Variation of the metric entropy with respect to the SRB measure for - PowerPoint PPT Presentation

Variation of the metric entropy with respect to the SRB measure for hyperbolic systems Miaohua Jiang Wake Forest University CIRM Luminy, France, July 8 -12 M Jiang CIRM Luminy, July 8 - July 12, 2019

Abstract For uniformly hyperbolic systems, it is well-known that its metric entropy with respect to the SRB measure depends on the system differentiably when the perturbation is sufficiently smooth. We present results on the possible values of the entropy when the system varies. In the Axiom A case, we present the derivative formula for the entropy with respect to the generalized SRB measure and in the dimension two case, the derivative formula for the Hausdorff dimension of the hyperbolic set. M Jiang CIRM Luminy, July 8 - July 12, 2019

Entropy w.r.t. the SRB measure f : M → M , diffeom. on a compact manifold: Two cases: (1) f has a hyperbolic attractor ∆ f . Or, (2) f is of Axiom A (locally maximal) on ∆ f , assuming topological transitivity. ρ f , the unique equilibrium state with respect to the potential function ϕ f = − log J u f on the hyperbolic set: the SRB measure. h ρ f , entropy of f w.r.t. ρ f , satisfying the variational principle: � � P ( ϕ f ) = h ρ f + ϕ f d ρ f (= max h µ + ϕ d µ ) . µ ∆ f ∆ f M Jiang CIRM Luminy, July 8 - July 12, 2019

Properties of the entropy function We now have a well-defined functional on the family of C 3 diffeomorphisms that are topologically conjugate to f : U ( f ). f ∈ U ( f ) → h ρ f ∈ R . The range of f → h ρ f : Answer: (0 , h 0 ( f )), h 0 ( f ) is the topological entropy of f . Start from f , we can perturb f successively along a C ∞ path so that h ρ f → 0 . (Hu, Jiang, J 2008). If f is measure preserving, the path can be constructed within the measure preserving family. (Hu, Jiang, J 2017). Question : Can the range be (0 , h 0 ( f )]? Given f , can we perturb f successively such that h ρ f → h 0 ( f )? M Jiang CIRM Luminy, July 8 - July 12, 2019

Differentiability and the derivative formula of f → h ρ f ? The entropy functional f → h ρ f is differentiable: In the attractor case: the topological pressure is zero, so � log J u fd ρ f . h ρ f = ∆ f � In the Axiom A case, h ρ f = P ( − log J u f ) + log J u fd ρ f . ∆ f f → log J u f is differentiable - when interpreted carefully, and an appropriate metric on M is chosen. f → ρ f is differentiable: the derivative formula is the linear response . We can also get the derivative formula in both cases, and more ... M Jiang CIRM Luminy, July 8 - July 12, 2019

Further Questions • How can we perturb the map f so that h ρ f is either increasing or decreasing? • Does the functional f → h ρ f have any local minimum or local maximum? • Does the gradient flow of the entropy functional mean anything? Can we treat the entropy as the energy of a hyperbolic system? The more interesting cases are when the system has weaker versions of hyperbolicity. Under what conditions, are these properties preserved within a reasonably large family of perturbations of f ? M Jiang CIRM Luminy, July 8 - July 12, 2019

Linear Response: the derivative of ρ f w.r.t. f We have a well-defined map: f → ρ f . This map is differentiable in f in the strong sense (Fr´ echet derivative) when f is in a sufficiently differentiable family of maps: C 3 . [Ruelle97] For a given smooth function ψ ( x ) on M , the derivative of the functional � f → ψ d ρ f ∆ f is the Linear Response Function of the dynamical system for the observable ψ ( x ). (Two approaches of the derivation: thermodynamic formalism & transfer operator.) M Jiang CIRM Luminy, July 8 - July 12, 2019

Calculation of the Linear Response Function based on Thermodynamics Formalism Most steps of the derivation are already complete. We just need to put them together carefully: ∆ f ψ d ρ f : a C 3 � (1) The domain of the functional, f → neighborhood of f 0 : U ( f 0 ). Every map of U ( f 0 ) is conjugate to f 0 via a H¨ older continuous map h f : f ◦ h f = h f ◦ f 0 . (2) h f will transport ρ f onto the hyperbolic set ∆ f 0 so that when f varies, h ∗ f ( ρ f ) is an invariant measure on ∆ f 0 . � � (3) f → ∆ f ψ d ρ f becomes f → ∆ f 0 ψ ( h f ) dh ∗ f ( ρ f ) (4) The measure h ∗ f ( ρ f ) is an equilibrium state of f 0 on ∆ f 0 for the potential function − log J u ( h f ( x )). M Jiang CIRM Luminy, July 8 - July 12, 2019

(5) An equilibrium state is the derivative of the topological pressure: d � dt P ( ϕ + t ψ ) | t =0 = ψ d µ ϕ . (6) The derivative formula of an equilibrium state µ ϕ with respect to the potential function ϕ is given by the correlation function series: ψ 1 d µ ϕ + s ψ 2 ) | s =0 = d 2 d � ds ( dsdt P ( ϕ + t ψ 1 + s ψ 2 ) | s , t =0 ∞ � � � � [ ψ 1 ◦ f n ] ψ 2 d µ ϕ − = ψ 1 d µ ϕ ψ 2 d µ ϕ . ∆ f ∆ f ∆ f n = −∞ M Jiang CIRM Luminy, July 8 - July 12, 2019

If the potential function ϕ = ϕ ( f ) is a function of f and we know its derivative: δ f ϕ , then, we have � δ f ( ψ d µ ϕ ( f ) ) ∞ � � � � [ ψ ◦ f n ] δ f ϕ d µ ϕ − ψ d µ ϕ δ f ϕ d µ ϕ . ∆ f ∆ f ∆ f n = −∞ (7) We only need to derive the derivative formulas of the conjugating map h f and the potential function − log J u ( h f ( x )) respect to the map f - a pure dynamical system problem, nothing to do with statistical mechanics. M Jiang CIRM Luminy, July 8 - July 12, 2019

f : attractor case [Ruelle97, Ruelle03, J12] Under a carefully chosen metric on the Riemannian manifold, δ log J u f u ( h f ( x )) | f 0 = div ρ X u ( f 0 ( x )) , the divergence taken with respect to the local volume form whose density function is n =1 J u f ( f − n ( y )) Π ∞ Π ∞ n =1 J u f ( f − n ( x )) with x fixed. M Jiang CIRM Luminy, July 8 - July 12, 2019

f : Axiom A case [J15] Under a carefully chosen metric on the Riemannian manifold, the same formula δ log J u f ( h f ( x )) | f 0 = div ρ X u ( f 0 ( x )) , holds if the unstable manifolds are 1D. M Jiang CIRM Luminy, July 8 - July 12, 2019

Difficulties: (1) X u is not differentiable along unstable manifold since the stable distribution E s is only H¨ older. (2) J u f ( y ) is not differentiable in y . (3) In the non-attractor case, X u is only defined on a (generalized) Cantor set. M Jiang CIRM Luminy, July 8 - July 12, 2019

Entropy’s Derivative Formula (1) [Ruelle 97, 03, J15] Derivative formula of the entropy of the SRB measure: � � log J u f ◦ f k � � div ρ X u d ρ f . δ h ρ f = − k ∈ Z Corollary: if J u f is a constant, then f is a critical value of the entropy function. Conjecture: the converse holds if M is a either a circle or a torus. Conjecture: the entropy functional does not have nontrivial critical values in U ( f ). M Jiang CIRM Luminy, July 8 - July 12, 2019

(2) Derivative formula of the Hausdorff dimension of the hyperbolic set ∆ f , when M is 2D. [J15] d H (∆ f ) = t u + t s , where t u and t s are Hausdorff dimensions of the intersections of the unstable and stable manifolds and the hyperbolic set. Let µ t u f be the equilibrium state for the potential function t u ϕ = − t u log J u f and H µ t u f be the entropy. We have � P ( − t u log J u f ) = H µ tuf + t u ϕ d µ t u f = 0 . M Jiang CIRM Luminy, July 8 - July 12, 2019

Composed with the conjugating map h f : � P ( − t u log J u f ( h f )) = H h ∗ ϕ ( h f ) dh ∗ h µ tuf + t u f µ t u f = 0 . Take the derivatives both sides of the equation with respect to f in the direction of δ f : δ ( 1 1 � div ρ X u d µ t u f . ) = t u H µ tuf M Jiang CIRM Luminy, July 8 - July 12, 2019

For t s , we have δ ( 1 1 � div ρ Df − 1 X s d µ t s f . ) = t s H µ ts f M Jiang CIRM Luminy, July 8 - July 12, 2019

How to get the derivative formula for the entropy Recall: Topological pressure is analytic on the space of H¨ older continuous functions on ∆ 0 . � δ P ( ϕ ) = δϕ d µ ϕ , ∆ 0 where µ ϕ is the unique equilibrium state of the potential function ϕ . Take the second order derivative of the topological pressure, we obtain the correlation function: � � [ ψ ◦ f k ] δϕ d µ ϕ . k ∈ Z Take the derivatives both sides of the equation of the variational principle: � ϕ ( h f ) dh ∗ δ P ( ϕ ( h f )) = δ H h ∗ f ρ f + δ f ρ f , ∆ 0 M Jiang CIRM Luminy, July 8 - July 12, 2019

We obtain � − div ρ f X u d ρ f = δ H h ∗ f ρ f + ∆ � � − div ρ f X u d ρ f + � ϕ ( f k )( − div ρ f X u d ρ f ) . + ∆ k ∈ Z Thus, � � log J u f ( f k )( div ρ f X u d ρ f ) . f ρ f = − δ H h ∗ k ∈ Z M Jiang CIRM Luminy, July 8 - July 12, 2019

For the derivative formula of t u (the unstable portion of the Hausdorff dimension): δ P ( ϕ ( h f ) t u ) = 0 . � δ ( ϕ ( h f ) t u ) dh ∗ f ρ t u f = 0 . ∆ 0 � ( − t u div ρ f X u + ϕδ t u ) d ρ t u f = 0 . ∆ Notice that � � t u log J u fd ρ t u f = t u log J u fd ρ t u f . H ρ tuf = ∆ ∆ We conclude δ ( 1 1 � div ρ X u d ρ t u f . ) = t u H ρ tuf M Jiang CIRM Luminy, July 8 - July 12, 2019