h top of interval maps Dual picture: Forward iterates Some kneading theory The Ruelle transfer operator. Backward iterates The dual Ruelle operator The Milnor-Thurston determinant and the Ruelle transfer operator Hans Henrik Rugh mailto:Hans-Henrik.Rugh@math.u-psud.fr Angers 2017 Hans Henrik Rugh The MT determinant and the Ruelle operator
h top of interval maps Dual picture: Forward iterates Some kneading theory The Ruelle transfer operator. Backward iterates The dual Ruelle operator Motivation and set-up We consider ( I , f ), a piecewise continuous and strictly monotone map of a 1 dimensional space. We may take I to be an Interval: Hans Henrik Rugh The MT determinant and the Ruelle operator
h top of interval maps Dual picture: Forward iterates Some kneading theory The Ruelle transfer operator. Backward iterates The dual Ruelle operator Analytic structures related to h top ( f ) The dynamical system ( I , f ) has a topological entropy h top ( f ). We are interested in related analytic structures. Here is the zoo: L ( t ) : Lap number generating function. D ( t ): Milnor-Thurston kneading determinant. ζ AM ( t ): Artin-Mazur topological zeta-function. L : Ruelle transfer operator for ( I , f ). ζ R ( t ): Ruelle dynamical zeta-function. Hans Henrik Rugh The MT determinant and the Ruelle operator
h top of interval maps Dual picture: Forward iterates Some kneading theory The Ruelle transfer operator. Backward iterates The dual Ruelle operator Backward iterates of critical points Crit ( f ) = { c 0 , c 1 , c 2 } ”Partition” into open intervals: I 1 = ( c 0 , c 1 ) , I 2 = ( c 1 , c 2 ) Z 1 = { I 1 , I 2 } Hans Henrik Rugh The MT determinant and the Ruelle operator
h top of interval maps Dual picture: Forward iterates Some kneading theory The Ruelle transfer operator. Backward iterates The dual Ruelle operator Backward iterates of critical points Crit ( f ) = { c 0 , c 1 , c 2 } ”Partition” into open intervals: I 1 = ( c 0 , c 1 ) , I 2 = ( c 1 , c 2 ) Z 1 = { I 1 , I 2 } Refinement by backward iteration of critical points: Z 2 = f − 1 Z 1 ∨ Z 1 Z 2 = { I 11 , I 12 , I 22 , I 21 } Hans Henrik Rugh The MT determinant and the Ruelle operator
h top of interval maps Dual picture: Forward iterates Some kneading theory The Ruelle transfer operator. Backward iterates The dual Ruelle operator Backward iterates of critical points Z n = f − ( n − 1) Z 1 ∨ · · · ∨ Z 1 Misiurewicz and Szlenk: 1 h top = lim n log # Z n n →∞ Lap number generating function: � t n # Z n ∈ Z + [[ t ]] L ( t ) = n ≥ 0 Hans Henrik Rugh The MT determinant and the Ruelle operator
h top of interval maps Dual picture: Forward iterates Some kneading theory The Ruelle transfer operator. Backward iterates The dual Ruelle operator An a priori simple analytic structure of the Lap generating function, example: L ( t ) = 1 + 2 t + 4 t 2 + 8 t 3 + 14 t 4 + ... Coefficients are non-negative integers. L analytic for: | t | < t ∗ = e − h top L diverges for t > t ∗ = e − h top Hans Henrik Rugh The MT determinant and the Ruelle operator
h top of interval maps Dual picture: Forward iterates Some kneading theory The Ruelle transfer operator. Backward iterates The dual Ruelle operator Analytic structures related to h top ( f ) Returning to the zoo: L ( t ): Lap number generating function. D ( t ) : Milnor-Thurston kneading determinant. ζ AM ( t ): Artin-Mazur topological zeta-function. L : Ruelle transfer operator for ( I , f ). ζ R ( t ): Ruelle dynamical zeta-function. Hans Henrik Rugh The MT determinant and the Ruelle operator
h top of interval maps Dual picture: Forward iterates Some kneading theory The Ruelle transfer operator. Backward iterates The dual Ruelle operator Dual picture: Forward iterates of critical points c 0 < c 1 < · · · c d < c d +1 . Intervals of monot.: I k = ( c k , c k +1 ), f k : I k → I = ( c 0 , c d +1 ) is strictly monotone and continuous. Need not be defined at c k and c k +1 . But f ( c + k ) and f ( c − k +1 ) are well-defined. Introduce ”directed” points to keep track of directed limits: x + = ( x , +1) , x − = ( x , − 1) Hans Henrik Rugh The MT determinant and the Ruelle operator
h top of interval maps Dual picture: Forward iterates Some kneading theory The Ruelle transfer operator. Backward iterates The dual Ruelle operator Lifting the map to a directed point map A directed point � x denotes either ( x , +1) or ( x , − 1) (limit from the right/left). On each directed interval [ c + k , c − k +1 ] the map either preserves or reverses orientation, also at endpoints. Set: s ( f , � x ) = +1 if f preserves the orientation at � x , s ( f , � x ) = − 1 if f reverses the orientation at � x . We ”lift” f to a map on the space of directed points: � x ) = � f ( � f (( x , ǫ )) = ( lim t → 0 + f ( x + ǫ t ) , s ( f , � x ) ǫ ) . Hans Henrik Rugh The MT determinant and the Ruelle operator
h top of interval maps Dual picture: Forward iterates Some kneading theory The Ruelle transfer operator. Backward iterates The dual Ruelle operator Kneading invariant When x < y , declare x < x + < y − < y and for � x ∈ � I and u ∈ R : � +1 / 2 if � x < u σ ( � x , u ) = − 1 / 2 if � x > u and for c ∈ Crit ( f ) the kneading invariant (coefficients = ± 1 2 ): � t n s ( f n , � x ) σ ( � f n ( � θ c ( � x , t ) = x ) , c ) . n ≥ 0 The kneading ”determinant” (in our unimodal case): D ( t ) = θ c 1 ( c + 1 , t ) − θ c 1 ( c − 1 , t ) . Hans Henrik Rugh The MT determinant and the Ruelle operator
h top of interval maps Dual picture: Forward iterates Some kneading theory The Ruelle transfer operator. Backward iterates The dual Ruelle operator Recall: The lap generating function L ( t ) = 1 + 2 t + 4 t 2 + 8 t 3 + 14 t 4 + ... diverges for t > t ∗ = e − h top while D ( t ) is analytic in D = {| t | < 1 } since coefficients are in {− 1 , 0 , 1 } . Cancellations of backward and forward orbit contributions: D ( t ) × L ( t ) is analytic in D with no roots! e − h top is a pole of L ( t ). e − h top is the smallest root of D ( t ). Hans Henrik Rugh The MT determinant and the Ruelle operator
h top of interval maps Dual picture: Forward iterates Some kneading theory The Ruelle transfer operator. Backward iterates The dual Ruelle operator With d critical points, Crit ( f ) = { c 0 , ..., c d +1 } one may introduce a ( d + 1) × ( d + 1) kneading matrix: � θ c k ( c + 0 , t ) + θ c k ( c − d +1 , t ) , j = 0 R jk ( t ) = θ c k ( c + j , t ) − θ c k ( c − j , t ) , 1 ≤ j ≤ d and a Milnor-Thurston kneading determinant: D ( t ) = det R jk ( t ) . Again a magic property (much harder to prove): D ( t ) L ( t ) is analytic in D and has no roots for | t | < e − h top . Once again: t ∗ = e − h top is the smallest root of D ( t ). Hans Henrik Rugh The MT determinant and the Ruelle operator
h top of interval maps Dual picture: Forward iterates Some kneading theory The Ruelle transfer operator. Backward iterates The dual Ruelle operator Analytic structures related to h top ( f ) Back at the zoo ...: L ( t ): Lap number generating function. D ( t ): Milnor-Thurston kneading determinant. ζ AM ( t ) : Artin-Mazur topological zeta-function. L : Ruelle transfer operator for ( I , f ). ζ R ( t ): Ruelle dynamical zeta-function. Hans Henrik Rugh The MT determinant and the Ruelle operator
h top of interval maps Dual picture: Forward iterates Some kneading theory The Ruelle transfer operator. Backward iterates The dual Ruelle operator The Artin-Mazur topological zeta-function (tacitly assuming finitely many fixed points): � t n n # Fix ( f n ) . ζ AM ( t ) = exp n ≥ 1 Yet again magic cancellations: D ( t ) × ζ AM ( t ) is analytic in D with no roots! Hans Henrik Rugh The MT determinant and the Ruelle operator
h top of interval maps Dual picture: Forward iterates Some kneading theory The Ruelle transfer operator. Backward iterates The dual Ruelle operator Analytic structures related to h top ( f ) Continuing the tour at the zoo ...: L ( t ): Lap number generating function. D ( t ): Milnor-Thurston kneading determinant. ζ AM ( t ): Artin-Mazur topological zeta-function. L : Ruelle transfer operator for ( I , f ) . ζ R ( t ) : Ruelle dynamical zeta-function. Hans Henrik Rugh The MT determinant and the Ruelle operator
h top of interval maps Dual picture: Forward iterates Some kneading theory The Ruelle transfer operator. Backward iterates The dual Ruelle operator The Ruelle transfer operator and zeta-function For φ ∈ BV ( I ), a function of bounded variation on I , we set: � L φ ( y ) = φ ( x ) x : f ( x )= y Acting on the constant function we simply count pre-images: Card { x : f n ( x ) = y } = L n 1 ( y ) . One has: r sp ( L ) = lim n →∞ �L n � 1 / n BV = e h top . L is a positive operator ⇒ ( r sp ( L ) − L ) is non-invertible. Hans Henrik Rugh The MT determinant and the Ruelle operator
Recommend
More recommend
