Characterizing Endpoints of Generalized Inverse Limits Lori Alvin Bradley University Nipissing Topology Workshop 2018 Lori Alvin (Bradley U.) Endpoints lalvin@bradley.edu 1 / 44
Motivation There has long been a push to understand the topological structure of inverse limit spaces generated by unimodal maps. Although such maps are simple to define, their inverse limits can exhibit quite complicated structures, and it is often difficult to distinguish the inverse limits associated with distinct maps. Lori Alvin (Bradley U.) Endpoints lalvin@bradley.edu 2 / 44
Motivation There has long been a push to understand the topological structure of inverse limit spaces generated by unimodal maps. Although such maps are simple to define, their inverse limits can exhibit quite complicated structures, and it is often difficult to distinguish the inverse limits associated with distinct maps. One of the things we have focused on in the study of these classical inverse limits is the set of endpoints, as they are a topological invariant. Thus, it is logical to investigate the properties of endpoints and use them to explore and distinguish inverse limits. Lori Alvin (Bradley U.) Endpoints lalvin@bradley.edu 2 / 44
Motivation There has long been a push to understand the topological structure of inverse limit spaces generated by unimodal maps. Although such maps are simple to define, their inverse limits can exhibit quite complicated structures, and it is often difficult to distinguish the inverse limits associated with distinct maps. One of the things we have focused on in the study of these classical inverse limits is the set of endpoints, as they are a topological invariant. Thus, it is logical to investigate the properties of endpoints and use them to explore and distinguish inverse limits. As we have transitioned to the study of generalized inverse limits, it remains helpful to study the collection of endpoints; however they can arise differently from set-valued functions than from continuous maps. Lori Alvin (Bradley U.) Endpoints lalvin@bradley.edu 2 / 44
Motivation In this talk, we will demonstrate how the collection of endpoints arises from set-valued inverse limits by looking at various projection mappings. Lori Alvin (Bradley U.) Endpoints lalvin@bradley.edu 3 / 44
Motivation In this talk, we will demonstrate how the collection of endpoints arises from set-valued inverse limits by looking at various projection mappings. We note that although the standard definitions of endpoints of continua are equivalent when working with chainable continua, because our inverse limits can contain triods, we must be careful which definition we work with. In fact, characterizations of endpoints under one definition may not hold in our setting under a different definition. Lori Alvin (Bradley U.) Endpoints lalvin@bradley.edu 3 / 44
Motivation In this talk, we will demonstrate how the collection of endpoints arises from set-valued inverse limits by looking at various projection mappings. We note that although the standard definitions of endpoints of continua are equivalent when working with chainable continua, because our inverse limits can contain triods, we must be careful which definition we work with. In fact, characterizations of endpoints under one definition may not hold in our setting under a different definition. After presenting several examples of endpoints for generalized inverse limits satisfying a given property, we discuss our conjecture for the characterization of endpoints and the progress/stumbling blocks in proving the conjecture. Lori Alvin (Bradley U.) Endpoints lalvin@bradley.edu 3 / 44
Generalized Inverse Limits Let X be a continuum; 2 X is the space of all non-empty, compact subsets of X . Lori Alvin (Bradley U.) Endpoints lalvin@bradley.edu 4 / 44
Generalized Inverse Limits Let X be a continuum; 2 X is the space of all non-empty, compact subsets of X . Given a function F : X → 2 X we define its graph to be the set Γ( F ) = { ( x , y ) : y ∈ F ( x ) } . Lori Alvin (Bradley U.) Endpoints lalvin@bradley.edu 4 / 44
Generalized Inverse Limits Let X be a continuum; 2 X is the space of all non-empty, compact subsets of X . Given a function F : X → 2 X we define its graph to be the set Γ( F ) = { ( x , y ) : y ∈ F ( x ) } . � � ∞ � lim − F = x ∈ X : x i − 1 ∈ F ( x i ) for all i ∈ N ← i =0 Lori Alvin (Bradley U.) Endpoints lalvin@bradley.edu 4 / 44
Generalized Inverse Limits Let X be a continuum; 2 X is the space of all non-empty, compact subsets of X . Given a function F : X → 2 X we define its graph to be the set Γ( F ) = { ( x , y ) : y ∈ F ( x ) } . � � ∞ � lim − F = x ∈ X : x i − 1 ∈ F ( x i ) for all i ∈ N ← i =0 Γ 1 = X and for all n ≥ 2 we define the projection Γ n by: � n − 1 � � Γ n = x ∈ X : x i − 1 ∈ F ( x i ) for all 1 ≤ i < n i =0 Lori Alvin (Bradley U.) Endpoints lalvin@bradley.edu 4 / 44
Inverse and Forward Limits − F is to look at the graph of F − 1 . If One of the keys to understanding lim ← Γ( F ) = { ( x , y ) : y ∈ F ( x ) } , then Γ( F − 1 ) = { ( y , x ) : y ∈ F ( x ) } . We are interested in set-valued functions F : [0 , 1] → 2 [0 , 1] such that Γ( F − 1 ) = ∪ α ∈ A Γ( f α ), where each f α : [0 , 1] → [0 , 1] is a continuous function. → F − 1 = lim lim − F = lim → ( ∪ α ∈ A f α ) ← − − So really, these techniques can be used to study the forward dynamics of set-valued functions. Lori Alvin (Bradley U.) Endpoints lalvin@bradley.edu 5 / 44
Definitions of Endpoints Bing’s Definition : The point p is an endpoint of the continuum X if for any two subcontinua H , K ⊆ X both containing p , either H ⊆ K or K ⊆ H . Lori Alvin (Bradley U.) Endpoints lalvin@bradley.edu 6 / 44
Definitions of Endpoints Bing’s Definition : The point p is an endpoint of the continuum X if for any two subcontinua H , K ⊆ X both containing p , either H ⊆ K or K ⊆ H . Lelek’s Definition The point p is an endpoint of the continuum X if and only if p is an endpoint of every arc in X that contains p . Lori Alvin (Bradley U.) Endpoints lalvin@bradley.edu 6 / 44
Definitions of Endpoints Bing’s Definition : The point p is an endpoint of the continuum X if for any two subcontinua H , K ⊆ X both containing p , either H ⊆ K or K ⊆ H . Lelek’s Definition The point p is an endpoint of the continuum X if and only if p is an endpoint of every arc in X that contains p . Miller’s Definition The point p is an endpoint in the continuum X if every irreducible continuum in X containing p is irreducible between p and some other point. Lori Alvin (Bradley U.) Endpoints lalvin@bradley.edu 6 / 44
Motivating Question Theorem (J. Kelly (2016)) Let F : X → 2 X . Suppose there exists a collection { f α } α ∈ A of continuous functions such that Γ( F − 1 ) = ∪ α ∈ A Γ( f α ) . Then for every p ∈ lim − F the following are equivalent using Bing’s ← definition of an endpoint. 1 p is an endpoint of lim − F. ← 2 ( p 0 , p 1 , . . . , p n − 1 ) is an endpoint of Γ n for infinitely many n ∈ N . 3 ( p 0 , p 1 , . . . , p n − 1 ) is an endpoint of Γ n for all n ∈ N . Note that it always follows (even with Lelek’s and Miller’s definitions) that if p is an endpoint of lim − F , then ( p 0 , . . . , p n − 1 ) is an endpoint of Γ n for all ← n ∈ N . Lori Alvin (Bradley U.) Endpoints lalvin@bradley.edu 7 / 44
Motivating Question Does Kelly’s result hold if we assume Lelek’s (Miller’s) definition of an endpoint? If not, what is the proper characterization for Lelek’s (Miller’s) definition of an endpoint in this setting? Lori Alvin (Bradley U.) Endpoints lalvin@bradley.edu 8 / 44
Our Counterexample: F Let F : [0 , 1] → 2 [0 , 1] be the set-valued function obtained by attaching the √ line segment connecting the points ( 1 2 , 1+ 5 ) and (1 , 1) to the tent map 4 √ T s with slope s = 1+ 5 . 2 Lori Alvin (Bradley U.) Endpoints lalvin@bradley.edu 9 / 44
Our Counterexample: F Let F : [0 , 1] → 2 [0 , 1] be the set-valued function obtained by attaching the √ line segment connecting the points ( 1 2 , 1+ 5 ) and (1 , 1) to the tent map 4 √ T s with slope s = 1+ 5 . 2 Lori Alvin (Bradley U.) Endpoints lalvin@bradley.edu 9 / 44
Our Counterexample: F − 1 = f ∪ g We can decompose F − 1 into the union of two functions F − 1 = f ∪ g : g f √ √ √ � − 2 x +1+ � 2 x x ≤ 1+ 5 5 x ≤ 1+ 5 √ , √ , 4 4 1+ 5 1+ 5 f ( x ) = g ( x ) = √ √ √ √ 2 x +1 − 5 x ≥ 1+ 5 2 x +1 − 5 x ≥ 1+ 5 √ . √ . 4 4 3 − 5 3 − 5 Lori Alvin (Bradley U.) Endpoints lalvin@bradley.edu 10 / 44
Our Counterexample: Γ 2 (1 , 1) √ ( 1+ 5 , 1 2 ) 4 (0 , 0) (0 , 1) Lori Alvin (Bradley U.) Endpoints lalvin@bradley.edu 11 / 44
Our Counterexample: Γ 3 √ √ ( 3 5 − 3 , 1+ 5 , 1 (1 , 1 , 1) 2 ) 4 4 √ √ √ √ ( 1+ 5 , 1 5 − 1 ( 1+ 5 , 1 2 , 5 − 5 ) ) 2 , 4 4 4 4 √ √ 5 − 1 , 1+ 5 , 1 ( 2 ) 4 4 (0 , 0 , 0) (0 , 1 , 1) (0 , 0 , 1) Lori Alvin (Bradley U.) Endpoints lalvin@bradley.edu 12 / 44
Our Counterexample: Γ 4 Lori Alvin (Bradley U.) Endpoints lalvin@bradley.edu 13 / 44
Recommend
More recommend
