Gradient Flow of the Sinai-Ruelle-Bowen Entropy

We study both the local and global existence of a gradient flow of the Sinai-Ruelle-Bowen entropy functional on a Hilbert manifold of expanding maps of a circle equipped with a Sobolev norm in the tangent space of the manifold. We show that, under a slightly modified metric, starting from any initial value, every trajectory of the gradient flow converges to the map with a constant expanding rate where the entropy attains the maximal value. In a simple case, we obtain an explicit formula for the flow's ordinary differential equation representation. This gradient flow has a close connection to a nonlinear partial differential equation: a gradient-dependent diffusion equation.


Introduction
Let f 0 be a C r , r ≥ 3 transitive Anosov map or an expanding map on a compact Riemannian manifold M .Let U (f 0 ) be the connected open component of all C r maps in either family topologically conjugate to f 0 .It is well-known that there exists a unique Sinai-Ruelle-Bowen (SRB) measure ρ f for every map f ∈ U (f 0 ) [20] and the entropy of the map with respect to the SRB measure ρ f is given by the formula H(f ) = M ln J u f dρ f [16,17], where J u f is the Jacobian of f along the unstable subspace.We call it the SRB entropy of f .This entropy is a Fréchet differentiable functional in f with respect to the C r topology on U (f 0 ) [2,12,18].
Recently, it has been proven that in low dimensional cases, this entropy functional H(f ) does not have non-trivial critical points: if the value of the functional is less than its global maximum, then, there exists a direction in which the value of the functional increases when the map is perturbed.The global maximum of the entropy functional is only reached at maps that are smoothly conjugate to a linear map [13,14,19].In view of Gallavotti-Cohen Chaotic Hypothesis [6,7,8,9], this property can be regarded as the realization of the Second Law of Thermodynamics in mathematical models of chaotic dynamical systems.If we hypothesize that each map f in U (f 0 ) represents a possible state of a thermodynamic system, the SRB entropy of f corresponds to the Boltzmann entropy, and a thermodynamic system not at its equilibrium evolves in the direction along which the entropy increases the fastest, we encounter naturally the following two related questions: One, does the fastest increasing direction, or the direction of the gradient vector, of the SRB entropy exist?Two, if the gradient vector field exists, does the SRB entropy induce a gradient flow ?The answers to these two questions are not automatic since the SRB entropy functional is defined on a Banach manifold U (f 0 ) with an infinite dimensional tangent space.In this article, we give affirmative answers to these two questions in a simple case where maps are just expanding maps on a circle.Note that in this simple case U (f 0 ) consists of all C r expanding maps with the same degree n ≥ 2.
In next section, we first extend the family U (f 0 ) (still denoted by U (f 0 )) to include all expanding maps whose rth (r ≥ 2) derivative is L 2 and show that U (f 0 ) is then equipped with a natural Hilbert manifold structure with a Sobolev norm in the tangent space.Under this Hilbert manifold structure, we show that the SRB entropy functional H(f ) remains Fréchet differentiable and thus the functional gives a gradient vector field over U (f 0 ).In Section 2, we show the gradient vector field is at least Lipschitz continuous, which guarantees the existence of a local gradient flow.We note that a differentiable function's gradient is metric dependent.Under a slightly modified metric, we show in Section 4 the global existence of a gradient flow of the SRB entropy functional and the convergence of the flow to the linear expanding map as time approaches infinity.In the last section, via harmonic analysis, we obtain an ordinary differential equation representation of the gradient flow over the Hilbert manifold equipped with the Sobolev norm and give an example of a typical orbit using numerical approximation.The gradient flow also leads to a gradient-dependent diffusion equation on the circle.

Hilbert manifold structure on the family of circle expanding maps
The SRB entropy is a differentiable functional in the space of C r , (r ≥ 3) expanding maps on the unit circle.But when we consider the gradient of the entropy functional, the C r norm may not be the most convenient one.So, a Sobolev norm becomes a more natural choice instead of the C r -norm.We first give the definition of a gradient vector for any Gateaux differentiable functional on the Hilbert manifold M with a tangent space T p M = T at each point p ∈ M and a Hilbert metric (inner product) < • > M .Definition 1.A vector V ∈ T p M is called a gradient vector of a Gateaux differentiable functional H if H ′ s Gateaux derivative defines a bounded linear functional on T p M with its Riesz representation equal to V .
We now describe a Hilbert manifold structure on a family of circle expanding maps.

2.1.
Hilbert manifold of expanding maps on the circle.First of all, by considering its lift, we identify every continuous map f on S 1 = {e i2πx , x ∈ [0, 1)} with a function f defined on the real line satisfying the conditions f (0) ∈ [0, 1) and f (x+1) = f (x)+n, where n is the degree of the map.We only consider orientation preserving maps.The case for orientation reversing maps is essentially the same.Since expanding maps are defined on the circle, there must be a fixed point and we may assume 0 is a fixed point: f (0) = 0 = f (0) and f (1) = n.Thus, each C r , r ≥ 1 map f on the circle is identified with its lift f on R, the universal covering of S 1 , with the following properties: We now define a family of C r−1 expanding maps F r , r ≥ 2 via properties of their lift maps where we consider the gradient flow of the entropy functional: The family F r is slightly larger than the C r family of expanding maps since we only require the rth derivative f (r) to be L 2 , instead of being continuous.F r is a Hilbert manifold modeled on Sobolev space H r .For any given map f ∈ F r , its open neighborhood is identified with an open neighborhood of the origin of the following Hilbert space Notice that Φ r can be identified with a Sobolev sequence space equipped with the corresponding Sobolev norm where {a n } ∞ n=0 and {b n } ∞ n=1 are Fourier coefficients of ϕ.
When r ≥ 3, each expanding map f ∈ F r possesses a unique invariant probability measure absolutely continuous with respect to the Lebesgue measure on S 1 .Its probability density function ρ f (x) is at least C 1 ( [1] ) and depends on f differentiably in C r−1 topology.It yields the Fréchet differentiability of the entropy of f with respect to the measure ρ f dx, i.e., H(f ) = S 1 ln f ′ (x)ρ f (x)dx.We want to prove that H(f ) is also a Fréchet differentiable functional with respect to the new Hilbert metric on F r .This can be done by considering the transfer operator over the Sobolev space of density functions.We leave the proof in this general case to another article since in this paper, we restrict our study to a simpler case when the SRB measure ρ f is preserved by the perturbation of f .When ρ f (x) is independent of f , the differentiability with respect to the Hilbert metric is much easier to prove.We can directly calculate the derivative operator and prove the Fréchet differentiability with respect to the Sobolev norm.
2.2.Hilbert manifold of expanding maps preserving the Lebesgue measure.We now define a Hilbert manifold F r (ρ), r ≥ 2 to be the subset of F r consisting of maps that preserve the same invariant measure with a density function ρ(x).We may assume ρ(x) = ρ 0 (x) = 1 by changing the Riemannian metric on the circle [3,13].The corresponding subset is denoted by F r (ρ 0 ).
Given any map f ∈ F r (ρ 0 ), the invariance of the Lebesgue measure under f is characterized by the equation We see that not only the equation ( 1) is nonlinear in f , the points {y i }, preimages of x, also depend on f .Thus, it is not convenient when we calculate Gateaux derivatives of the entropy functional with respect to f .Instead, we now identify the subset F r (ρ 0 ) with another Hilbert manifold with a Sobolev tangent space where the same SRB entropy functional's properties are much easier to study.
For each map f ∈ F r (ρ 0 ), we consider its lift f 's inverse map g(y), y and the invariance of the Lebesgue measure becomes an equation linear in g: We now define the Hilbert manifold where we consider the SRB entropy's gradient for r ≥ 2.
For each g ∈ G r , its open neighborhood is identified with an open neighborhood in the Sobolev space, still denoted by Φ r : The Sobolev norm on Φ r is defined in the same way: For convenience, we denote the even larger family of functions without the constraint of preservation of Lebesgue measure by Ḡr and Φr : Indeed, G r is a submanifold of Ḡr and Φ r a subspace of Φr .
On the Hilbert manifold Ḡr , there is a nature metric d(g 1 , g 2 ) that is consistent with the Sobolev norm in the tangent space of Ḡr : Remarks.
(1) We point out an interesting connection between the SRB entropy of measure-preserving expanding maps and the Gibbs entropy of a probability measure with a density.Any function g(y) ∈ Ḡr can be considered as a probability measure on [0, n] with a density function 0 < g ′ (y) < 1. H(g) is then precisely the Gibbs entropy of a probability measure.
(2) We also point out similarities and differences between our approach and the approach of Jordan, Kinderleherer, and Otto (JKO) [10,11] in their study of the gradient flow of the entropy functional (see also [15]).The main similarity is that both approaches start from the Gibbs entropy.But two approaches have major differences: In JKO's approach, the entropy (or the Gibbs-Boltzmann entropy as it is called in [10,11].For more discussions on Gibbs and Boltzmann entropy, see [5] ) is defined for probability density functions on R n not associated with any dynamical system.They use a discretized process to obtain an approximate orbit from an initial density and then show that the orbit converges to an orbit from the heat equation as the step-size approaches zero.In our approach, the SRB entropy is defined for a chaotic dynamical system.That the entropy formula (2) taking the Gibbs entropy form seems to be coincidental.For Anosov systems on a higher dimensional manifold, the SRB entropy is defined for a higher dimensional map.The entropy formula may not take this particular form.The SRB entropy changes as the underlying chaotic dynamical system varies and we directly calculate the entropy functional's gradient under the Sobolev norm.Our approach leads to a system of countably many ordinary differential equations where the vector field is defined via integrals.The system does have close connection to a nonlinear partial differential equation, a gradient-dependent diffusion equation on the unit circle.See Section 5 for details.
We now state main results on the SRB entropy functional H(g) on the Hilbert manifold G r .Theorem 1. (1) The SRB entropy functional H(g) is Fréchet differentiable on G r .
(2) The gradient vector field of H(g) is well-defined: for each g ∈ G r , there exists a unique vector X(g) ∈ Φ r such that the directional derivative < DH(g), X(g)/∥X∥ H r > is the unique maximum among all directional derivatives.
(3) The gradient vector field X(g) ∈ G r is Lipschitz continuous in g.
(4) There is a unique critical point for the gradient vector field X(g) at the point where g is a linear function on [0, n].
An immediate consequence of Theorem 1 is that the differential equation defined on G r by dFt dt t=0 = X(g) has a unique local solution for t ∈ (−ϵ, ϵ): F t (g) = F(t, g) is a local flow defined on (−ϵ g , ϵ g ) × G r .

Proof of Theorem 1 (1)
We need to show that H(g) is Fréchet differentiable.Since G r is a submanifold of Ḡr .We can just prove differentiability in Ḡr .That means we do not need to consider the constraint of preservation of the Lebesgue measure.
We first calculate the first order term in ϵ of the difference H(g + ϵϕ) − H(g).
Thus, the first order term of H(g + ϵϕ) − H(g) in ϵ is We now show that H(g) is Fréchet differentiable on Ḡr , i.e, for any given g ∈ Ḡr , According to our earlier calculation, We now need a simple lemma on the upper bound of |ϕ ′ (y)|: , where 1 ≤ k < r and M is constant independent of ϕ.

Proof of Lemma 1
Since ϕ is of period n and ϕ (r) ∈ L 2 [0, n], we have ϕ's Fourier expansion where the Fourier coefficients a k , b k , k ≥ 1 satisfy the condition For convenience, we may assume a 0 = 0. Thus, .
and apply the inequality above to ϕ ′ (y).We have □ Since g ′ (y) > 0 is bounded away from 0 and ϕ

Proof of Theorem 1 (2) & (4)
We pick an orthonormal basis of Φ r : where Given any ϕ ∈ Φ r with ∥ϕ∥ H r = 1 and ϕ = ∞ i=1 < ϕ, e i > H r e i , where < • > H r denotes the inner product of Φ r , we have Thus, the gradient vector field X(g) is well-defined on Hilbert manifold G r .Since we know that X(g) = 0 if and only if g is linear [13], X(g) ̸ = 0 for all g ∈ G r except for g linear.

Proof of Theorem 1 (3)
We now prove that the derivative operator DH g is Lipschitz, which leads to the local existence of the gradient flow [14b].
Given any two maps g 1 , g 2 ∈ G r , denote ψ = g 2 − g 1 ∈ Φ r .We now estimate the distance between two derivative operators DH g 1 and DH g 1 +ψ .When g 1 and g 2 are close, their open neighborhoods overlap.Thus, we can assume two derivative operators DH g 1 and DH g 2 are acting on the same Sobolev space Φ r .For simplicity of notation, we drop the subscript in g 1 .We have Since g ′ > 0 is bounded from below and ψ ∈ Φ r , r ≥ 2, we may assume that |ψ ′ | is sufficiently small and | ψ ′ g ′ | < δ < 1.So, there is a constant K such that Thus, we have g ′ (y) ] 1/2 .We conclude that DH g is Lipschitz continuous over G r .
Remark Since ln x is an analytic function in x in a small neighborhood of any x 0 > 0, we can in fact show that H(g) is analytic in g on Hilbert manifold G r .

Global Existence of the Gradient Flow
We now study the global existence of the gradient flow of the SRB entropy for t ∈ [0, ∞) and prove that every trajectory converges to the unique equilibrium where the expansion rate of the map is a constant.We note that under the Sobolev norm, the gradient vector at point g ∈ G r is defined by the integral n 0 g ′′ (y) g ′ (y) ϕ(y)dy.While the integral does define a linear functional in the tangent space Φ r for each g ∈ G r .Its Riesz representation in Φ r is, in general, not g ′′ (y) g ′ (y) since it is in general not a vector in Φ.This poses an obstacle for proving the global existence, even though, the global existence is likely true.We instead reconsider the global existence of the gradient flow in a different Hilbert metric on a slightly different Hilbert manifold.
We first expand the domain of the entropy functional to a larger Hilbert space where the gradient vector's Riesz representation can be obtained explicitly.
For the Hilbert manifold each map g(y) ∈ G r is uniquely defined by its derivative: g(y) = y 0 g ′ (τ )dτ .So, we can embed G r into another Hilbert manifold G ′ : Given any g(y) ∈ G r , the embedding map is defined by G ′ is clearly a Hilbert manifold with a tangent space Note also that the entropy functional, which is well defined on entire G ′ and the Gateaux derivative of H(h) in the direction of ψ ∈ Ψ exists and has the same formula (see ( 3)): This Gateaux derivative defines a bounded linear functional on the tangent space Ψ.
A direct calculation will confirm that this linear functional's Riesz representation is given by where h(y) is extended into a period n function over [0, ∞).Indeed, we can easily verify that R h (y) ∈ Ψ.We only need to verify that for all y ∈ [0, 1] clearly holds due to the periodicity of h(y).
To see that n 0 R h (y)ψdy = − n 0 ln h(y)ψdy for all ψ ∈ Ψ, we first extend ψ to a period n function and calculate the following integral applying integration by substitution and periodicity of both functions h(y) and ψ(y): We summarize the properties of the entropy functional H(h) = − n 0 ln h(y) h(y)dy over the Hilbert manifold G ′ in the following proposition.Proposition 1. (1) H(h) is Gateaux differentiable at every h ∈ G ′ and the derivative formula in the direction of ψ ∈ Ψ is given by a continuous linear functional on Ψ: (2) The Riesz representation of the derivative operator DH h over Ψ is where h(y) is extended periodically to [0, ∞).
(3) The maximum value of DH h (ψ) over ψ ∈ Ψ with We denote this gradient vector field over G ′ by It is Lipschitz continuous in terms of h under the L 2 norm, thus, locally integrable.
We now prove the following theorem on the global existence of the gradient flow of the SRB entropy and the convergence of every flow trajectory to a unique equilibrium as t → ∞.
Theorem 2. For Lebesgue measure preserving C 1+α expanding maps on the circle, the SRB entropy functional H(f ) = 1 0 ln f ′ (x)dx induces a gradient flow on the space of derivatives of inverse maps under the L 2 norm.This gradient flow exists globally for all t ∈ [0, ∞) and every trajectory converges to the unique equilibrium corresponding to the linear expanding map.
Proof.For any fixed initial map h(y) ∈ G ′ , let G t (h) = g(t, h) denote the local flow defined by the gradient vector field Y (h) on G ′ for t ∈ (−ϵ h , ϵ h ).For any y ∈ [0, 1], We have By periodicity of h(y), for all k = 1, 2, • • • , n − 1, we also have Introduce new variables The solution to the system exists globally for all initial values in the region 0 and all solutions converge to the unique equilibrium

Differential equation representation of the gradient flow
We now explore the possibility of representing the gradient flow F t (g) from Section 2 as explicit differential equations.
Let F t denote the gradient flow defined by the vector field X(g) over G r ,i.e, F t (g) is a map from (−ϵ, ϵ) × G r → G r differentiable in t and F 0 (g) = g and where X(g) ∈ Φ r is defined by an integral operator We see that maps in Ḡr can be easily represented as a series.In the simple case when n = 2, we can obtain a system of ordinary differential equations that generates the flow.Numerical methods such as Euler's method [4] can then be used to obtain typical approximate trajectories of the flow.
For any given g ∈ Ḡr , g(y) − y n ∈ Φr is a continuous periodic function of period n and its derivative is bounded.Thus, its Fourier series converges to itself both pointwise and in the Sobolev norm.Thus, Hilbert manifold Ḡr can be represented as where a k , b k are Fourier coefficients of g(y) − y n satisfying the condition Notice that we have replaced the condition g(0) = 0 by dropping the constant term in the Fourier series since the entropy is a function of g ′ (x).This adjustment is also made to the tangent space Φr .
Maps in the submanifold G r will have to satisfy an addition linear equation: Assume that we have an orthonormal basis of Φ r : {e k } ∞ k=1 .Then, any trajectory of the flow u(t, y) = F t (g) can be written in the form We have a system of countably many ordinary differential equations: While it is easy to obtain a set of orthonormal basis for Φr since the set {cos 2πk n y, sin 2πk n y} ∞ k=1 is clearly an orthogonal basis, the linear constraint ( 6) poses an to finding orthogonal basis for Φ r .Fortunately, in the simple case when n = 2, an orthonormal basis for Φ r can be obtained directly from this set.That will allow us to obtain a system of countably many ordinary differential equations explicitly and thus, to approximate numerically typical trajectories of the flow.
5.1.Ordinary differential equation representation when n = 2.We now look at the case when n = 2. r ≥ 2 can be any number.In this case, the linear constraint (6) becomes Let u(t, y) be a trajectory of the flow F t (g).For each t, For a fixed value of t, the gradient vector at u(x, t) is Notice that We obtain explicitly a system of ordinary differential equations defined on G 2 that generates the gradient flow.i.e., the local flow F t (g) is the solution to the system of differential equations: The systems in (10) and ( 8) differ only by a constant coefficient in front of each equation.

5.3.
Numerical approximation of a flow trajectory.We limit the scope of our numerical exploration to the case when g ′ is an even function: We have ȧ2m−1 = 0 for all m ∈ N. Thus, the system of ODEs in ( 9) is reduced to   We see that the diffusion process from this gradient flow is different from that of the heat equation.Due to the linearity, the flow from the heat equation does not create higher frequency terms if the initial heat distribution does not have them.In the gradient flow induced by the SRB entropy, the higher frequency terms appear immediately when t increases even though the amplitudes of these high frequency terms are very small.Numerical evidence suggests that the gradient flow F t (g) is also defined globally over G 2 and lim t→∞ F t (g) = y 2 .However, a rigorous proof is not available at the moment.

Acknowledgement
The author thanks John Gemmer, Sarah Raynor, Yang Yun and Yunping Jiang for many beneficial discussions.

Figure 1 .
Figure 1.Graphs of the deviation of the derivative of the inverse map from its equilibrium for various values of t along a trajectory of the gradient flow.

Figure 2 .
Figure 2. The dotted line is the graph of a cosine curve.The solid line is the graph of the deviation of the derivative of the inverse map from its equilibrium when t is large.The verical axis is re-scaled with a factor of 1000.