Kakutani equivalence of unipotent flows Adam Kanigowski Montreal, - PowerPoint PPT Presentation

Kakutani equivalence of unipotent flows Adam Kanigowski Montreal, 07.27.2018 (joint w. K. Vinhage and D. Wei) 1 / 11

General setting ( X , B , µ ) – probability standard Borel space; ( T t ) : ( X , B , µ ) → ( X , B , µ ) – measure-preserving, ergodic flow. Isomorphism ( T t ) : ( X , B , µ ) → ( X , B , µ ), ( S t ) : ( Y , C , ν ) → ( Y , C , ν ) are isomorphic, if R ◦ T t = S t ◦ R for t ∈ R , where R : ( X , B , µ ) → ( Y , C , ν ) is invertible and ν ( C ) = µ ( R − 1 C ), C ∈ C . Classification up to isomorphism NOT possible in general (M. Foreman, D. Rudolph, B. Weiss, 2011) 2 / 11

General setting ( X , B , µ ) – probability standard Borel space; ( T t ) : ( X , B , µ ) → ( X , B , µ ) – measure-preserving, ergodic flow. Isomorphism ( T t ) : ( X , B , µ ) → ( X , B , µ ), ( S t ) : ( Y , C , ν ) → ( Y , C , ν ) are isomorphic, if R ◦ T t = S t ◦ R for t ∈ R , where R : ( X , B , µ ) → ( Y , C , ν ) is invertible and ν ( C ) = µ ( R − 1 C ), C ∈ C . Classification up to isomorphism NOT possible in general (M. Foreman, D. Rudolph, B. Weiss, 2011) 2 / 11

General setting ( X , B , µ ) – probability standard Borel space; ( T t ) : ( X , B , µ ) → ( X , B , µ ) – measure-preserving, ergodic flow. Isomorphism ( T t ) : ( X , B , µ ) → ( X , B , µ ), ( S t ) : ( Y , C , ν ) → ( Y , C , ν ) are isomorphic, if R ◦ T t = S t ◦ R for t ∈ R , where R : ( X , B , µ ) → ( Y , C , ν ) is invertible and ν ( C ) = µ ( R − 1 C ), C ∈ C . Classification up to isomorphism NOT possible in general (M. Foreman, D. Rudolph, B. Weiss, 2011) 2 / 11

General setting ( X , B , µ ) – probability standard Borel space; ( T t ) : ( X , B , µ ) → ( X , B , µ ) – measure-preserving, ergodic flow. Isomorphism ( T t ) : ( X , B , µ ) → ( X , B , µ ), ( S t ) : ( Y , C , ν ) → ( Y , C , ν ) are isomorphic, if R ◦ T t = S t ◦ R for t ∈ R , where R : ( X , B , µ ) → ( Y , C , ν ) is invertible and ν ( C ) = µ ( R − 1 C ), C ∈ C . Classification up to isomorphism NOT possible in general (M. Foreman, D. Rudolph, B. Weiss, 2011) 2 / 11

Orbit and Kakutani equivalence Orbit equivalence ( T t ) and ( S t ) are orbit equivalent, if there exists a invertible transformation that maps orbits of ( T t ) to orbits of ( S t ). (Dye’s theorem, 1959) Every two ergodic flows are orbit equivalent. Kakutani equivalence, 1943 ( T t ) : ( X , B , µ ) → ( X , B , µ ) and ( S t ) : ( Y , C , ν ) → ( Y , C , ν ) are K Kakutani equivalent (denoted ( T t ) ∼ ( S t )), if they have isomorphic sections, i.e. can be represented as special flows over the same transformation. Standard flows t ) : T 2 → T 2 , R α Let ( R α t ( x , y ) = ( x + t , y + t α ). Then, for every ∼ ( R β K ∈ Q , ( R α α, β / t ) t ) (Katok, 1976; Ornstein, Rudolph, Weiss, 1982); K ∼ ( R α ( T t ) is standard, if ( T t ) t ) for some α / ∈ Q . 3 / 11

Orbit and Kakutani equivalence Orbit equivalence ( T t ) and ( S t ) are orbit equivalent, if there exists a invertible transformation that maps orbits of ( T t ) to orbits of ( S t ). (Dye’s theorem, 1959) Every two ergodic flows are orbit equivalent. Kakutani equivalence, 1943 ( T t ) : ( X , B , µ ) → ( X , B , µ ) and ( S t ) : ( Y , C , ν ) → ( Y , C , ν ) are K Kakutani equivalent (denoted ( T t ) ∼ ( S t )), if they have isomorphic sections, i.e. can be represented as special flows over the same transformation. Standard flows t ) : T 2 → T 2 , R α Let ( R α t ( x , y ) = ( x + t , y + t α ). Then, for every ∼ ( R β K ∈ Q , ( R α α, β / t ) t ) (Katok, 1976; Ornstein, Rudolph, Weiss, 1982); K ∼ ( R α ( T t ) is standard, if ( T t ) t ) for some α / ∈ Q . 3 / 11

Orbit and Kakutani equivalence Orbit equivalence ( T t ) and ( S t ) are orbit equivalent, if there exists a invertible transformation that maps orbits of ( T t ) to orbits of ( S t ). (Dye’s theorem, 1959) Every two ergodic flows are orbit equivalent. Kakutani equivalence, 1943 ( T t ) : ( X , B , µ ) → ( X , B , µ ) and ( S t ) : ( Y , C , ν ) → ( Y , C , ν ) are K Kakutani equivalent (denoted ( T t ) ∼ ( S t )), if they have isomorphic sections, i.e. can be represented as special flows over the same transformation. Standard flows t ) : T 2 → T 2 , R α Let ( R α t ( x , y ) = ( x + t , y + t α ). Then, for every ∼ ( R β K ∈ Q , ( R α α, β / t ) t ) (Katok, 1976; Ornstein, Rudolph, Weiss, 1982); K ∼ ( R α ( T t ) is standard, if ( T t ) t ) for some α / ∈ Q . 3 / 11

Orbit and Kakutani equivalence Orbit equivalence ( T t ) and ( S t ) are orbit equivalent, if there exists a invertible transformation that maps orbits of ( T t ) to orbits of ( S t ). (Dye’s theorem, 1959) Every two ergodic flows are orbit equivalent. Kakutani equivalence, 1943 ( T t ) : ( X , B , µ ) → ( X , B , µ ) and ( S t ) : ( Y , C , ν ) → ( Y , C , ν ) are K Kakutani equivalent (denoted ( T t ) ∼ ( S t )), if they have isomorphic sections, i.e. can be represented as special flows over the same transformation. Standard flows t ) : T 2 → T 2 , R α Let ( R α t ( x , y ) = ( x + t , y + t α ). Then, for every ∼ ( R β K ∈ Q , ( R α α, β / t ) t ) (Katok, 1976; Ornstein, Rudolph, Weiss, 1982); K ∼ ( R α ( T t ) is standard, if ( T t ) t ) for some α / ∈ Q . 3 / 11

Orbit and Kakutani equivalence Orbit equivalence ( T t ) and ( S t ) are orbit equivalent, if there exists a invertible transformation that maps orbits of ( T t ) to orbits of ( S t ). (Dye’s theorem, 1959) Every two ergodic flows are orbit equivalent. Kakutani equivalence, 1943 ( T t ) : ( X , B , µ ) → ( X , B , µ ) and ( S t ) : ( Y , C , ν ) → ( Y , C , ν ) are K Kakutani equivalent (denoted ( T t ) ∼ ( S t )), if they have isomorphic sections, i.e. can be represented as special flows over the same transformation. Standard flows t ) : T 2 → T 2 , R α Let ( R α t ( x , y ) = ( x + t , y + t α ). Then, for every ∼ ( R β K ∈ Q , ( R α α, β / t ) t ) (Katok, 1976; Ornstein, Rudolph, Weiss, 1982); K ∼ ( R α ( T t ) is standard, if ( T t ) t ) for some α / ∈ Q . 3 / 11

Orbit and Kakutani equivalence Orbit equivalence ( T t ) and ( S t ) are orbit equivalent, if there exists a invertible transformation that maps orbits of ( T t ) to orbits of ( S t ). (Dye’s theorem, 1959) Every two ergodic flows are orbit equivalent. Kakutani equivalence, 1943 ( T t ) : ( X , B , µ ) → ( X , B , µ ) and ( S t ) : ( Y , C , ν ) → ( Y , C , ν ) are K Kakutani equivalent (denoted ( T t ) ∼ ( S t )), if they have isomorphic sections, i.e. can be represented as special flows over the same transformation. Standard flows t ) : T 2 → T 2 , R α Let ( R α t ( x , y ) = ( x + t , y + t α ). Then, for every ∼ ( R β K ∈ Q , ( R α α, β / t ) t ) (Katok, 1976; Ornstein, Rudolph, Weiss, 1982); K ∼ ( R α ( T t ) is standard, if ( T t ) t ) for some α / ∈ Q . 3 / 11

Some results on Kakutani equivalence Standard systems finite rank systems (Ornstein, Rudolph, Weiss, 1982); closed under factors, inverse limits, compact extensions (Katok, 1976; Ornstein, Rudolph, Weiss, 1982) horocycle flows (Ratner, 1978) Non-standard systems first example due to Feldman, 1975; uncountably many pairwise non Kakutani equivalent systems (Ornstein, Rudolph, Weiss, 1982); Let ( h t ) denote the horocycle flow on SL (2 , R ) / Γ. Then ≁ ( h t ) l for k � = l (Ratner, 1980). ( h t ) k K 4 / 11

Some results on Kakutani equivalence Standard systems finite rank systems (Ornstein, Rudolph, Weiss, 1982); closed under factors, inverse limits, compact extensions (Katok, 1976; Ornstein, Rudolph, Weiss, 1982) horocycle flows (Ratner, 1978) Non-standard systems first example due to Feldman, 1975; uncountably many pairwise non Kakutani equivalent systems (Ornstein, Rudolph, Weiss, 1982); Let ( h t ) denote the horocycle flow on SL (2 , R ) / Γ. Then ≁ ( h t ) l for k � = l (Ratner, 1980). ( h t ) k K 4 / 11

Some results on Kakutani equivalence Standard systems finite rank systems (Ornstein, Rudolph, Weiss, 1982); closed under factors, inverse limits, compact extensions (Katok, 1976; Ornstein, Rudolph, Weiss, 1982) horocycle flows (Ratner, 1978) Non-standard systems first example due to Feldman, 1975; uncountably many pairwise non Kakutani equivalent systems (Ornstein, Rudolph, Weiss, 1982); Let ( h t ) denote the horocycle flow on SL (2 , R ) / Γ. Then ≁ ( h t ) l for k � = l (Ratner, 1980). ( h t ) k K 4 / 11

Some results on Kakutani equivalence Standard systems finite rank systems (Ornstein, Rudolph, Weiss, 1982); closed under factors, inverse limits, compact extensions (Katok, 1976; Ornstein, Rudolph, Weiss, 1982) horocycle flows (Ratner, 1978) Non-standard systems first example due to Feldman, 1975; uncountably many pairwise non Kakutani equivalent systems (Ornstein, Rudolph, Weiss, 1982); Let ( h t ) denote the horocycle flow on SL (2 , R ) / Γ. Then ≁ ( h t ) l for k � = l (Ratner, 1980). ( h t ) k K 4 / 11

Some results on Kakutani equivalence Standard systems finite rank systems (Ornstein, Rudolph, Weiss, 1982); closed under factors, inverse limits, compact extensions (Katok, 1976; Ornstein, Rudolph, Weiss, 1982) horocycle flows (Ratner, 1978) Non-standard systems first example due to Feldman, 1975; uncountably many pairwise non Kakutani equivalent systems (Ornstein, Rudolph, Weiss, 1982); Let ( h t ) denote the horocycle flow on SL (2 , R ) / Γ. Then ≁ ( h t ) l for k � = l (Ratner, 1980). ( h t ) k K 4 / 11