Measure rigidity of Anosov flows via the factorization method

Abstract

Using the factorization method of Eskin and Mirzakhani, we show that generalized u-Gibbs states over quantitatively non-integrable partially hyperbolic systems have absolutely continuous disintegrations on unstable manifolds. As an application, we show a pointwise equidistribution theorem analogous to the equidistribution results of Kleinbock–Shi–Weiss and Chaika–Eskin.

Appendices

Appendix A: Technical Construction of Factorization

1.1 Basic facts and definitions.

We assume throughout that V is a finite-dimensional vector bundle over M, may be measurable, which is smooth along stable leaves. Moreover, we fix some $x\in M$ and $y\in ~W^{s}(x)$. We will assume that x is generic, in light of [KS17, Theorem 2.5], we may endow $T(W^{s}(x))=E^{s}(x)$ with normal forms coordinates. If $y\in ~W^{s}(x)$ is sufficiently close to x we can write $y=exp_{x}({\underline{y}})$ for some vector ${\underline{y}} \in E^{s}(x)=T(W^{s}(x))$. Using the normal forms coordinates, we may calculate $g_{\ell }.y$ by calculating the action of $g_{\ell }.{\underline{y}} \in E^{s}(g_{\ell }.x)$ by of sub-resonant polynomials. Throughout the section we will write $g_{\ell }.y$ as the result of this computation. We note that this coordinates immediately generalize to higher tangent spaces, $T^{k}(E^{s}(x))$, for all $k\ge 1$.

Definition A.1

Assume V is a finite dimensional vector bundle over M. Let $x\in M$ be given and $y\in W^{s}(x)$. Assume that $T(y):~V(x)\rightarrow ~V(y)$ is a family of linear operators. Moreover we assume that x, y both belong to the same local trivialization of V.

We say that T is factorizable if for any $\beta >0$ there exists a vector bundle $V_{ext}$, a map $F_{x}:W^{s}(x)\rightarrow V_{ext}(x)$

and a linear map $A(x,\ell ):~V_{ext}(x)\rightarrow ~Hom(V(g_{\ell }.x),V(g_{\ell }.x))$, $A=A(T)$, such that

$$\begin{aligned} \left\Vert \xi \circ T(y)-A(x,\ell )F_{x}(y) \right\Vert _{V(g_{\ell }.x),op} \ll e^{-\beta \cdot \ell }, \end{aligned}$$

(A.1)

provided that $g_{\ell }.x,g_{\ell }.y$ lie in the same local trivialization, so we can do the following identification of vector spaces $V(g_{\ell }.x)\simeq V(g_{\ell }.y)$ by means of the isomorphism $\xi$.

Theorem A.2

(Smooth holonomy map is factorizable). Assume that V is a vector bundle over M which has smooth stable holonomy map $H_{V}(x,y):V(x)\rightarrow V(y)$, then $H_{V}$ is factorizable.

Proof

As the holonomy is smooth, let $A(x,\ell )$ be the prolongation of $H_{V}(g_{\ell }.x,\star )$, of some order k which will be determined later, as a function with domain being $W^{s}(g_{\ell }.x)$. Using Taylor expansion over $E^{s}(g_{\ell }.x)$ we may write

$$\begin{aligned} H_{v}(g_{\ell }.x,z) = \sum _{0\le |\alpha |\le k}\partial ^{\alpha }H_{V}(g_{\ell }.x,g_{\ell }.x)\cdot \log (z)^{\alpha }+E_{g_{\ell }.x,k}(z) \end{aligned}$$

(A.2)

where $\log (z)\in E^{s}(g_{\ell }.x)\le TM(g_{\ell }.x)$ is the inverse image of z under the exponential map at $g_{\ell }.x$ (given in terms of the normal forms coordinates) and $E_{g_{\ell }.x,k}(z)$ stands for the reminder term. Using Lagrange’s reminder formula, noticing that M is compact, we may bound

$$\begin{aligned} \Vert E_{g_{\ell }.x,k}(z) \Vert \ll _{M} dist(g_{\ell }.x,z)^{k+1}. \end{aligned}$$

Specializing to the case of $z=g_{\ell }.y$ one gets

$$\begin{aligned} \begin{aligned} \Vert E_{g_{\ell }.x,k}(z) \Vert&\ll _{M} dist(g_{\ell }.x,g_{\ell }.y)^{k+1}\\&\ll _{M} e^{-(k+1)\lambda _{C}\cdot \ell }, \end{aligned} \end{aligned}$$

for some $\lambda _{C}$ which depends on the Lyapunov spectrum of M. Taking k large enough so that

$$\begin{aligned} k+1>\frac{\beta }{\lambda _{C}} \end{aligned}$$

yields the result, where the map $F_{x}(y)$ amounts to calculating the values of the various $\log (y)^{\alpha }$, $V_{ext}=T^{\le k}E^{s}$ - the k’th higher-order tangent space to $E^{s}$, and $A(x,\ell )$ matches the k’th prolongation of $H_{V}(g_{\ell }.x,\star )$ with $g_{\ell }.F_{x}(y)$. $\square$

Definition A.3

Let V be a finite dimensional vector bundle over M, and endows V with the Pesin inner-product. Assume $S\le V$ is a subspace. Given some $\ell >0$ we say that a basis ${\textsf{B}}(g_{\ell }.x)$ of $S(g_{\ell }.x)$ is called almost orthonormal if for any $v\in {\textsf{B}}(g_{\ell }.x)$ we have

$$\begin{aligned} |\Vert v \Vert -1|\le o(\ell ) \end{aligned}$$

and for any two distinct vectors $v,v' \in {\textsf{B}}(g_{\ell }.x)$,

$$\begin{aligned} |\langle v,v' \rangle |\le o(\ell ). \end{aligned}$$

Definition A.4

A subspace $S\le V$ is factorizable if for any $\beta >0$ there is a bundle $V_{ext}$ and a measurable map $F_{x}:W^{s}(x)\rightarrow V_{ext}(x)$, and a linear map

$$\begin{aligned} A(x,\ell ):V_{ext}(x)\rightarrow ~\oplus _{i=1}^{\dim S}V(g_{\ell }.x), \end{aligned}$$

$A=A(S)$ such that

$$\begin{aligned} \Vert \mathfrak {s} - A(x,\ell )F_{x}(y) \Vert _{\oplus _{i=1}^{\dim S}V(g_{\ell }.x)} \ll e^{-\beta \cdot \ell }, \end{aligned}$$

(A.3)

for some almost orthonormal basis $\mathfrak {s}$ of $S(g_{\ell }.y)$, where we assume $g_{\ell }.x, g_{\ell }.y$ lie in the same local trivialization.

We will extend the definition of factorization given above to say that a subspace is factorizable if it is factorizable over a subset of arbitrarily large measure of M.

Corollary A.5

(Preliminary factorization). Assume that V is a bundle admitting smooth stable holonomies which we denote $H_{V}(x,y)$. Assume that $S\le ~V$ is a subspace which is preserved under stable holonomies. Then S is factorizable over a subset of arbitrarily large measure of M.

Proof

Fix $G_{S}:M\rightarrow \oplus _{i=1}^{\dim S}V$ some measurable choice of orthonormal bases for S according to the Pesin inner-product. The map G has the following equivarience property $g_{\ell }.G_{S}(x):=G_{S}(g_{\ell }.x)$. On the level of subspaces we have

$$\begin{aligned} H_{V}(g_{\ell }.x,g_{\ell }.y)S(g_{\ell }.x)=S(g_{\ell }.y) \end{aligned}$$

hence on the level of bases we get that $H_{V}(g_{\ell }.x,g_{\ell }.y).G_{S}(g_{\ell }.x)$ is a basis for $S(g_{\ell }.y)$. Moreover, by a theorem of of Araujo-Bufetov-Filip [ABF16],

$$\begin{aligned} \Vert H_{V}(x,y)-Id \Vert \ll dist(x,y)^{\gamma }, \end{aligned}$$

(A.4)

for some $\gamma$ depending on the Lyapunov spectrum of M on a set of arbitrarily large measure. Consider this set. Hence the basis given by $H_{v}(g_{\ell }.x,g_{\ell }.y).G_{S}(g_{\ell }.x)$ of $S(g_{\ell }.y)$ is an almost orthonormal basis. As $H_{V}$ is smooth map, it is factorizable by Theorem A.2. Combining the above yields the result. We remark here that the dependence in $\beta$ (namely the error of the factorization) relates to the order of the Taylor expansion of $H_{v}$ been used, while the error of the almost orthonormal basis $\mathfrak {s}$ produced is related both to $\beta$ and to the estimate regarding $\gamma$ in (A.4). $\square$

Corollary A.6

(Future factorization). Given some fixed $\alpha >0$, define $T_{\max }=\alpha \cdot \ell$. Using the previous notations, we may factorize $g_{t}.S(g_{\ell }.y)$ in the following manner—for all $\beta ,\ell , t\le T_{\max }(\ell )$, there exists a bundle $V_{ext}$, a map $F_{x}:W^{s}(x)\rightarrow V_{ext}(x)$ and a linear map $A(x,\ell ):V_{ext}(x)\rightarrow ~\oplus _{i=1}^{\dim S}V(g_{\ell }.x)$ such that

$$\begin{aligned} \Vert g_{t}.\mathfrak {s}-g_{t}A(x,\ell )F_{x}(y) \Vert _{\oplus _{i=1}^{\dim S}V(g_{t+\ell }.y)} \ll e^{-\beta \cdot \ell }, \end{aligned}$$

(A.5)

for some “almost orthonormal” basis $\mathfrak {s}$ of $S(g_{\ell }.y)$.

Proof

Apply Corollary A.5 with some $\beta '>0$ to be specified later.

Consider the vector $u=\mathfrak {s}-A(x,\ell )F_{x}(y) \in \oplus _{i=1}^{\dim S}V(g_{\ell }.y)$.

By Oseledets theorem

$$\begin{aligned} \Vert g_{t}.u \Vert \le e^{\lambda _{M}\cdot t}\cdot \Vert u\Vert \ll _{v} e^{\lambda _{M}\cdot T_{\max }}\cdot e^{-\beta '\cdot \ell } \end{aligned}$$

for all $0\le t\le T_{\max }$, for some $\lambda _{M}>0$.

Given $\beta >0$, one may take $\beta '>0$ large enough so that

$$\begin{aligned} \lambda _{M}\cdot T_{max}-\beta '\cdot ~\ell ~\le ~-\beta \cdot \ell . \end{aligned}$$

With that choice of $\beta '$, we recover the required factorization. $\square$

1.2 The $P^{-}$ operator.

Assume V is a vector bundle over M, smooth along stables. For a set of full measure ${\hat{M}}\subset M$ we have a splitting of V by Oseledets theorem as $V=\oplus _{i} V^{\lambda _{i}}$ .

Definition A.7

The attached bundle to V, ${\hat{V}}$, is the (measurable) bundle ${\hat{V}}=\oplus _{i} V^{\le \lambda _{i}}/V^{<\lambda _{i}}$ . In view of Oseledets theorem, for any biregular point $x\in {\hat{M}}$ we have the (measurable) translating map $i_{x}:~V(x)\rightarrow {\hat{V}}(x)$. Writing the Osceledets splitting as $V(x)=\oplus _{i} V^{\lambda _{i}}$, we can write each $v\in V(x)$ as $v=\sum _{i} v^{\lambda _{i}}$ with $v^{\lambda _{i}}\in V^{\lambda _{i}}$ for all i. Hence we can we may send each subspace to $V^{\lambda _{i}}$ to its associated quotient, namely $V^{\lambda _{i}} \mapsto V^{\le \lambda _i}/V^{<\lambda _i}$. In particular for each $v^{\lambda _{i}}$ we have

$$\begin{aligned} i_{x}(v^{\lambda _i})=v^{\lambda _i}+V^{<\lambda _{i}} = v^{\lambda _{i}}+\oplus _{j<i}V^{\lambda _j}. \end{aligned}$$

As each component of ${\hat{V}}$ has a single Lyapunov exponent, the attached bundle ${\hat{V}}$ admits smooth stable holonomies (c.f. [ASV13, Proposition 3.4], [KS13, Proposition 4.2]) which we denote as $H_{{\hat{V}}}(x,y):{\hat{V}}(x)\rightarrow ~{\hat{V}}(y)$.

Definition A.8

Let V be a finite dimensional vector bundle over M which is smooth along stables. For any two points $x,y\in M$ such that $y\in W^{s}(x)$ we define the measurable operator $P^{-}_{V}(x,y):V(x)\rightarrow ~V(y)$ between two biregular points x, y to be the composition

$$\begin{aligned} P^{-}(x,y) = i^{-1}_{y} \circ H_{{\hat{V}}}(x,y) \circ i_{x}, \end{aligned}$$

(A.6)

where $i_{x}:V(x)\rightarrow {\hat{V}}(x)$ and $i_{y}^{-1}:{\hat{V}}(y)\rightarrow V(y)$ are the translating maps given by Oseledets theorem as in Definition A.7.

This operator is analogous to the operator defined in [EM18, §4.2].

A fundamental result due to F. Ledrappier [Led86, Theorem 1] shows that $P^{-}_{V}$ preserves any measurable equivariant subspace $S\le V$.

Theorem A.9

(General subspace factorization theorem). Assume V is a vector bundle smooth along stables. Let $S\le V$ be any measurable equivariant subspace. Then S is factorizable.

The factorization can be achieved by considering the equation

$$\begin{aligned} S(g_{\ell }.y)=P^{-}_{V}(g_{\ell }.x,g_{\ell }.y)S(g_{\ell }.x), \end{aligned}$$

(A.7)

Which follows from Ledrappier’s theorem. We now need to show that we may factorize the operator $P^{-}_{V}(g_{\ell }.x,g_{\ell }.y)$. Examining the definition of $P^{-}$ in (A.6), we see that

$$\begin{aligned} P^{-}_{V}(g_{\ell }.x,g_{\ell }.y) = i_{g_{\ell }.y}^{-1} \circ H_{{\hat{V}}}(x,y)\circ i_{g_{\ell }.x}. \end{aligned}$$

The term $H_{{\hat{V}}}(x,y)\circ i_{g_{\ell }.x}$ is factorizable (as an operator with its image in the attached bundle ${\hat{V}}$, which has smooth holonomies) by Corollary A.5, as $H_{{\hat{V}}}$ is smooth along stables, and i preserves spaces of dynamical definition.

We consider $i^{-1}_{\star }$ as a vector inside a bundle in the following manner

$$\begin{aligned} i^{-1}_{\star }\in Hom({\hat{V}}(\star ),V(\star )), \end{aligned}$$

or alternatively, one may consider $i^{-1}$ as section $M\rightarrow Hom({\hat{V}}(\star ),V(\star ))$, hence belong to the vector space of such sections.

Unfortunately, the bundle of sections, does not necessarily admit smooth holonomies. Nevertheless, it is smooth along stables (as V is). The next subsection introduces a technique based on a theorem of Brown–Eskin–Filip–Rodriguez–Hertz to overcome this issue and factorize the section $i^{-1}_{\star }$.

1.3 Overcoming non-existence of holonomies.

We will need to discuss cases when the vector bundle V is only smooth along stables and does not necessarily admit smooth holonomies. We note the following theorem.

Theorem A.10

(Brown–Eskin–Filip–Rodriguez–Hertz [BEFH], §A.4, Corollary A.4.6). Let V be a bundle which is smooth along stables, there exists an embedding $j_{\star }:~V(\star )\rightarrow ~V'(\star )$ such that $V'$ admits smooth stable holonomies. Moreover the embedding $j_{\star }$ is $g_{t}$-equivarient.

As part of the proof of the above theorem, using cocycle normal forms, we have the following crucial observation:

Observation A.11

(Explicit construction of cocycle normal forms) The map $j_{\star }$ is an explicit analytic map.

Due to this nature of the map $j_{\star }$, one may recover factorizability of S, by applying the simple factorization theorem A.5 for the image of the vector $\mathfrak {s}$, $j_{g_{\ell }.y}(\mathfrak {s})\in \oplus _{i=1}^{\dim S}V'(g_{\ell }.y)$, and inverting $j_{g_{\ell }.y}$ by calculating a polynomial approximation by applying the Lagrange-Good inversion formula for power series (c.f. [Hof79]).

The following proposition finishes the proof of Theorem A.9.

Proposition A.12

$i_{g_{\ell }.y}^{-1}:{\hat{V}}(g_{\ell }.y) \rightarrow V(g_{\ell }.y)$ is factorizable as a vector, namely consider the subspace $L(g_{\ell }.y)$ spanned by $i_{g_{\ell }.y}^{-1}$, then this space is factorizable.

The bundle $Hom({\hat{V}}(\star ),V(\star ))$ is smooth along stables, as the attached bundle ${\hat{V}}$ is. Using Theorem A.10 of Brown–Eskin–Filip–Rodriguez–Hertz, we may consider the image of $i_{g_{\ell }.y}^{-1}$ inside a bundle which admits smooth stable holonomies $j_{g_{\ell }.y}(i^{-1}_{g_{\ell }.y}) \in Hom({\hat{V}}(g_{\ell }.y),V(g_{\ell }.y))'$. We note the following equation

$$\begin{aligned} j_{g_{\ell }.y}(i^{-1}_{g_{\ell }.y}) = H(g_{\ell }.x,g_{\ell }.y).g_{\ell }.H(y,x)j_{y}(i^{-1}_{y}), \end{aligned}$$

(A.8)

where H is the stable holonomy of the bundle $Hom({\hat{V}},V)'$. We may define a map $G_{x}(y)$ to be

$$\begin{aligned} G_{x}(y)=H(y,x)j_{y}(i^{-1}_{y}). \end{aligned}$$

(A.9)

The map $G_{x}(y)$ can be seen a section from $W^{s}(x)$ to some bundle $Hom({\hat{V}}(x),V(x))'$. Using Theorem A.5, expanding the smooth holonomy $H(g_{\ell }.x,\star )$, we may factorize the subspace spanned by $j_{g_{\ell }.y}(i^{-1}_{g_{\ell }.y})$ in the form of the factorization of the holonomy H, acting on the section $G_{x}(y)$.

Using Observation A.11, we may factorize (the one-dimensional subspace defined by) $i^{-1}_{g_{\ell }.y}$, by using the Lagrange-Good formula in order to extract $i_{g_{\ell }.y}^{-1}$ out of the $j_{\star }$ embedding. This concludes the proof of Proposition A.12. Combining the factorizations of $i^{-1}_{g_{\ell }.y}$ from Proposition A.12 and $H_{{\hat{V}}}(g_{\ell }.x,g_{\ell }.y)\circ i_{g_{\ell }.x}$ which follow from Corollary A.5 yields Theorem A.9.

Observation A.13

For any vector bundle V, the translating map defined by Oseledets’ theorem, $i^{-1}_{x}:~{\hat{V}}(x)~\rightarrow ~V(x)$ between the attached bundle and V is “almost-isometry” in the following meaning: for every $v\in {\hat{V}}(x)$ and $|t |\gg 0$ we get

$$\begin{aligned} \frac{\left\Vert g_{t}.i^{-1}_{x}(v) \right\Vert _{V(g_{t}.x)}}{\left\Vert g_{t}.v \right\Vert _{{\hat{V}}(g_{t}.x)}} \ll e^{\varepsilon \cdot |t|}. \end{aligned}$$

(A.10)

This follows immediately from the definition of the quotients and Pesin norms used.

Corollary A.14

(Approximation of $E^{u}(g_{\ell }.y)$). The subspace $E^{u}(g_{\ell }.y)$ is factorizable, for any generic point $g_{\ell }.y$.

Proof

Consider the tangent bundle $V=T_{p}M$. The subspace $E^{u}\le ~TM$ is measurable equivariant, hence by Theorem A.9 it is factorizable, by approximating $P^{-}_{V}(g_{\ell }.x,g_{\ell }.y){\textsf{B}}(g_{\ell }.x)$, where ${\textsf{B}}(g_{\ell }.x)$ is any orthonormal basis of $E^{u}(g_{\ell }.x)$. $\square$

1.4 Approximation of a Taylor polynomial.

Consider the unstable bundle $(M,E^{u})$. Assume that $f:W^{u}(p)\rightarrow \mathbb {R}^{n}$ is some smooth function. We will be interested in studying its Taylor expansion (over $W^{u}(p)$), for example in order to do the computation discussed in Section 4.3. One may think of the Taylor coefficients of such a function as a element in some jet bundle.

Ideally, we would to read of the coefficients from the jet bundle $J(E^{u}(p))$. As the unstable bundle $(M,E^{u})$ is not smooth along stables but only Holder-continuous, we may not be able to use the previous construction. Using the partially hyperbolic splitting we have an embedding of bundle $E^{u}\hookrightarrow TM$ and as a result we get an embedding of the corresponding jet bundles $J(E^{u}(p))\hookrightarrow J(M)$.

Proposition A.15

For any finite k, the sub-bundle of jets of order less or equal to k $J^{\le k}(M)$ of J(M) is factorizable.

Proof

The full jet bundle J(M) is smooth along stables, as TM is, so is $J^{\le k}(M)$. Hence the result follows from Theorem A.9, considering $S=V=J^{\le k}(M)$. $\square$

Once we have a vector in V in hand, one may evaluate the jet in any given direction of TM in order to recover the needed derivatives.

Corollary A.16

Combining the factorizability of $E^{u}$ and J(M), we may factorize the various Taylor coefficients of any smooth function defined over $W^{u}(g_{\ell }.y)$.

In view of the formula for directional derivative, we have the following:

Lemma A.17

Fix some smooth $f:W^{u}(p)\rightarrow \mathbb {R}^{n}$. Assume that $v,v'\in ~TM$ then $\Vert D_{v}f(p)-D_{v'}f(p) \Vert \ll _{f}~ \Vert v-v' \Vert$, where the dependence of f is by some $C^{\star }$-norm, locally around p, where the order of the norm relates to the order of the differential.

This has an immediate corollary showing we may actually recover the Taylor approximation, as we have a good approximation to both the coefficients and the actual subspace.

Corollary A.18

(Factorization of Taylor polynomial). Let $TP_{f,R}$ denote the Taylor polynomial of $f:W^{u}(g_{\ell }.y)\rightarrow ~\mathbb {R}^{n}$ of degree R and $\widetilde{TP}_{f,R}$ denote the factorized Taylor polynomial, then

$$\begin{aligned} \Vert TP_{f,R}(z)-\widetilde{TP}_{f,R}(z) \Vert \ll e^{-\alpha \cdot \ell }\cdot \max \{1,\Vert z\Vert ^R\}. \end{aligned}$$

As all the coefficients’ differences are bounded by $e^{-\alpha \cdot \ell }$, the corollary follows at once.

1.5 Approximation of $W^{u}$ and the transfer function.

Consider $W^{u}$ as a function to some ambient Euclidean space $f:W^{u}\rightarrow \mathbb {R}^{n}$. Then the Taylor polynomial for $W^{u}(q'_1)$ is factorizable due to Corollary A.18, where the degree of the polynomial is determined by the Lyapunov exponent with respect to the error estimate needed, as explain in Section 4.3.

Moreover, the Taylor polynomial for $W^{cs}(u.q_{1/2})$ is trivially factorizable (but in terms of an operator $A(q,u,\ell )$ depending on u as well), as it depends only on u.q ($W^{u}(q)$) and is independent from $q'$, given u.

We may now solve, approximately, the equation $W^{cs}(u.q_{1/2})\cap W^{u}(q'_{1/2})$ by applying the Lagrange-Good inversion formula [Hof79] to the factorized expression, in order to calculate approximation of z in terms of $u.q_{1/2},q'_{1/2},\ell /2$ by polynomials up to the required precision.

Corollary A.19

The point z is factorizable in the following sense - we identify $z\in W^{u}(q'_1)$, by means of the exponential map, with a vector $z\in E^{u}(q'_1)\le T_{q'_1}M$ dependent on $q,q',u,\ell$. For all $\beta >0$ there exists a vector bundle V, a map $F_{q}:W^{u}(q)\rightarrow ~V_{ext}(q)$, a linear operator $A(q,u,\ell ):~V_{ext}(q)\rightarrow ~T_{q'_1}M$ such that

$$\begin{aligned} \Vert z-A(q,u,\ell )F_{q}(q') \Vert _{T_{q'_1}M} \ll e^{-\beta \cdot \ell }. \end{aligned}$$

For the transfer function $T^{u}_{q'_{1/2}\rightarrow z}$, we may consider the backward flag as an element of the bundle $\oplus _{i}\left( \bigwedge ^{\dim V^{\lambda _i}} TM\right)$. Using Lemma A.17 again for $E^{u}$ and the jet bundle $J(\oplus _{i}\bigwedge ^{\dim V^{\lambda _{i}}}TM)$, yields factorization of the Taylor polynomial for the transfer function.

Knowing the backward flag allows us to measure the distance between $E^{2}\oplus E^{1}$ from $E^{1}$ (in some generalized Grassmannian). As this distance function is smooth, we recover a factorized expression for the distance. Using the procedure described in Section 4.5, as both of our bundles of interest are of dimension 1.

Remark A.20

On a more general framework, one may calculate the “distance” $E^{1}\oplus E^{2}/E^{1}$ in terms of the normal forms coordinates of $W^{u}(q'_1/2)$. This requires showing that the normal forms coordinates themselves are factorizable, and will be done in the upcoming work of Brown–Eskin–Filip–Rodriguez–Hertz [BEFH].

Corollary A.21

The distance $hd_{z}(W^{uu}(q'_1), W^{uu}(q_1))$ is factorizable, namely there exists a bundle $V_{ext}$, a map $F_{q}:W^{s}(q)\rightarrow V_{ext}(q)$ and an operator $A(q,u,\ell ):~V_{ext}(q_{1})\rightarrow ~\mathbb {R}$ such that that

$$\begin{aligned} |\text {hd}_{u.q_1}(W^{uu}(q'_1),W^{uu}(q_1)) - A(q,u,\ell )F_{q}(q') |\ll e^{-\beta \cdot \ell }. \end{aligned}$$

Using the corollary A.6 we may calculate the needed distance for any $t\le \alpha \cdot \ell$. We may identify the range of the operator $A(q,u,\ell ,t)$ with $\mathcal {Q}$ as defined in (4.11). This concludes the proof of the factorization theorem 4.4 in Section 4.

1.6 Contraction property.

Fix a biregular point $q\in M$. Recall we fixed some subspaces ($E^{u}$ and $\text {span}\left( T^{u}_{q'_{1/2}}(z)\right)$) which we factorized, along with their Taylor expansions associated to them. Let $V_{total}$ be a finite dimensional vector bundle, which consists of the data needed for the calculation (which includes the Oseledets decomposition and the related copies of the jet bundles). Define a vector bundle $V_{ext}$ to be a bundle consisting the required to factorize $P^{-}_{V_{total}}$. This bundle consists of the higher tangent bundles of M together with the data required in order to calculate the translation map over the embedding of the extended vector bundle of $V_{total}$, given by the theorem of Brown–Eskin–Filip–Rodriguez–Hertz as in (A.8),(A.9).

Let $F_{q}:~W^{s}(q)~\rightarrow ~V_{ext}(q)$ be the measurable map amounting to expanding the data needed to expand operator $P^{-}_{V_{total}}(g_{\ell }.q,\star )$ into a Taylor polynomial of the required degree and evaluating it at $\star =g_{\ell /2}.q'=q'_{1/2}$. The operator A then will amount to appending the operator $P^{-}_{V_{total}}$ with a measurable choice of a subspace for the required subspace we are interested in factorizing in the following sense $A(q,u,\ell )F_{q}(q') = P^{-}_{V_{total}}(g_{\ell /2}.q,g_{\ell /2}.q').\textbf{B}(g_{\ell /2}.q)+\text {ERROR}$, where the error is bounded exponentially in $\ell$ and $\textbf{B}$ is some measurable choice of basis.

Lemma A.22

The following estimate holds:

$$\begin{aligned} |A(q,u,\ell ,t)F_{q}(q) |\ll e^{-\beta \cdot \ell }. \end{aligned}$$

This follows naturally by plugging $q'_1=q_1$ into the expression for the Hausdorff distance $hd_{u.q_1}(W^{uu}_{\text {loc}}(q'_1),W^{uu}_{\text {loc}}(q_1))$ give 0. Essentially, this estimate basically says that in the case that $q=q'$, the computation described above (without the errors arising from the truncations of the Taylor expansions) gives that

$$\begin{aligned} g_{-\ell /2}.z=W^{cs}(u.q_{1/2})\cap W^{u}(q_{1/2})=u.q_{1/2}. \end{aligned}$$

Then we have the immediate corollary from the triangle inequality:

Corollary A.23

$$\begin{aligned} |\text {hd}_{g_{t}.u.q_1}(W^{uu}_{\text{ loc }}(g_{t}.q'_1),W^{uu}_{\text{ loc }}(g_{t}.q_1))- A(q,u,\ell ,t)\left( F_{q}(q')-F_{q}(q)\right) |\ll e^{-\alpha \cdot \ell }. \end{aligned}$$

This is analogous to the construction appearing in Eskin–Mirzakhani [EM18, Proposition 6.11, Proof of Lemma 5.1].

Lemma A.24

The $g_{t}$ action over the subspace of $V_{ext}$ defined by $\text {ess-span } \Big (\bigcup _{q'\in W^{s}_{\text {loc}}(q)}\left( F_{q}(q')-F_{q}(q)\right) \Big )$ is contracting (when we think about those two vectors in a given local trivialization), hence the $g_{-t}$ action is expanding.

Proof

By definition of $\sigma$, it consists of data in the higher tangent bundle and some Oseledets splittings of various bundles. The image of $\sigma (q)$ in the higher tangent bundle amounts to $\overline{0}$, while the image of $\sigma (q')$ amounts to a vector in the stable part of space hence contracted (in an exponential manner) under the $g_{t}$-action, namely

$$\begin{aligned} \begin{aligned} \Vert F_{g_{t}.q}(g_{t}.q')\mid _{T^{\le k}M}-F_{g_{t}.q}(g_{t}.q)\mid _{T^{\le k}M} \Vert _{V_{ext}}&\ll dist(g_{t}.q',g_{t}.q) \\&\ll e^{-\lambda _{C}\cdot t}. \end{aligned} \end{aligned}$$

(A.11)

For the Oseledets data, using the theorem by [ABF16], we see that the Oseledets decomposition is dominated, in a Holder fashion, by the distance, namely

$$\begin{aligned} \begin{aligned} \Vert F_{g_{t}.q}(g_{t}.q')\mid _{\text {Oseledets}}-F_{g_{t}.q}(g_{t}.q)\mid _{\text {Oseledets}} \Vert _{V_{ext}}&\ll dist(g_{t}.q',g_{t}.q)^{\eta } \\&\ll e^{-\lambda _{C}\cdot t\cdot \eta }, \end{aligned} \end{aligned}$$

(A.12)

for some $\eta >0$ depending on the Oseledets decomposition. $\square$

Appendix B: Proofs of Lemma 3.12 and Auxiliary Lemmata

We start with an easy Lemma.

Lemma B.1

For any $\rho > 0$ there is a constant $c(\rho )>0$ with the following property: Let $A: V \rightarrow W$ be a linear map between Euclidean spaces. Then there exists a proper subspace $\mathfrak {M} \subset V$ such that for any v with $\Vert v\Vert =1$ and $d(v,\mathfrak {M}) > \rho$, we have

$$\begin{aligned} \Vert A\Vert \ge \Vert A v\Vert \ge c (\rho ) \Vert A\Vert . \end{aligned}$$

Proof of Lemma B.1

The matrix $A^t A$ is symmetric, so it has a complete orthogonal set of eigenspaces $W_1, \dots , W_m$ corresponding to eigenvalues $\mu _1~>~\mu _2~>~\dots ~>~\mu _m$. Let $\mathfrak {M} = W_1^{\perp }$. $\square$

1.1 Proof of Lemma 3.12.

Let $\mu _{\star }^{s}$ denote the conditional measure along the partition $\mathcal {B}^{s}$. We push this measure forward by the measurable map $F_{x}$ to a measure $(F_{x})_{\star }(\mu ^{s}_{x})$ supported in $F_{x}(W^{s}(x))\subset V(x)$. We define $\mathcal {L}(x)$ as the linear span of $\textrm{supp}\left( F_{x}(\mu ^{s}_{x})\right) \subset V(x)$.

We note that as $W^{s}(x)$ is a sub-bundle of V(x) and the projection of $F_{x}(W^{s}(x))$ is the identity map, the map $F_{x}$ is actually injective, hence choosing an appropriate point in $F_{x}(W^{s}(x))$ in particular yields a point in $W^{s}(x)$.

Remark B.2

As we assume that $\mu ^{>1}_{x}$ is absolutely continuous with respect to the Riemannian volume on $W^{>1}_{loc}(x)$, using Ledrappier-Young entropy formula [LY82a] we see that $h_{\mu }(g_{t})>0$. We also have that $h_{\mu }(g_{t})=~h_{\mu }(g_{-t})$ as $g_{t}$ is invertible flow. Hence $h_{\mu }(g_{-t})>0$ as well. Using the Ledrappier-Young entropy formula again, we see that we must have that $\dim _{H}(\mu ^{s}_{x})>0$ for $\mu$ almost every x. In particular, we see that for $\mu$ almost every x, $\mathcal {L}(x)$ is not the trivial subspace, namely $\dim \mathcal {L}(x)>0$, as $W^{s}(x)$ is embedded isometrically into V(x).

Lemma B.3

For $\mu$ almost every $x\in M$, for any $\epsilon >0$, the restriction of $(F_{x})_{\star }(\mu ^{s}_{x})$ to the ball $B_{\epsilon }(\overline{0})\subset V(x)$ is not supported on a finite union of proper affine subspaces.

Proof of Lemma B.3

Suppose not. Denote by N(x) the minimal number of subspaces such that for some $\epsilon =\epsilon (x)>0$, the restriction of $(F_{x})_{\star }\mu ^{s}_{x}$ to the ball $B_{\epsilon }(0)$ is supported on N affine subspaces with non-zero $W^{s}$ components.

As the projection to $F_{x}(W^{s}(x))$ to $W^{s}(x)$ is the identity, and the action of $g_{-t}$ is expanding over $W^{s}(x)$, we have that the $g_{-t}$ action over $\mathcal {L}(x)$ is expanding in the following sense: For every affine subspace U of V(x) with non-zero $W^{s}(x)$ component, we have that $g_{-t}.U \rightarrow \infty$. Then N(x) is invariant under $g_{-t}$, as for $g_{-t}.x$ we may pick $\epsilon (g_{-t}.x):=C(t)\cdot \epsilon (x)$ for some $C\ge 1$ which amounts to the expansion factor. Hence N(x) is constant almost-surely as in invariant function over an ergodic system. Now we claim that $N(x)=0$ almost surely. Assume not, so $N(x)=k>0$ almost surely. Take $y\in W^{s}(x)$ such that $F_{x}(y)$ is inside some affine subspace which is contained inside the support of $F_{x}.\mu ^{s}_{x}$. Hence as $N(y)=N(x)$ we must have that there are k subspaces which are supported by $(F_{y})_{\star }\mu ^{s}_{y}$. As $\mu ^{s}_{x}=\mu ^{s}_{y}$, and as x and y are stably-related we have $V(x)=V(y)$. Now we see that the conditional measure is supported over at-least $k+1$ subspaces, which yields a contradiction.

Therefore we see that $(F_{x})_{\star }.\mu ^{s}_{x}$ can only be supported over subspaces with 0 as their $W^{s}$ component, which are also proper. This contradicts the definition of $\mathcal {L}(x)$. $\square$

Lemma B.4

For every $\omega >0$ and $N>0$ there exists $\beta _{1}=\beta _{1}(\,N)>~0$, $\rho _{1}=\rho _{1}(\,N)>0$ and a compact subset $K_{\omega ,N}\subset M$ of measure at-least $1-\omega$ such that for all $x\in K_{\omega ,N}$ and any proper subspaces $\mathfrak {M}_{1},\ldots ,\mathfrak {M}_{N}$ we have

$$\begin{aligned} (F_{x})_{\star }\mu ^{s}_{x}\left( F_{x}.\mathcal {B}^{s}[x] \setminus \bigcup _{k=1}^{N}Nbhd(\mathfrak {M}_{k},\rho _{1}) \right) \ge \beta _{1}\cdot (F_{x})_{\star }\mu ^{s}_{x}\left( F_{x}.\mathcal {B}^{s}[x] \right) \end{aligned}$$

Proof of Lemma B.4

By Lemma B.3, for $\mu$ almost every x there exists $\beta _{x}(N)>0$ and $\rho _{x}(N)>0$ such that for any subspaces $\mathfrak {M}_1(x),\ldots ,\mathfrak {M}_{N}(x)$ we have

$$\begin{aligned} (F_{x})_{\star }\mu ^{s}_{x}\left( F_{x}.\mathcal {B}^{s}[x] \setminus \bigcup _{k=1}^{N}Nbhd(\mathfrak {M}_{k},\rho _{x})\right) \ge \beta _{x}\cdot (F_{x})_{\star }\mu ^{s}_{x}(F_{x}.\mathcal {B}^{s}[x]). \end{aligned}$$

(B.1)

Taking the union over decreasing values of $\rho ,\beta$ give that

$$\begin{aligned} \mu \left( \bigcup _{\rho>0, \beta >0}\left\{ x \in M \mid \text {(B.1) holds for } x \right\} \right) =1. \end{aligned}$$

Now for any $\omega$, choosing sufficiently small $\rho _{1},\beta _{1}$ yields the subset $K_{\omega ,N}$ of $\bigcup _{\rho>0, \beta >0}\left\{ x \in M \mid \text {(B.1) holds for } x \right\}$ of measure larger than $1-\omega$. $\square$

Lemma B.5

For every $\omega >0$ and $\epsilon _{1}>0$ there exists $\beta =\beta (\omega ,\epsilon _{1})>~0$ and $K_{\omega }\subset M$ a compact subset of measure larger than $1-\omega$ and $\rho =~\rho (\omega ,\epsilon _{1})>0$ such that the following holds - suppose that for each $u\in \mathcal {B}$ we have an proper subspace $\mathfrak {M}_{u}(x)$ of V(q). Let

$$\begin{aligned} \begin{aligned} J_{\text {good}}(x) = \left\{ y\in F_{x}.\mathcal {B}^{s}(x) \bigg \vert \text {at least } (1-\epsilon _{1}) \text { fraction of } u \text { in } \mathcal {B}, d_{0}(y,\mathfrak {M}_{u}(x))>\rho /2 \right\} . \end{aligned} \end{aligned}$$

Then for any $x\in K_{\omega }$

$$\begin{aligned} (F_{x})_{\star }\mu ^{s}_{x}(J_{\text {good}}(x)) \ge \beta \cdot (F_{x})_{\star }\mu ^{s}_{x}(F_{x}.\mathcal {B}^{s}[x]). \end{aligned}$$

Proof of Lemma B.5

Let $n=\dim V(q)$. By considering determinants, it is easy to show that for any $C > 0$ there exists a constant $c_n = c_n(C) > 0$ depending on n and C such that for any $\omega > 0$ and any points $v_1, \ldots , v_n$ in a ball of radius C with the property that $\Vert v_1\Vert \ge \omega$ and for all $1 < i \le n$, $v_i$ is not within $\omega$ of the subspace spanned by $v_{1},\ldots , v_{i-1}$, then $v_1,\ldots ,v_n$ are not within $c_{n}\cdot \omega ^{n}$ of any $n-1$ dimensional subspace. Let $k_{\max }\in \mathbb {N}$ be the smallest integer greater than $1+n/\epsilon _{1}$ and let $N=N(\epsilon _{1})=\left( {\begin{array}{c}k_{\max }\\ n-1\end{array}}\right)$. Let $\beta _{1},\rho _1$ and $K_{\omega ,N}$ as in Lemma B.4. Let $\beta =\beta (\omega ,\epsilon _{1})=\beta _{1}(\omega ,N(\epsilon _{1}))$, $\rho =\rho (n,\epsilon _{1})=c_{n}\cdot \rho _{1}^{n}$, $K_{\omega }(\epsilon _{1})=K_{\omega ,N(\epsilon _{1})}$. Define

$$\begin{aligned} J_{\text {bad}}(x)=\mathcal {B}^{s}[x]\setminus J_{\text {good}}(x). \end{aligned}$$

We claim that $J_{\text {bad}}(x)$ is contained in the union of $\rho _{1}$-neighborhoods of at-most N subspaces. Suppose this is not true. Then, for $1 \le k \le k_{max}$ we can inductively pick points $v_1,\ldots , v_k \in J_{\text {bad}}(x)$, such that $v_{j}$ is not inside a $\rho _{1}$-neighborhood of any of the subspaces spanned by $v_{i_1},\ldots ,v_{i_r}$ for $1\le i_{1}< \cdots \le i_{r} \le j-1$. Then, any r-tuple of the points $v_{i_1},\ldots , v_{i_r}$ is not contained within $\rho =c_{n}\rho _{1}$ of a single subspace. As $v_{i}\in J_{\text {bad}}(x)$ there exists $U_{i}\subset \mathcal {B}$ with $|U_{i} |\ge \epsilon _{i}|\mathcal {B}|$ such that for all $u\in U_{i}$, $d_{0}(v_{i},\mathfrak {M}_{u})<\rho /2$. We claim that for any $1\le i_{1}< \cdots \le i_{r} \le j-1$ we have

$$\begin{aligned} U_{i_1}\cap \cdots \cap U_{i_r} = \emptyset . \end{aligned}$$

Assume that $u\in U_{i_1}\cap \cdots \cap U_{i_r}$, then each of $v_{i_1},\ldots ,v_{i_r}$ is within $\rho /2$ of the single subspace $\mathfrak {M}_{u}$, in contradiction to the choice of the $v_{i}$’s. Now we calculate

$$\begin{aligned} \epsilon _{1}\cdot k_{\max }\cdot |\mathcal {B}|\le \sum _{i=1}^{k_{\max }} |U_{i} |\le n\cdot |\bigcup _{i=1}^{k_{\max }} U_{i} |\le n\cdot |\mathcal {B} |, \end{aligned}$$

Which is a contradiction to the choice of $k_{\max }$.

Now by applying Lemma B.4 we get

$$\begin{aligned} \begin{aligned} (F_{x})_{\star }\mu ^{s}_{x}(J_{\text {good}}(x))&\ge (F_{x})_{\star }\mu ^{s}_{x}\left( F_{x}.\mathcal {B}^{s}[x] \setminus \bigcup _{k=1}^{N}Nbhd(\mathfrak {M}_{k},\rho _{1}) \right) \\&\ge \beta \cdot (F_{x})_{\star }\mu ^{s}_{x}\left( F_{x}.\mathcal {B}^{s}[x] \right) \end{aligned} \end{aligned}$$

$\square$

We are now ready for the proof of Lemma 3.12.

Proof of Lemma 3.12

Define

$$\begin{aligned} M'_{\text {dense}}=\left\{ x\in M' \mid \mu ^{s}_{x}(M'\cap \mathcal {B}^{s}[x]) \ge (1-\delta ^{1/2})\mu ^{s}_{x}(\mathcal {B}^{s}[x]) \right\} . \end{aligned}$$

Since $\mathcal {B}^{s}$ is a partition, we must have that $\mu (M'_{dense})\ge (1-\delta ^{1/2})$, by Markov’s inequality. For $x\in M''$ we have that

$$\begin{aligned} (F_{x})_{\star }\mu ^{s}_{x}(F_{x}.(M'\cap \mathcal {B}^{s}[x])) \ge (1-\delta ^{1/2})(F_{x})_{\star }\mu ^{s}_{x}(F_{x}.\mathcal {B}^{s}[x]) . \end{aligned}$$

Let $\beta (\omega ,\epsilon _1)$ as in Lemma B.5. Set

$$\begin{aligned} c(\delta )=\delta +\inf \left\{ (\omega ^2+\epsilon _{1}^{2})^{1/2} \mid \beta (\omega ,\epsilon _{1})\ge 8\cdot \delta ^{1/2} \right\} . \end{aligned}$$

Clearly we have that $c(\delta )\rightarrow 0$ as $\delta \rightarrow 0$. By definition of $c(\delta )$, we may choose $\omega =\omega (\delta )<c(\delta )$ and $\epsilon _{1}=\epsilon _{1}(\delta )<c(\delta )$ such that $\beta (\omega ,\epsilon _{1})\ge ~8\cdot ~\delta ^{1/2}$. By Lemma B.5, for $x\in K_{\omega }$ we have

$$\begin{aligned} (F_{x})_{\star }\mu ^{s}_{x}(J_{\text {good}}(x)) \ge \beta \cdot (F_{x})_{\star }\mu _{x}^{s}(F_{x}.\mathcal {B}^{s}[x]) \ge 8\cdot \delta ^{1/2}\cdot (F_{x})_{\star }\mu _{x}^{s}(F_{x}.\mathcal {B}^{s}[x]). \end{aligned}$$

Let $M''=M'_{\text {dense}}\cap K_{\omega }$. We have $\mu (M'')\ge 1-\delta -\delta ^{1/2}-~c(\delta )$, so $\mu (M'')\rightarrow 1$ as $\delta \rightarrow 0$. Also if $q\in M''$ then we have that $M''\cap ~J_{\text {good}}(q)~\ne ~\emptyset$, by measure considerations. Hence we may choose $q'=s.q\in \mathcal {B}^{s}[q]$ such that $F_{q}(s.q)\in J_{\text {good}}(q)$. The upper bound for $dist(q,q')$ follows trivially by the bound over the diameter of each atom in the partition. The lower bound follows from the fact that up to using the exponential map and its inverse (from $W^{s}(q)$ to the tangent space), we see that $\Vert s \Vert ^{2} \gg dist(F_{q}(s.q),p)^{2} + dist(p,\overline{0}) \ge \rho ^{2}(\delta )$, where p stands for the orthogonal projection from $F_{q}(s.q)$ to $\mathfrak {M}_{u}(q)$. $\square$

Appendix C: QNI Condition

For this appendix we will assume $(M,g_{t},\mu )$ is an Anosov system. The definition can be easily modified to accommodate other situations.

Assume $x,x'\in M$ are two points which are stably related meaning $x'\in W^{s}(x)$. We will use the notation u.x to denote points over the unstable leaf of $x, W^{u}(x)$. This notation is convenient and derived from homogeneous dynamics. Given $u.x\in W^{u}(x)$, we define the center stable projection of u.x to $W^{u}(x')$ as the unique point $u'.x'$ such that $u'.x'\in W^{u}(x')\cap W^{cs}(x)$. This projection defines a continuous mapping $\textrm{Pr}^{cs}_{x,x'}$ between $W^{u}(x)$ and $W^{u}(x')$. We remark here that Hasselblat–Wilkinson [HW99] showed that the map is only Hölder continuous in general.

Definition C.1

A dynamical quadrilateral $\mathfrak {Q}$ is an ensemble of four points, $x,x',u.x, z \in M$ such that x is a Pesin regular point and the following holds:

• $x'\in W^{s}(x)$,

• $u.x \in W^{uu}_{\text {loc}}(x)$,

• $y \in W^{u}(x')\cap W^{cs}(u.x)$.

So the quadrilateral $\mathfrak {Q}$ is defined by a base point x, translation along $W^{uu}_{\text {loc}}(x)$ and translation along $W^{s}(x)$. As the point y is defined by $x,x',u.x$ via means of $y=\textrm{Pr}^{cs}_{x,x'}(u.x)$, we will denote $\mathfrak {Q}=\mathfrak {Q}(x,x',u.x)$.

Consider a dynamical quadrilateral $\mathfrak {Q}(x,x',u.x)$ for which $x'$ is a Pesin regular point as well. Inside $W^{u}(x')$, we may consider the embedded disk $W^{>1}_{\text {loc}}(x')$. The following notion of non-integrability means that this projection does not close on itself, in a quantitative manner. We refer the reader to Section 3.2 to the definition of the conditional measures used in the next definition.

Definition C.2

We say that a dynamical quadrilateral $\mathfrak {Q}(x,x',u.x)$ formed of Pesin regular points $x\in M,x'\in W^{s}(x)$ and $u.x\in W^{>1}_{\text {loc}}(x)$ satisfies quantitative non-integrability (abbreviated QNI from now on) of order $\alpha$ and constant C, for some fixed $\alpha >0$ and fixed $C>0$, if the following estimate holds:

$$\begin{aligned} dist_{W^{u}}(y,W^{uu}_{\text {loc}}(x')) \ge C\cdot \min \left\{ 1,\textrm{dist}(x,x')^{\alpha }, \textrm{dist}(x,u.x)^\alpha \right\} . \end{aligned}$$

(C.1)

Estimate (C.1) allows one to get an a-priori bound over the growth of $dist_{W^{u}}(g_{t}.y,W^{uu}_{\text {loc}}(g_{t}.x'))$ using the minimal growth ensure by the Lyapunov exponent $\lambda _{1}$. In particular, one sees that the stopping time when this distance grows to size O(1), is polynomial in $dist(x,x')$. This polynomial depends will play a crucial role in the construction of the factorization operator in Section 6.

We say that an Anosov system $(M,g_{t},\mu )$ satisfies quantitative non-integrability if there exists some $\alpha >0$ and $C>0$ such that for a set of positive $\mu$-measure of points $\mathfrak {P}\subset M$, for every $x\in \mathfrak {P}$, there exists a subset $\mathfrak {S}(x)\subset W^{s}_{\text {loc}}(x)$ of positive $\mu ^{s}_{x}$-density, not including x, and for $x'\in ~\mathfrak {S}(x)$, there exist subsets $Q_{QNI}(x,x')\subset W^{uu}_{\text {loc}}(x)$, $Q_{QNI}(x',x)\subset W^{uu}_{\text {loc}}(x')$ satisfying

$$\begin{aligned} \liminf _{r\rightarrow 0}\frac{{\textsf{m}}_{x}^{uu}(Q_{QNI}(x,x') \cap B_{r}^{uu}(x))}{{\textsf{m}}_{x}^{uu}(B^{uu}_{r}(x))}>0 , \end{aligned}$$

and

$$\begin{aligned} \liminf _{r\rightarrow 0}\frac{{\textsf{m}}_{x'}^{uu}(Q_{QNI}(x',x) \cap B^{uu}_{r}(x'))}{{\textsf{m}}_{x'}^{uu}(B^{uu}_{r}(x'))}>0, \end{aligned}$$

where $B^{uu}_{r}(\star )$ denotes a ball of radius r in the embedded disk $W^{uu}_{\text {loc}}(\star )$, such that for all $u.x\in Q_{QNI}(x,x')$ and all $u'.x'\in Q_{QNI}(x',x)$ the dynamical quadrilaterals $\mathfrak {Q}(x,x',u.x)$, $\mathfrak {Q}(x',x,u'.x')$ satisfy QNI of order greater or equal to $\alpha$ and constants greater or equal to C.

Fig. 5

Illustration of the dynamical quadrilateral $\mathfrak {Q}$ and the QNI condition.

As $x,x'$ are stably-related, flowing forward shrinks the distance between them in an exponential manner, hence we may assume that $\mathfrak {S}(x)$ accumulates at x (Fig. 5).

As the set satisfying QNI is of positive $\mu$ measure, and as the set of $\mu$-generic points is of full measure, we may assume that the set of points which satisfy QNI is formed of generic points.

The main difference between this definition and Definition 1.6 is the requirement regarding the the trajectories of $x'$ under f to return to the set $\mathcal {X}$, but the previous definition had to ensure the non-integrability “in all scales”. In the proof itself, we actually going to use the second definition. We will indicate where we are using that definition in due course.