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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

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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

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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

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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. ρf , the unique equilibrium state with respect to the potential function ϕf = − log Juf on the hyperbolic set: the SRB measure. hρf , entropy of f w.r.t. ρf , satisfying the variational principle: P(ϕf ) = hρf +

  • ∆f

ϕf dρf (= max

µ

hµ +

  • ∆f

ϕdµ).

M Jiang CIRM Luminy, July 8 - July 12, 2019

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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 ) → hρf ∈ R. The range of f → hρf : Answer: (0, h0(f )), h0(f ) is the topological entropy of f . Start from f , we can perturb f successively along a C ∞ path so that hρf → 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, h0(f )]? Given f , can we perturb f successively such that hρf → h0(f )?

M Jiang CIRM Luminy, July 8 - July 12, 2019

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Differentiability and the derivative formula of f → hρf ?

The entropy functional f → hρf is differentiable: In the attractor case: the topological pressure is zero, so hρf =

  • ∆f

log Jufdρf . In the Axiom A case, hρf = P(− log Juf ) +

  • ∆f

log Jufdρf . f → log Juf is differentiable - when interpreted carefully, and an appropriate metric on M is chosen. f → ρf 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

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Further Questions

  • How can we perturb the map f so that hρf is either increasing or

decreasing?

  • Does the functional f → hρf 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

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Linear Response: the derivative of ρf w.r.t. f

We have a well-defined map: f → ρf . 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 →

  • ∆f

ψdρf is the Linear Response Function of the dynamical system for the

  • bservable ψ(x).

(Two approaches of the derivation: thermodynamic formalism & transfer operator.)

M Jiang CIRM Luminy, July 8 - July 12, 2019

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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: (1) The domain of the functional, f →

  • ∆f ψdρf : a C 3

neighborhood of f0: U(f0). Every map of U(f0) is conjugate to f0 via a H¨

  • lder continuous map hf :

f ◦ hf = hf ◦ f0. (2) hf will transport ρf onto the hyperbolic set ∆f0 so that when f varies, h∗

f (ρf ) is an invariant measure on ∆f0.

(3) f →

  • ∆f ψdρf becomes f →
  • ∆f0 ψ(hf )dh∗

f (ρf )

(4) The measure h∗

f (ρf ) is an equilibrium state of f0 on ∆f0 for the

potential function − log Ju(hf (x)).

M Jiang CIRM Luminy, July 8 - July 12, 2019

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(5) An equilibrium state is the derivative of the topological pressure: d dt P(ϕ + tψ)|t=0 =

  • ψdµϕ.

(6) The derivative formula of an equilibrium state µϕ with respect to the potential function ϕ is given by the correlation function series: d ds (

  • ψ1dµϕ+sψ2)|s=0 = d2

dsdt P(ϕ + tψ1 + sψ2)|s,t=0 =

∞

  • n=−∞
  • ∆f

[ψ1 ◦ f n] ψ2dµϕ −

  • ∆f

ψ1dµϕ

  • ∆f

ψ2dµϕ.

M Jiang CIRM Luminy, July 8 - July 12, 2019

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If the potential function ϕ = ϕ(f ) is a function of f and we know its derivative: δf ϕ, then, we have δf (

  • ψdµϕ(f ))

∞

  • n=−∞
  • ∆f

[ψ ◦ f n] δf ϕdµϕ −

  • ∆f

ψdµϕ

  • ∆f

δf ϕdµϕ. (7) We only need to derive the derivative formulas of the conjugating map hf and the potential function − log Ju(hf (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

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f : attractor case

[Ruelle97, Ruelle03, J12] Under a carefully chosen metric on the Riemannian manifold, δ log Juf u(hf (x))|f0 = divρX u(f0(x)), the divergence taken with respect to the local volume form whose density function is Π∞

n=1Juf (f −n(y))

Π∞

n=1Juf (f −n(x))

with x fixed.

M Jiang CIRM Luminy, July 8 - July 12, 2019

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f : Axiom A case

[J15] Under a carefully chosen metric on the Riemannian manifold, the same formula δ log Juf (hf (x))|f0 = divρX u(f0(x)), holds if the unstable manifolds are 1D.

M Jiang CIRM Luminy, July 8 - July 12, 2019

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Difficulties: (1) X u is not differentiable along unstable manifold since the stable distribution E s is only H¨

  • lder.

(2) Juf (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

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Entropy’s Derivative Formula

(1) [Ruelle 97, 03, J15] Derivative formula of the entropy of the SRB measure: δhρf = −

  • k∈Z

log Juf ◦ f k divρX udρf . Corollary: if Juf 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

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(2) Derivative formula of the Hausdorff dimension of the hyperbolic set ∆f , when M is 2D. [J15] dH(∆f ) = tu + ts, where tu and ts are Hausdorff dimensions of the intersections of the unstable and stable manifolds and the hyperbolic set. Let µtuf be the equilibrium state for the potential function tuϕ = −tu log Juf and Hµtuf be the entropy. We have P(−tu log Juf ) = Hµtuf +

  • tuϕdµtuf = 0.

M Jiang CIRM Luminy, July 8 - July 12, 2019

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Composed with the conjugating map hf : P(−tu log Juf (hf )) = Hh∗

hµtuf + tu

  • ϕ(hf )dh∗

f µtuf = 0.

Take the derivatives both sides of the equation with respect to f in the direction of δf : δ( 1 tu ) = 1 Hµtuf

  • divρX udµtuf .

M Jiang CIRM Luminy, July 8 - July 12, 2019

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For ts, we have δ( 1 ts ) = 1 Hµts f

  • divρDf −1X sdµtsf .

M Jiang CIRM Luminy, July 8 - July 12, 2019

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How to get the derivative formula for the entropy

Recall: Topological pressure is analytic on the space of H¨

  • lder

continuous functions on ∆0. δP(ϕ) =

  • ∆0

δϕdµϕ, where µϕ is the unique equilibrium state of the potential function ϕ. Take the second order derivative of the topological pressure, we

  • btain the correlation function:
  • k∈Z
  • [ψ ◦ f k]δϕdµϕ.

Take the derivatives both sides of the equation of the variational principle: δP(ϕ(hf )) = δHh∗

f ρf + δ

  • ∆0

ϕ(hf )dh∗

f ρf ,

M Jiang CIRM Luminy, July 8 - July 12, 2019

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We obtain

  • ∆

−divρf X udρf = δHh∗

f ρf +

+

  • ∆

−divρf X udρf +

  • k∈Z
  • ϕ(f k)(−divρf X udρf ).

Thus, δHh∗

f ρf = −

  • k∈Z
  • log Juf (f k)(divρf X udρf ).

M Jiang CIRM Luminy, July 8 - July 12, 2019

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For the derivative formula of tu (the unstable portion of the Hausdorff dimension): δP(ϕ(hf )tu) = 0.

  • ∆0

δ(ϕ(hf )tu)dh∗

f ρtuf = 0.

  • ∆

(−tudivρf X u + ϕδtu)dρtuf = 0. Notice that Hρtuf =

  • ∆

tu log Jufdρtuf = tu

  • ∆

log Jufdρtuf . We conclude δ( 1 tu ) = 1 Hρtuf

  • divρX udρtuf .

M Jiang CIRM Luminy, July 8 - July 12, 2019

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Differentiating the potential function

Ideas: (1) divX u exists because of the good properties of the holonomy map: divX u is closely related to the Jacobian of the holonomy

  • map. In dimension 2, the holonomy map is in fact differentiable.

(2) It is possible to modify the metric on the manifold so that Juf (y) is differentiable in y. (3) If the unstable manifolds are dimension one, divX u can be calculated uniquely using interpolation: the holonomy map can be extended to the local unstable and stable manifold.

M Jiang CIRM Luminy, July 8 - July 12, 2019

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Numerical Exploration of fλ(x) = λx(1 − x)

Another way to look at the SRB measure: for Lebesgue a.e. x, 1 N

N−1

  • n=0

ϕ(f n(x)) =

  • ϕdρf .

When φ(x) = x, we have 1 N

N−1

  • n=0

f n(x) =

  • xdρf .

Thus, the spatial average exists: SA(fλ) = 1 N

N−1

  • n=0
  • f n(x)dx.

M Jiang CIRM Luminy, July 8 - July 12, 2019

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Numerical Exploration

(1) SA(fλ) exists for all λ ∈ (0, 4]. (2) SA(fλ) is continuous in λ. (3) Let S be the set of parameters λ where fλ has an attracting periodic orbit, we have for any λ0 ∈ S SA(fλ0) = sup

λ∈A,λ→λ0

SA(fλ).

M Jiang CIRM Luminy, July 8 - July 12, 2019

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Merci beaucoup

J’aime C.I.R.M.

M Jiang CIRM Luminy, July 8 - July 12, 2019