s paper Inverse limit construction The End Why this may be interesting to us? Going back to the commuting diagram, ˜ f U U Ξ Ξ f X X starting with X we are provided with the map Ξ . If we could lift the map f to a map ˜ f on U then finding CGs of ( X , f ) corresponds to finding CGs of ( U , ˜ f ) that are constant on the level sets of Ξ .

Motivation Smyth’s paper Inverse limit construction The End Problems? Not every Ξ will work. It needs to be tuned with respect to the given f . Eg, the aforementioned binary expansion doesn’t allow lifting of the map x �→ 3 2 x (hence, 1 3 �→ 1 2 ). Can we always find Ξ that allows lifting?

Motivation Smyth’s paper Inverse limit construction The End Construction Inverse system f 2 , 1 f 3 , 2 f 4 , 3 X 1 ← − X 2 ← − X 3 ← − X 4 . . . Inverse limit � lim − X i = { ( x i ) i ∈ X i | f n +1 , n ( x n +1 ) ∈ x n } . ← i

Motivation Smyth’s paper Inverse limit construction The End Construction Given a space X , take a sequence ǫ n ↓ 0 , construct finite (or countable) families of open sets C n st. diameter of each set in C n is less then ǫ n sets from each C n cover X each set from C n +1 is contained in the unique set from C n * image under f of each set in C n +1 is contained in the unique set from C n We think of the sets in C n as symbols. The binding maps are inclusions. Assuming that X has no isolated points, the inverse − C n is homeomorphic to the Cantor set or ω ω depending limit lim ← on whether X was compact or not. And f induces a lifted map ˜ f .

Motivation Smyth’s paper Inverse limit construction The End A different perspective Having constructed U = lim − C n it is clear that the projection Ξ should map ( x i ) i to � x i where x i ∈ C i is a decreasing sequence of ← sets. Also, the map is far from being 1-1 since sets in C i are not disjoint. Rather than constructing an universal covering space and then quotienting it to obtain initial space, Smyth encodes additional information within each finite approximation C i in form of a directed graph with the vertices set C i and with an edge connecting each pair of intersecting sets. This gives rise to a graph structure on the inverse limit. We put edge between ( x i ) i and ( y i ) i iff there is an edge connecting x i and y i for every i .

Motivation Smyth’s paper Inverse limit construction The End A different perspective Note that ( x i ) i and ( y i ) i are connected by an edge iff they represent the same point in X . Hence, space X is now given as an quotient with respect to this relation. The triplet ( X , τ, R ) is a topological graph which captures properties of both topological spaces and graphs. For a topological space one can take R to be the trivial (diagonal) relation, and for a finite graph, the topology is trivial (discrete) and the relation is one given by the graph. Idea is that on top of having a topology, the relation R tells us which are our the closest neighbours of a given point. Using this one can construct a coarser topology on X consisting of only those open sets such that they have their close neighbours inside that set (R-invariant). The intermediate step is taking all such (not just open) sets and that family needs not to form a topology but it is a neighbourhood space .

Motivation Smyth’s paper Inverse limit construction The End This essentially means that graphs are represented as non Hausdorff finite spaces and their inverse limit is a non Hasdorff space whose T 0 quotient is initial space. ǫ - approximation by a finite graph then amounts to taking C n consisting of subsets having small enough diameters. The trouble is that our lifted function is not mapping on C n . It is mapping from C n +1 to C n which means that mapping is increasing our error. After n iterations we will not know where in the space we are.

Motivation Smyth’s paper Inverse limit construction The End The End! Thank you!