Measures of maximal entropy that are SRB

The Main Theorem is proved in the last section.

Theorem A (Section 3) provides a complete answer to Question 1.1 in the particular case of DA-diffeomorphisms of $\mathbb{T}^{3}$. This question is relevant in more general contexts: different manifolds, higher dimensions, and more general dynamics. However, for general partially hyperbolic diffeomorphisms in three-dimensional manifolds, this issue is open and has not been addressed yet. We consider this paper an analysis of a toy example that can provide hints for future generalizations.

Condition (3.1) in Theorem A is very fragile in the sense that it can be easily broken by a $C^{\infty}$-perturbation, both in the conservative and non-conservative case, see Theorem B. On the other hand, the set of diffeomorphisms $f\in{\rm{DA}}$ for which condition (3.1) fails is a $C^{1}$-open set.

As stated in the Introduction, this article is inspired by Conjecture 1.2, which states that the entropy-maximizing measure of the geodesic flow of a manifold with negative curvature coincides with the Liouville measure if and only if the manifold is locally symmetric (constant curvature in the 2-dimensional manifold case). The conjecture was proved true for surfaces, but it remains open in the general case despite successive advances.

We aim to obtain a result of rigidity in the spirit of Conjecture 1.2. This is the content of Theorem A. Theorem B shows that this rigidity is not as “rigid” as expected in A. Katok’s setting (in fact, people in the geodesic flows area were more surprised by Theorem B than by Theorem A). One can have a diffeomorphism for which the entropy-maximizing measure is SRB, or even Lebesgue, but the diffeomorphism is not a linear Anosov diffeomorphism - the analogous of “constant curvature”-.

From Theorems A and B it follows that, even though there is some degree of rigidity, this rigidity is not enough to guarantee $f$ to be an Anosov diffeomorphism. This panorama may change if we increase our requirements.

The authors thank Pablo Carrasco for his support. The authors also acknowledge anonymous reviewers for their feedback that helped improve this presentation.

We will see that the cohomological equation (3.2) admits a continuous solution $\phi$ on $\operatorname{supp}\mu$. That is, there exists a continuous $\phi$ satisfying:

For any periodic point $p$ in the support of $\mu$,   $\mu_{p}$ is the ergodic invariant measure supported on the orbit of $p$. If there is a continuous function $\phi$ satisfying (3.2), we can integrate (3.2) with respect to the measure $\mu_{p}$ and get $\lambda_{f}^{+}(p)=h_{top}(A)$. This proves Theorem A.

To obtain a continuous solution of the cohomological equation (3.2) we start with a “formal solution” of (3.2) and then prove that this formal solution is measurable and coincides with a continuous function $\phi$ $\mu$-almost everywhere in $\operatorname{supp}\mu$.

We introduce a formal solution of (3.2).

Since $f$ is a DA-diffeomorphism, there exists a semi conjugacy $h:\mathbb{T}^{3}\to\mathbb{T}^{3}$ such that

Assume that $Jh^{+}$ is formally defined. Then, from (3.3), we get

By the rules of the Jacobian, $Jh^{+}(f(x)).Jf^{+}(x)=JA^{+}.Jh^{+}(x)$. Taking $\log$,

So, $\log Jh^{+}$ is a formal solution of (3.2).

We will see that $Jh^{+}$ is measurable and coincides with a continuous function $\mu$-almost everywhere.

Assume that the measure of maximal entropy $\mu$ is an SRB measure. Then $\lambda_{\mu}^{c}(f)<0$ (Theorem 3.3.) This implies that $W^{+}(x)=W^{u}(x)$ $\mu$-almost every $x\in\mathbb{T}^{3}$, since $\dim W^{+}(x)=$number of positive Lyapunov exponents=1.

Given a point $z$, take a cubic box $C(h(z))$ with a side $L>0$ centered on $h(z)$ and subfoliated by the local “linear foliations ” $E^{s}_{A},E^{c}$, and $E^{u}_{A}$. For simplicity, assume that the axes $E^{\sigma}_{A}$ are orthogonal. Denote by $D(z)=h^{-1}(C(h(z)))$. $L$ can be taken to be uniform for all $z\in\mathbb{T}^{3}$.

$D(z)$ is a neighborhood of $z$ partitioned by bounded subarcs of $W^{+}(x)$, with $x\in D(z)$. For each $x\in D(z)$, $\mu^{+}_{x}$ is the conditional measure associated with the restriction of $\mu$ to $D(z)$, and $m^{+}_{x}$ is the measure induced by the Riemannian structure of $W^{+}(x)$ of the connected component of $W^{+}(x)$ intersected with $D(z)$ containing $x$.

Since $\mu^{+}_{x}<<m^{+}_{x}$ for $\mu$-almost every point $x$, there exists an $L^{1}$ density function $\rho_{x}^{+}$ such that for every interval $[a,b]^{+}\subset W^{+}_{D}(x)$, we have:

$h$ takes unstable leaves with respect to $f$ into unstable leaves with respect to $A$ ([Ure12]), and sends the measure $\mu^{+}(x)$ - equivalent to volume on $W^{u}(x)$ - into the normalized volume measure on $E_{A}^{u}(h(x))$. Then, it sends the volume on $W^{u}(x)$ into a measure equivalent to Lebesgue on $E_{A}^{u}(h(x))$. (Exercise: Check that the claim is true from this point on.)

  [Claim 3.6]

The claim 3.6 implies that $h$ has an unstable Jacobian $Jh^{+}$. $Jh^{+}$ only depends on the Riemannian metric. So, it does not depend on the choice of the box $C(h(z))$ (see Definition 2.9). This implies the following

The formulas (3.4) and (3.7) together imply that

Since $\log\rho^{+}_{x}$ is continuous, $\log Jh^{+}$ extends to a continuous function on $D(z)\cap\operatorname{supp}\mu$. So, $Jh^{+}$ extends to a continuous function on $\operatorname{supp}\mu$.

  [Theorem A, case $\lambda^{c}_{A}<0$]

The above proposition is a particular case of [RHUY22, Proposition 5.2]. We include a proof of it in our simpler setting, to improve readability.

The easy part is to prove that $W^{+}(x)$ is contained in $W^{cu}(x)$ for $\mu$-almost every $x$, and it is left to the reader.

The difficult part is the other inclusion. $W^{+}(x)$ contains a 2-disc centered at $x$. If this disc does not ”colapse” under the action of $h$, and $h(W^{+}(x))$ contains a disc centered at $h(x)$, then the linear action of $A$ will make this disc grow uniformly - playing with successive $A^{n}\circ h\circ A^{-n}$ -. The fact that $h$ is proper (the reader can look for this meaning) will show that $W^{+}(x)$ occupies the whole immersed plane $W^{cu}(x)$.

$h$ is injective $\mu$-almost everywhere (Lemma 3.7). Then, we can consider that $h$ is injective in every $x\in{\mathcal{P}}$ (Notation 2.1). This implies that the $h$-image of $W^{+}(x)\cap W^{c}(y)$ contains a neighborhood of $h(y)$ - a positive radius center segment centered at $h(y)$ - in $E^{c}_{A}(h(y))$ for all $x\in{\mathcal{P}}$ and $y\in W^{+}(x)$ (recall that $h$ only identifies points in the same center leaf, see Lemma 3.7.)

There is a full measure set ${\mathcal{R}}$ on which $h$ is injective. $h$ only collapses center segments, so $h(W^{+}(x)\cap W^{c}(x))$ contains a neighborhood in $E^{c}_{A}(x)$ for all $x\in{\mathcal{R}}$. Because $m(h({\mathcal{R}}))=1$, there is an $m$-positive measure set $B$ so that $h(W^{+}(x)\cap W^{c}(c))$ contains a segment of radius $\delta>0$ centered at $x$ for all $x\in B$.

The ergodicity and invariance of $m$ with respect to $A$ imply that $h(W^{+}(x)\cap W^{c}(x))=E^{c}_{A}(h(x))$. Then $W^{+}(x)\cap W^{c}(x)=W^{c}(x)$ for $\mu$-almost every $x\in{\mathcal{R}}$.

The reader may complete the details.

  [Proposition 3.9]

In Subsection 3.3 ($\lambda^{c}_{A}<0$) the continuity of $\rho_{x}^{+}$ follows directly from formulas (3.5) and from uniform hyperbolicity.

This case is more elaborated. The continuity of $\rho^{+}_{x}$ is only clear when restricted to a local Pesin unstable manifold $W^{+}_{D}(x)$, with $x\in{\mathcal{P}}$. Its continuity on $\mathbb{T}^{3}$ is not trivial.

We build a continuous extension of $\rho^{+}_{x_{0}}$ that coincides with $\mu$-almost everywhere with $\frac{1}{L^{2}}Jh^{+}$. We then extend $Jh^{+}$ continuously on $D$.

If $x\in D$ and $H$ is the stable holonomy map between $x$ and $x_{0}$, define

where $JH:D\to\mathbb{R}^{+}$ is the Jacobian of $H$. The Jacobian $JH$ is continuous and positive [PS72, Theorem 2.1], and so is $\rho^{+}$. $JH$ is continuous on $D$. The formula for $JH$ on $D$ is

The uniform convergence of the formula in $D$ is proved in [PS72, Theorem 2.1] (3.10). (The continuity of $JH$ on a given transversal appears more frequently in the literature, but the continuity of $JH$ on $D$ follows from formula (3.10). See below.)

The continuity of $JH$ on $D$ follows from the Hölder continuity of $Jf^{cu}$:

If $y\in W^{s}_{D}(x)$, then

because $H(x)=H(y)$. Since $\log Jf^{cu}$ is Hölder, there are positive constants $K,C$, and $\alpha$ such that

The last inequality is because $y\in W^{s}_{D}(x)$, so $JH$ is Hölder continuous on stable plaques.

A computation as in the proof of Proposition 3.10 implies that $\frac{1}{L^{2}}Jh^{+}$ coincides $\mu$-almost everywhere with $\rho^{+}$ in $D$. This implies that $Jh^{+}$ extends to a continuous function on $\mathbb{T}^{3}$. To avoid an excess of notation, we call this continuous extension $Jh^{+}$. $\phi=\log Jh^{+}$ is a continuous function that satisfies the cohomological equation (3.2). This implies that $h_{top}(f)=\lambda^{+}_{f}(p)$ for all periodic points $p$, and finishes the proof of Theorem A.

  [Theorem A]