Persistent Markov Partitions in Complex Dynamics Mary Rees University of Liverpool Postgraduate Conference in Complex Dynamics, 11-13 March 2015, De Morgan House, London
For some specific dynamical systems, the use of Markov partitions in dynamics dates back to the early twentieth century.
For some specific dynamical systems, the use of Markov partitions in dynamics dates back to the early twentieth century. Markov partitions for hyperbolic dynamical systems were developed systematically by Rufus Bowen (and others) in the 1960’s and ’70’s.
For some specific dynamical systems, the use of Markov partitions in dynamics dates back to the early twentieth century. Markov partitions for hyperbolic dynamical systems were developed systematically by Rufus Bowen (and others) in the 1960’s and ’70’s. Because interest then was concentrated on invertible dynamical systems, the development was primarily for these.
For some specific dynamical systems, the use of Markov partitions in dynamics dates back to the early twentieth century. Markov partitions for hyperbolic dynamical systems were developed systematically by Rufus Bowen (and others) in the 1960’s and ’70’s. Because interest then was concentrated on invertible dynamical systems, the development was primarily for these. There is surprisingly little early literature on the simpler case of expanding dynamical systems.
Partitions Since we are interested in partitions of topological spaces, our definition is as follows.
Partitions Since we are interested in partitions of topological spaces, our definition is as follows. A partition P of a topological space X satisfies
Partitions Since we are interested in partitions of topological spaces, our definition is as follows. A partition P of a topological space X satisfies 1. P = { P i : 1 ≤ i ≤ r } for some integer r ≥ 1, where P i ⊂ X is closed for 1 ≤ i ≤ r ,
Partitions Since we are interested in partitions of topological spaces, our definition is as follows. A partition P of a topological space X satisfies 1. P = { P i : 1 ≤ i ≤ r } for some integer r ≥ 1, where P i ⊂ X is closed for 1 ≤ i ≤ r , 2. X = ∪ r i = 1 P i ,
Partitions Since we are interested in partitions of topological spaces, our definition is as follows. A partition P of a topological space X satisfies 1. P = { P i : 1 ≤ i ≤ r } for some integer r ≥ 1, where P i ⊂ X is closed for 1 ≤ i ≤ r , 2. X = ∪ r i = 1 P i , 3. int ( P i ) ∩ int ( P j ) = ∅ for i � = j , 1 ≤ i , j ≤ r ,
Partitions Since we are interested in partitions of topological spaces, our definition is as follows. A partition P of a topological space X satisfies 1. P = { P i : 1 ≤ i ≤ r } for some integer r ≥ 1, where P i ⊂ X is closed for 1 ≤ i ≤ r , 2. X = ∪ r i = 1 P i , 3. int ( P i ) ∩ int ( P j ) = ∅ for i � = j , 1 ≤ i , j ≤ r , 4. P i = int ( P i ) for 1 ≤ i ≤ r .
Partitions Since we are interested in partitions of topological spaces, our definition is as follows. A partition P of a topological space X satisfies 1. P = { P i : 1 ≤ i ≤ r } for some integer r ≥ 1, where P i ⊂ X is closed for 1 ≤ i ≤ r , 2. X = ∪ r i = 1 P i , 3. int ( P i ) ∩ int ( P j ) = ∅ for i � = j , 1 ≤ i , j ≤ r , 4. P i = int ( P i ) for 1 ≤ i ≤ r .
The intersection of any finite set of open dense sets is itself open and dense, and the union of any finite set of nowhere dense sets is nowhere dense. So from any set P which satisfies the first three conditions we can find a set of (possibly smaller) sets which satisfies all four conditions, simply by replacing P by { int ( P ) : P ∈ P} .
The intersection of any finite set of open dense sets is itself open and dense, and the union of any finite set of nowhere dense sets is nowhere dense. So from any set P which satisfies the first three conditions we can find a set of (possibly smaller) sets which satisfies all four conditions, simply by replacing P by { int ( P ) : P ∈ P} . A set satisfying the first two conditions is a covering of X by closed sets. If P is a covering by closed sets then we can make a set of sets Q satisfying condition 3 also by P i 1 ∩ P i 2 ∩ · · · ∩ P i s : Q = int ( P i 1 ) ∩ · · · ∩ Int ( P i s ) � = ∅ , s maximal
The intersection of any finite set of open dense sets is itself open and dense, and the union of any finite set of nowhere dense sets is nowhere dense. So from any set P which satisfies the first three conditions we can find a set of (possibly smaller) sets which satisfies all four conditions, simply by replacing P by { int ( P ) : P ∈ P} . A set satisfying the first two conditions is a covering of X by closed sets. If P is a covering by closed sets then we can make a set of sets Q satisfying condition 3 also by P i 1 ∩ P i 2 ∩ · · · ∩ P i s : Q = int ( P i 1 ) ∩ · · · ∩ Int ( P i s ) � = ∅ , s maximal
Any Hausdorff limit of a sequence of coverings by closed sets is also a covering by closed sets. But the Hausdorff limit of a sequence of partitions might not be a partition.
Any Hausdorff limit of a sequence of coverings by closed sets is also a covering by closed sets. But the Hausdorff limit of a sequence of partitions might not be a partition. If P and Q are partitions of X then P ∨ Q is also a partition of X where P ∨ Q = { int ( P ) ∩ int ( Q ) : P ∈ P , Q ∈ Q} .
The Markov property We shall say that a partition or covering P = { P i : 1 ≤ i ≤ r } of X , is Markov respect to a map f : X → X if whenever int ( P i ) ∩ f ( P j ) � = ∅ then
The Markov property We shall say that a partition or covering P = { P i : 1 ≤ i ≤ r } of X , is Markov respect to a map f : X → X if whenever int ( P i ) ∩ f ( P j ) � = ∅ then 1. int ( P i ) ⊂ f ( int ( P j )) and
The Markov property We shall say that a partition or covering P = { P i : 1 ≤ i ≤ r } of X , is Markov respect to a map f : X → X if whenever int ( P i ) ∩ f ( P j ) � = ∅ then 1. int ( P i ) ⊂ f ( int ( P j )) and 2. int ( P i ) ∩ f ( ∂ P j ) = ∅ .
The Markov property We shall say that a partition or covering P = { P i : 1 ≤ i ≤ r } of X , is Markov respect to a map f : X → X if whenever int ( P i ) ∩ f ( P j ) � = ∅ then 1. int ( P i ) ⊂ f ( int ( P j )) and 2. int ( P i ) ∩ f ( ∂ P j ) = ∅ . This is not a useful definition if f is invertible.
The Markov property We shall say that a partition or covering P = { P i : 1 ≤ i ≤ r } of X , is Markov respect to a map f : X → X if whenever int ( P i ) ∩ f ( P j ) � = ∅ then 1. int ( P i ) ⊂ f ( int ( P j )) and 2. int ( P i ) ∩ f ( ∂ P j ) = ∅ . This is not a useful definition if f is invertible. If int ( P i ) is connected then 2 implies 1.
The Markov property We shall say that a partition or covering P = { P i : 1 ≤ i ≤ r } of X , is Markov respect to a map f : X → X if whenever int ( P i ) ∩ f ( P j ) � = ∅ then 1. int ( P i ) ⊂ f ( int ( P j )) and 2. int ( P i ) ∩ f ( ∂ P j ) = ∅ . This is not a useful definition if f is invertible. If int ( P i ) is connected then 2 implies 1. If f is injective on P i then 1 implies 2.
The Markov property We shall say that a partition or covering P = { P i : 1 ≤ i ≤ r } of X , is Markov respect to a map f : X → X if whenever int ( P i ) ∩ f ( P j ) � = ∅ then 1. int ( P i ) ⊂ f ( int ( P j )) and 2. int ( P i ) ∩ f ( ∂ P j ) = ∅ . This is not a useful definition if f is invertible. If int ( P i ) is connected then 2 implies 1. If f is injective on P i then 1 implies 2. Given a Markov covering, the partition constructed from the process outlined above is Markov.
Expanding maps The definition of Markov partition given is particularly useful for expanding maps of compact metric spaces. A map f : ( X , d ) → ( X , d ) is expanding if there is λ > 1 and δ > 0 such that d ( f ( x ) , f ( y )) ≥ λ d ( x , y ) whenever x , y ∈ X with d ( x , y ) ≤ δ .
Expanding maps The definition of Markov partition given is particularly useful for expanding maps of compact metric spaces. A map f : ( X , d ) → ( X , d ) is expanding if there is λ > 1 and δ > 0 such that d ( f ( x ) , f ( y )) ≥ λ d ( x , y ) whenever x , y ∈ X with d ( x , y ) ≤ δ . An expanding map is locally injective.
Expanding maps The definition of Markov partition given is particularly useful for expanding maps of compact metric spaces. A map f : ( X , d ) → ( X , d ) is expanding if there is λ > 1 and δ > 0 such that d ( f ( x ) , f ( y )) ≥ λ d ( x , y ) whenever x , y ∈ X with d ( x , y ) ≤ δ . An expanding map is locally injective. An expanding map of a compact space is boundedly finite-to-one.
Let P = { P i : 1 ≤ i ≤ r } .
Let P = { P i : 1 ≤ i ≤ r } . If f : ( X , d ) → ( X , d ) is expanding on balls of diameter 3 δ 0 , and the sets in P have diameter ≤ δ 0 , then P is Markov for f if, for all i and j , x ∈ int ( P i ) ∩ f − 1 ( P j ) ∧ d ( x , y ) < δ 0 ∧ y ∈ f − 1 ( P j )
Let P = { P i : 1 ≤ i ≤ r } . If f : ( X , d ) → ( X , d ) is expanding on balls of diameter 3 δ 0 , and the sets in P have diameter ≤ δ 0 , then P is Markov for f if, for all i and j , x ∈ int ( P i ) ∩ f − 1 ( P j ) ∧ d ( x , y ) < δ 0 ∧ y ∈ f − 1 ( P j ) ⇒ y ∈ P i .
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