# Tubular dimension: Leaf-Wise Asymptotic Local Product Structure, and Entropy
and Volume Growth

Snir Ben Ovadia

###### Abstract.

We introduce the notion of tubular dimension, and give a formula for it. As an
application we show that every invariant measure of a $C^{1+\gamma}$
diffeomorphism of a closed Riemannian manifold admits an asymptotic local
product structure for conditional measures on intermediate foliations of
unstable leaves. As a second application, we prove a bound on the gap between
any two consecutive conditional entropies, in the form of volume growth. As a
third application, for certain $C^{\infty}$ maps we compute all conditional
entropies for the measure of maximal entropy; And in particular as a
consequence, in a follow-up paper we compute the Hausdorff dimension of the
equilibrium measure of holomorphic endomorphisms of
$\mathbb{C}\mathbb{P}^{k}$, $k\geq 1$, giving a solution to the Binder-DeMarco
conjecture, and answering a question of Fornæss and Sibony.

###### Contents

  1. 1 Introduction and main results
    1. 1.1 Motivation
    2. 1.2 Main results
  2. 2 Setup and definitions
  3. 3 Covers, differentiation, and the intermediate entropy
    1. 3.1 Covering and differentiation lemmas
    2. 3.2 The intermediate entropy
  4. 4 Tubes and tubular dimension
    1. 4.1 Tubular covers and differentiation
    2. 4.2 Tubular dimension
  5. 5 Leaf-wise asymptotic local product structure
  6. 6 Entropy gap and volume growth

##  1\. Introduction and main results

###  1.1. Motivation

Given a dynamical system $(M,f)$, invariant probability measures are an
important object, which represents the system in its different “equilibria”,
where the probability of an event does not change with time.

In particular, smooth dynamical system (i.e $M$ is a closed Riemannian
manifold, and $f\in\mathrm{Diff}^{1+}(M)$) are important as models of physical
systems, where we expect the time evolution map to be smooth. Such systems are
endowed with a geometric structure, and a natural question is to understand
the interplay between the geometry of the space, and “the geometry of
invariant measures”. For example, are invariant measures exact dimensional?
Can we find singular sub-manifolds on which an invariant measure disintegrates
in a non-atomic manner? What is the relationship between Lyapunov exponents,
which are a dynamical quantity which measures chaos, and entropy, a measure
theoretic measurement of chaos, and the dimension of invariant measures? This
question relates to the understanding of the geometry of Bowen balls and
balls.

Thanks to Pesin theory [Pes77] we know that every measure which admits a
positive Lyapunov exponent a.e admits a singular smooth sub-manifold a.e,
called an unstable leaf. Studying the measure theoretic properties of the
conditional measures on unstable leaves has physical importance (see for
example the study of SRB and physical measures [You02]).

In [LY85b], Ledrappier and Young proved that for any measure with positive
entropy, the conditional measures on unstable leaves have a rich structure,
which includes exact point-wise dimension. Moreover, conditional measures on
unstable leaves can be disintegrated further into “stronger” unstable leaves,
corresponding to larger Lyapunov exponents. Ledrappier and Young gave a
complete formula to compute the point-wise dimension of each conditional
measure on a strong unstable foliation, where the formula is in terms of
Lyapunov exponents and conditional entropy. The notion of conditional entropy
is a natural extension of the metric entropy: In [BK83] Brin and Katok showed
that the entropy of a measure can be computed by the asymptotic exponential
decay rate of the measure of Bowen balls; Conditional entropy extends this
notion by considering asymptotic exponential decay rate of the conditional
measure of Bowen balls. Thus Ledrappier and Young put together in one formula
three of the most important quantities of smooth dynamical systems: dimension,
entropy, and Lyapunov exponents.

A remarkable aspect of this result is the fact that in particular the
conditional measures on unstable foliations are non-atomic in the presence of
positive entropy (and similarly for strong unstable foliation and conditional
entropy). This implies that the invariant measure is “localized” on smooth
geometric objects.

To illustrate this, consider the strong unstable foliation inside an unstable
leaf, on which the conditionals admit a positive dimension- i.e “many” typical
points inside such a leaf, but not full dimension. Now consider any other
typical foliation, for example a foliation by planes in the chart of the
unstable leaf. Then each plane intersects each strong leaf at finitely many
points (typically), and the less probable case is that this intersection
happens at a typical point.

Another heuristic of the non-triviality of that fact, is the lack of
transverse foliations which correspond to weakly expanding directions in an
unstable leaf. Their lacking implies the difficulty to find any foliations
transverse to the strong unstable leaves on which the conditionals are non-
atomic.

The proof of Ledrappier and Young involves computing the measure of transverse
balls (see Definition 2.4), which are balls saturated by leaves of the strong
unstable foliation. One can think of transverse balls as “long” in the
direction of the strong expansion, and “narrow” in the direction of the weak
expansion.

Q1: What dynamically significant object can we define which is “long” in the
weak direction? Q2: How can we compute the measure of such an object? Q3: Can
we find such an object with a nice geometric description as well?

We call such an object a tube (see Definition 4.1), referring to its geometric
proportions, and to its desired nice geometric description. Questions 1-3 are
particularly interesting in light of the lacking of dynamical foliations
transverse to the strong unstable leaves.

The idea behind the notion of tubes, is to find an object such that if the
weakly expanding direction were to be integrable, the tube would be almost a
pluck of that invariant foliation. However, even in that case, there is no
guarantee that the conditional measures on that foliation would be non-atomic.
Moreover, even in the case that there is such a foliation, and that the
conditional measures on this foliation have non-trivial structure, there is
still no guarantee that they would admit an asymptotic local product
structure, as illustrated in Figure 3 below.

In [BPS99], Pesin, Barreire, and Schmeling showed that hyperbolic measures
(i.e no zero Lyapunov exponents, and at least one positive exponent and one
negative exponent) are exact dimensional (rather than only their
conditionals). They showed further that hyperbolic measures admit asymptotic
local product structure, which we explain below.

Fig.1

Fig. 2

Fig. 3

Assume that we have a discrete measure in a chart where each atom has equal
mass. We can then cover the full measure set of atoms by 3 different covers:
vertical cylinders (called verticals), horizontal cylinders (called
horizontals), and intersections of both (called boxes). In the three figures
above we illustrate the distribution of a measure with, and without a product
structure.

In figure 1 we see nine total atoms, covered by at least nine boxes. Each
vertical contains exactly three atoms, and at least three verticals are
required in order to cover the measure. Similarly for the horizontals. The
notion of product structure in this context means that the number of boxes
needed to cover the measure is the product of the number verticals needed and
horizontals needed. The intersection of every vertical which contains an atom
with a horizontal which contains an atom, is a box which contains an atom.

In figure 2, we see the extreme opposite of a product structure: a diagonal
measure. Each cylinder contains exactly one atom, and the number of
horizontals needed to cover the measure, times the number of verticals needed
to cover the measure, is three times bigger than the number of boxes needed.
For every horizontal which contains an atom, only one vertical intersects it
in an atom. However, diagonal measures are not the only case with no product
structure. In figure 3 we see another example with no product structure, but
which is not diagonal.

The notion of asymptotic local product structure is that for almost every
point, at small enough scale, the number of boxes needed to cover the measure
(or a large measure subset of “good” points) is exponentially large (in the
presence of positive entropy), while the number of boxes is up to a sub-
exponential factor the product of the number of verticals and the number of
horizontals.

In [BPS99] the authors prove that every hyperbolic measure has an asymptotic
local product structure, where the horizontals are Bowen balls to the past,
and the verticals are Bowen balls to the future. This statement gives useful
information on the geometry of the localization of hyperbolic measures. Their
proof relies on the fact that both families of horizontals and verticals
consist of dynamically meaningful objects, and on the fact that Bowen balls to
the past are saturated by unstable leaves (resp. stable leaves for Bowen balls
to the future).

Q4: Can we also show an asymptotic local product structure by tubes and
transverse balls?

###  1.2. Main results

In this paper we answer questions Q1-Q4 from §1.1, and offer a couple of
applications.

  1. (1)

We introduce tubes (see Definition 4.1), which are elongated in the weakly
expanding direction, with exponential eccentricity. In §4 we show geometric
measure theoretic properties of tubes (e.g they form a differentiation basis),
and we compute their measure as a function of their eccentricity, which is
called the tubular dimension (see Definition 4.5).

  2. (2)

As a first application to the formula of the tubular dimension, we prove that
for every $f$-invariant measure, the conditional measures on unstable leaves
admit asymptotic local product structure (see §5).

  3. (3)

As an additional application, in §6 we give a bound to the entropy gap
$h_{i+1}(x)-h_{i}(x)$ for any two consecutive conditional entropies (see
Definition 2.2) for any $f$-invariant measure, where the gap is bounded by the
volume growth (see Definition 6.1).

  4. (4)

In Corollary 6.5, using the bounds on the entropy gap, for $C^{\infty}$
diffeomorphisms which satisfy a certain super-additive relationship between
the volume growth of disks in different dimensions, we compute the conditional
entropies of the measure of maximal entropy (see Definition 2.1).

In particular, in a follow up paper we show that for holomorphic endomorphisms
of $\mathbb{C}\mathbb{P}^{k}$, $k\geq 1$, the assumptions of Corollary 6.5
hold, and as a consequence we provide a formula for the Hausdorff dimension of
the measure of maximal entropy. This gives an answer to a question of Fornæss
and Sibony in their list of fundamental open problems in higher dimensional
complex analysis and complex dynamics ([FsS01, Question 2.17]), and proves the
Binder-DeMarco conjecture ([BD03, Conjecture 1.3]).

##  2\. Setup and definitions

Let $M$ be a closed Riemannian manifold of dimension $d\geq 2$, and let
$f\in\mathrm{Diff}^{1+\gamma}(M)$ with $\gamma>0$.

######  Definition 2.1 (Pesin blocks).

Let $\mu$ be an ergodic $f$-invariant probability measure which admits $u\geq
1$ distinct positive Lyapunov exponents. Set
$\underline{\chi}:=((\chi_{1},k_{1}),\ldots,(\chi_{u},k_{u}))$, where $k_{i}$
is the dimension of the Oseledec subspace corresponding to $\chi_{i}$ for all
$1\leq i\leq u$, and $\chi_{i}>\chi_{i+1}$ for all $1\leq i\leq u-1$.

  1. (1)

Let
$0<\tau\leq\tau_{\underline{\chi}}:=\frac{1}{100d}\min\\{\chi_{u},\chi_{i}-\chi_{i+1}:i\leq
u-1\\}$, and let $C_{\underline{\chi},\tau}(\cdot)$ be the Lyapunov change of
coordinates for points in $\mathrm{LR}_{\underline{\chi}}=\\{\text{Lyapunov
regular points with an index }\underline{\chi}\\}$ (see [KM95]).

  2. (2)

Let
$\mathrm{PR}_{\underline{\chi}}=\\{x\in\mathrm{LR}_{\underline{\chi}}:\limsup_{n\to\pm\infty}\frac{1}{n}\log\|C_{\underline{\chi},\tau}^{-1}(f^{n}(x))\|=0,\forall
0<\tau\leq\tau_{\underline{\chi}}\\}$, the set of $\underline{\chi}$-Pesin
regular points which carries $\mu$.
$\mathrm{PR}:=\bigcup_{\underline{\chi}}\mathrm{PR}_{\underline{\chi}}$ is
called the set of Pesin regular points.

  3. (3)

Given $x\in\mathrm{PR}_{\underline{\chi}}$, let $E_{j}(x)$ be the Oseledec
subspace of $x$ corresponding to $\chi_{j}$.

  4. (4)

A Pesin block $\Lambda^{(\underline{\chi},\tau)}_{\ell}$ is a subset of
$\bigcup_{|\underline{\chi}^{\prime}-\underline{\chi}|_{\infty}<\tau}\mathrm{PR}_{\underline{\chi}^{\prime}}$
which is a level set $[q_{\tau}\geq\frac{1}{\ell}]$ of a measurable function
$q_{\tau}:\bigcup_{|\underline{\chi}^{\prime}-\underline{\chi}|_{\infty}<\tau}\mathrm{PR}_{\underline{\chi}^{\prime}}\to(0,1)$
s.t (a) $\frac{q_{\tau}\circ f}{q_{\tau}}=e^{\pm\tau}$, (b)
$q_{\tau}(\cdot)\leq\frac{1}{\|C^{-1}_{\underline{\chi},\tau}(\cdot)\|^{\frac{d}{\gamma}}}$.
Often we omit the subscript $\ell$ when the dependence on $\ell$ is clear from
the context.

  5. (5)

Given $i\leq u$, there exists a strong unstable leaf $W^{i}(x)$ tangent to
$\oplus_{j\leq i}E_{j}(x)$, for $\mu$-a.e $x$ (see [Rue79] for example). The
laminations $W^{1}>\ldots>W^{u}$ are called the intermediate foliations.

######  Definition 2.2 (Conditional entropy and dimension [LY85b]).

Let $\mu$ be an ergodic $f$-invariant probability measure which admits $u\geq
1$ distinct positive Lyapunov exponents. Let $\xi_{i}$, $i=1,\ldots,u$ be the
increasing measurable partitions subordinated to the intermediate foliations
constructed in [LY85b, § 9], and let $\\{\mu_{\xi_{i}}(x)\\}$ be the system of
conditional measures given by the Rokhlin disintegration theorem.

  1. (1)

Given $\xi_{i}(x)$, for $y\in\xi_{i}(x)$ write $d_{n}^{i}(y,x):=\max_{0\leq
j\leq n}d_{f^{j}[\xi_{i}(x)]}(f^{j}(y),f^{j}(x))$.

  2. (2)

$B_{\xi_{i}(x)}(x,n,\epsilon):=\\{y\in\xi_{i}(x):d^{i}_{n}(y,x)\leq\epsilon\\}$.

  3. (3)

$h_{i}(x):=\lim_{\epsilon\to
0}\limsup\frac{-1}{n}\log\mu_{\xi_{i}(x)}(B_{\xi_{i}(x)}(x,n,\epsilon))$.

  4. (4)

$d_{i}(x):=\limsup_{r\to 0}\frac{\mu_{\xi_{i}(x)}B_{\xi_{i}(x)}(x,r)}{\log
r}$.

######  Theorem 2.3 ([LY85a, LY85b]).

Let $\mu$ be an ergodic $f$-invariant probability measure which admits $u\geq
1$ distinct positive Lyapunov exponents. Then for all $1\leq i\leq u$,

  1. (1)

$h_{i}(x)=\lim_{\epsilon\to
0}\liminf\frac{-1}{n}\log\mu_{\xi_{i}(x)}(B_{\xi_{i}(x)}(x,n,\epsilon))$ is
constant a.e,

  2. (2)

$d_{i}(x)=\liminf_{r\to 0}\frac{\mu_{\xi_{i}(x)}B_{\xi_{i}(x)}(x,r)}{\log r}$
is constant a.e,

  3. (3)

for $i\leq u-1$, $d_{i+1}-d_{i}=\frac{h_{i+1}-h_{i}}{\chi_{i+1}}$,

  4. (4)

$h_{u}=h_{\mu}(f)$.

######  Definition 2.4 (Transverse structure).

Let $\mu$ be an ergodic $f$-invariant Borel probability measure, and assume
that $\mu$ admits $u\geq 2$ distinct positive Lyapunov exponents.

  1. (1)

Given $1\leq i\leq u-1$ and a partition element $\xi_{i+1}(x)$, Ledrappier and
Young construct in [LY85b, § 11] a transverse metric
$d^{T}_{i+1}(\cdot,\cdot)$ on $\xi_{i+1}(x)$.111The transverse metric is
defined for $\eta_{i+1}/\eta_{i}$ in [LY85b, § 11], and extends to $\xi_{i+1}$
through the extension of the Lipschitz continuous projection map on a compact
set. Their construction uses the fact the strong foliation $\xi_{i}>\xi_{i+1}$
is Lipschitz, which they prove for $f\in\mathrm{Diff}^{2}(M)$. In [BPS99],
Barreira, Pesin, and Scmeling extend the proof to our setup, where
$f\in\mathrm{Diff}^{1+\gamma}(M)$, and the strong foliation $\xi_{i}$ is
restricted leaves of points in a Pesin block
$\Lambda^{(\underline{\chi},\tau)}_{\ell}$, where $\tau>0$ is sufficiently
small w.r.t $\chi_{u}$.

  2. (2)

A transverse ball is a ball in the transverse metric
$d_{i+1}^{T}(\cdot,\cdot)$.

  3. (3)

By [LY85b, § 8.3], for each $\xi_{i+1}$ and
$x\in\Lambda^{(\underline{\chi},\tau)}_{\ell}$ there exists a Lipschitz change
of coordinates $\mathcal{O}_{x}$ s.t all $W^{i}$-unstable leaves of
$\Lambda^{(\underline{\chi},\tau)}_{\ell}$ become planes. $\mathcal{O}_{x}$
depends only on the leaf $\xi_{i+1}(x)$. Let $\pi_{i+1}$ be the projection
onto the $\mathbb{R}^{\mathrm{dim}W^{i+1}-\mathrm{dim}W^{i}}$-coordinates and
$\pi_{i}$ be the projection onto the
$\mathbb{R}^{\mathrm{dim}W^{i}}$-coordinates.

Assume w.l.o.g once and for all throughout this paper that $\mu$ is ergodic
and $\chi_{u}>1$.

######  Definition 2.5 (Pucks, intermediate entropy, and scaling parameters).

Given $1\leq i\leq u-1$, fix $\Delta>0$, the window scale, and

  1. (1)

Let
$\beta\in(\frac{\Delta}{\chi_{i}},\frac{\Delta}{\chi_{i+1}})\cap\mathbb{Q}$,
and let
$\alpha:=(\Delta-\chi_{i+1}\beta)\in(0,\Delta\cdot(1-\frac{\chi_{i+1}}{\chi_{i}}))$
be the scaling parameters.

  2. (2)

Define the $i+1$-th $(\Delta,\beta,n,\epsilon)$-puck at $x$,

$P_{n,\epsilon}(x):=f^{-\lceil n\beta\rceil}[B^{T}(f^{\lceil
n\beta\rceil}(x),e^{-\alpha n-\epsilon n})],$

for all $n\in\mathbb{N}$. To ease notation, we refer to
$(\Delta,\beta,n,\epsilon)$-pucks as simply $(n,\epsilon)$-pucks.

  3. (3)

Set the $i+1$-th intermediate entropy of $\mu$ at $x$:

$h_{i+1}^{I}(x)=h^{I}_{i+1}(x,\Delta,\beta):=\lim_{\epsilon\to
0}\limsup_{n\to\infty}\frac{-1}{n}\log\mu_{\xi_{i+1}(x)}(P_{n,\epsilon}(x)).$

  4. (4)

The lower $i+1$-th intermediate entropy of $\mu$ at $x$ is defined by

$\underline{h}^{I}_{i+1}(x)=\underline{h}^{I}_{i+1}(x,\Delta,\beta):=\lim_{\epsilon\to
0}\liminf_{n\to\infty}\frac{-1}{n}\log\mu_{\xi_{i+1}(x)}(P_{n,\epsilon}(x)).$

Remarks:

  1. (1)

The limit over $\epsilon$ in the definition of the intermediate entropy exists
(similarly for the lower intermediate entropy), since it is monotone in
$\epsilon$.

  2. (2)

Note that it is clear from definition that the lower intermediate entropy is
always non-negative (and so does the intermediate entropy).

  3. (3)

The transverse metric on $\xi_{i+1}$ is canonical in the sense that all
transverse metric are equivalent due to the Lipschitz property of the
$\xi_{i}$ foliations, hence the intermediate entropy and the lower
intermediate entropy do not depend on the metric $d^{T}_{i+1}(\cdot,\cdot)$.

  4. (4)

The notion of intermediate entropy (and similarly lower intermediate entropy)
extends naturally to non-ergodic measures. By the point-wise ergodic theorem
each unstable leaf carries typical points of at most one ergodic component,
and so the conditional measures are always computed for an ergodic component.

  5. (5)

The definition extends trivially to the case where $\mu$ admits no positive
Lyapunov exponents: The conditional measures are Dirac delta measures, hence
$h^{I}_{i+1}(x)=0$ $\mu$-a.e; Therefore from here onwards we assume that $\mu$
admits a positive Lyapunov exponent a.e.

  6. (6)

The definition of the $i+1$-th intermediate entropy extends to the case $i=0$
where $\xi_{0}$ is the partition into points, and so the transverse metric is
simply the metric of $\xi_{1}$.

  7. (7)

The assumption $\chi^{u}>1$ does not make us lose generality, since if
$\chi_{u}\in(0,1)$ we may always consider
$f^{\lceil\frac{2}{\chi_{u}}\rceil}\in\mathrm{Diff}^{1+\gamma^{\prime}}(M)$
which preserves $\mu$ and
$\chi_{u}(\mu,f^{\lceil\frac{2}{\chi_{u}}\rceil})\geq 2>1$.

  8. (8)

Note, by the choice of $\alpha$ and $\beta$, when $\epsilon>0$ is small enough
so $\chi_{i}\cdot\beta>\Delta+\epsilon$,

$\displaystyle e^{-n\alpha-n\epsilon}\cdot e^{-n\beta\chi_{i+1}}=$
$\displaystyle e^{-\Delta n-n\epsilon}\gg e^{-n\beta\chi_{i}}.$ (1)

Therefore the pucks are “long” in the $\chi_{i+1}$-direction, but are very
“thin” in other directions.

##  3\. Covers, differentiation, and the intermediate entropy

###  3.1. Covering and differentiation lemmas

In this section we prove some useful geometric measure theoretic properties of
pucks, and give a formula to the intermediate entropy. This will be the
underlying foundations which allows us to later on compute the tubular
dimension (§4), prove the leaf-wise asymptotic local product structure (§5),
and bound the entropy gaps by volume growth (§6).

######  Lemma 3.1 (Puck covers).

There exists a constant $C_{d}$ which depends only on the dimension $d$, s.t
for a.e $x$, for every $n\in\mathbb{N}$, and for every $\epsilon>0$, every set
$A\subseteq\xi_{i+1}(x)$ can be covered by pucks centered at $A$,
$\mathcal{P}_{n}$ s.t for all $P\in\mathcal{P}_{n}$,
$\\#\\{P^{\prime}\in\mathcal{P}_{n}:P\cap P^{\prime}\neq\varnothing\\}\leq
C_{d}$.

###### Proof.

Assume w.l.o.g that $n\beta\in\mathbb{N}$. Notice that pucks refine the
partition $f^{-n\beta}[\xi_{i+1}]$, therefore it is enough to cover with a
bounded multiplicity $A\cap f^{-n\beta}[\xi_{i+1}(x_{0})]$ for $x_{0}\in A$.
Then we can cover by $f^{n\beta}[P_{n,\epsilon}(\cdot)]$ the set
$f^{n\beta}[A]\subseteq\xi_{i+1}(f^{n\beta}(x_{0}))$. Note, that
$f^{n\beta}[P_{n,\epsilon}(\cdot)]=B^{T}(f^{n\beta}(\cdot),e^{-\alpha
n-n\epsilon})$ by definition, hence we simply cover by transverse balls, which
admit a Besicovitch cover with a multiplicity bounded by $C_{d}$. ∎

######  Lemma 3.2 (Puck differentiation).

Let $A\in\mathcal{B}$, then for all $\epsilon>0$, $\mu$-a.e $x\in A$:

$\lim_{\epsilon\to
0}\limsup_{n\to\infty}\frac{-1}{n}\log\frac{\mu_{\xi_{i+1}(x)}(P_{n,\epsilon}(x)\cap
A)}{\mu_{\xi_{i+1}(x)}(P_{n,\epsilon}(x))}=0.$

###### Proof.

The proof is similar to [BORH, Lemma 2.3], where Lemma 3.1 is used in place of
[BORH, Lemma 2.2]. ∎

###  3.2. The intermediate entropy

######  Theorem 3.3.

Let $1\leq i\leq u-1$, then for $\mu$-a.e $x$,

$h^{I}_{i+1}(x)=\underline{h}^{I}_{i+1}(x)=\beta
h_{i+1}+\alpha\frac{h_{i+1}-h_{i}}{\chi_{i+1}}.$

###### Proof.

Let $n\in\mathbb{N}$ and let $\epsilon>0$, and assume w.l.o.g that
$n\beta\in\mathbb{N}$. The idea of the proof is to compute
$\mu_{\xi_{i+1}(f^{n\beta}(x))}(B^{T}(f^{n\beta}(x),e^{-\alpha
n-n\epsilon}))$, and then estimate the change of conditional measure when
pulling backwards by $f^{-n\beta}$.

Assume first that for all $\delta>0$ there exists $\epsilon_{\delta}>0$ small
enough, s.t for all $\epsilon\in(0,\epsilon_{\delta})$, there exists
$n_{\epsilon}$ s.t $\forall n\geq n_{\delta}$,

$\mu_{\xi_{i+1}(f^{n\beta}(x))}(B^{T}(f^{n\beta}(x),e^{-\alpha
n-n\epsilon}))=e^{-n\alpha\frac{h_{i+1}-h_{i}}{\chi_{i+1}}\pm\delta n}.$ (2)

Under this assumption, we continue to estimate
$\mu_{\xi_{i+1}(x)}(P_{n,\epsilon}(x))$: Write
$Q_{k}:=f^{k}[P_{n,\epsilon}(x)]$ and $\mu_{k}:=\mu_{\xi_{i+1}(f^{k}(x))}$,
then

$\displaystyle\mu_{\xi_{i+1}(x)}(P_{n,\epsilon}(x))=$
$\displaystyle\prod_{k=0}^{n\beta-1}\frac{\mu_{k}(Q_{k})}{\mu_{k+1}(Q_{k+1})}\cdot\mu_{n\beta}(Q_{n\beta}).$
(3)

Note, $f^{-1}[Q_{k+1}]=Q_{k}$, then by the invariance of $\mu$ and the
uniqueness of conditional measures:

$\frac{\mu_{k}(Q_{k})}{\mu_{k+1}(Q_{k+1})}=\mu_{k}(f^{-1}[\xi_{i+1}(f^{k+1}(x))])=:e^{-I_{i+1}(x)}.$

By the point-wise ergodic theorem, for $\mu$-a.e $x$,

$\lim_{n\beta\to\infty}\frac{1}{n\beta}\sum_{k=0}^{n\beta-1}I_{i+1}\circ
f^{k}(x)=\int I_{i+1}d\mu.$

By [LY85b, § 9.2, § 9.3], the r.h.s equals $h_{i+1}$, and so we are only left
to prove (2).

By [LY85b, § 11.4], for $\mu$-a.e $y$,

$\lim_{n\to\infty}\frac{-1}{n}\log\mu_{\xi_{i+1}(y)}(B^{T}(y,e^{-\alpha
n-n\epsilon}))=\frac{h_{i+1}-h_{i}}{\chi_{i+1}}(\alpha+\epsilon).$

Let $\delta>0$ small, and let $n_{\delta}^{\prime}\in\mathbb{N}$ large enough
so $\mu(\Omega_{\delta})\geq 1-\delta^{3}$ where

$\displaystyle\Omega_{\delta}:=\Big{\\{}y:\forall
n\geq\delta^{2}n_{\delta}^{\prime},$
$\displaystyle\mu_{\xi_{i+1}(y)}(B^{T}(y,e^{-\alpha
n-n\epsilon}))=e^{-n\frac{h_{i+1}-h_{i}}{\chi_{i+1}}(\alpha+\epsilon)\pm\delta^{2}n},$
$\displaystyle\frac{1}{n}\sum_{\ell=0}^{n-1}I_{i+1}(f^{-\ell}(y))=h_{i+1}\pm\delta^{2}\Big{\\}}.$

By the point-wise ergodic theorem, for $\mu$-a.e $x$ there exists
$n_{\delta}\geq n_{\delta}^{\prime}$ s.t $\forall n\geq n_{\delta}$, $\exists
j\in[n(1+\delta^{2}),n(1+2\delta^{2})]$ s.t $f^{j}(x)\in\Omega_{\delta}$, thus

$\displaystyle\mu_{n\beta}(Q_{n\beta})\geq$
$\displaystyle\mu_{n}(f^{-(j-n)}[B^{T}(f^{j}(x),e^{-\alpha n-\epsilon n})])$
$\displaystyle\geq$ $\displaystyle
e^{-n\frac{h_{i+1}-h_{i}}{\chi_{i+1}}(\alpha+\epsilon)\pm\delta^{2}n}\cdot\exp\left(-(j-n)\frac{1}{j-n}\sum_{\ell=n}^{j-1}I_{i+1}(f^{-\ell})(f^{j}(x))\right).$

Since $2\delta^{2}n\geq j-n\geq\delta^{2}n$, by the definition of
$\Omega_{\delta}$, for all $\delta>0$ small enough, we get
$\mu_{n\beta}(Q_{n\beta})\geq
e^{-n\alpha\frac{h_{i+1}-h_{i}}{\chi_{i+1}}-\delta n}$. The upper bound is
shown similarly. ∎

Remark: Note, the formula for the intermediate entropy can be also written as
$h^{I}_{i+1}(x)=\beta h_{i}+\Delta\frac{h_{i+1}-h_{i}}{\chi_{i+1}}$. This
alludes to the asymptotic local product structure which we prove in §5, since
it implies:

$\mu_{\xi_{i+1}(x)}(P_{n,\epsilon}(x))\approx\mu_{\xi_{i}(x)}(P_{n,\epsilon}(x))\cdot\mu_{\xi_{i+1}(x)}(B^{T}(x,e^{-\Delta
n})).$

######  Lemma 3.4.

For $\mu$-a.e $x$,

$h^{I}_{1}(x)=\underline{h}^{I}_{1}(x)=\frac{h_{1}}{\chi_{1}}(\alpha+\chi_{1}\beta).$

###### Proof.

The definition of pucks indeed extends to the case of $i=0$ (see the sixth
itemized remark after Definition 2.5):
$P_{n,\epsilon}(x)=f^{-n\beta}[B_{\xi_{1}(x)}(f^{n\beta}(x),e^{-n\alpha-n\epsilon})]$.
Therefore, for all $\delta>0$, for all $\epsilon>0$ small enough, for
$\mu$-a.e $x$, for all $n$ large enough so $n\beta\in\mathbb{N}$,

$B_{\xi_{1}(x)}(x,e^{-\Delta n-\delta n})\subseteq P_{n,\epsilon}(x)\subseteq
B_{\xi_{1}(x)}(x,e^{-\Delta n+\delta n}).$

Thus
$h^{I}_{1}(x)=\Delta\frac{h_{1}}{\chi_{1}}=\frac{\alpha+\chi_{1}\beta}{\chi_{1}}h_{1}$
$\mu$-a.e. ∎

Remark: The formula in Lemma 3.4 coincides with the formula in Theorem 3.3
when substituting $i=0$ and $h_{0}=0$.

##  4\. Tubes and tubular dimension

For $i=2$, the action of differential in the $\xi_{1}$ leaves is almost
conformal- in the sense that around good Pesin points, the differential acts
with the maximal Lyapunov exponent, up to a small exponential error.

However, for $i>2$, this is no longer the case, as the action of the
differential acts with different exponents in different Oseledec directions,
which breaks the conformality. In turn, this gives the pucks a “distorted”
shape in the $\xi_{i}$ direction.

The purpose of this section is to treat this issue, and define a proper object
with a nice geometric description, whose measure we can estimate. This is done
by saturating the pucks into tubes. Recall Definition 2.4.

######  Definition 4.1 (Tubes).

Let $1\leq i\leq u-1$.

  1. (1)

We restrict the range of values of the scaling parameter $\beta$ further- let

$\beta_{i+1}\in\left(\frac{\Delta}{\chi_{i}},\min\left\\{\frac{\Delta}{\chi_{i}\cdot(1-\theta)},\frac{\Delta}{\chi_{i+1}}\right\\}\right)\cap\mathbb{Q},$

where $\theta>0$ is the Hölder exponent of the $i+1$-th Oseledec direction on
$\Lambda^{(\underline{\chi},\tau)}$, whenever $0<\tau<\frac{1}{4}\min_{i\neq
j}\\{|\chi_{i}-\chi_{j}|,|\chi_{u}|\\}$. $\beta_{i+1}$ is called the tubular
scale. Since $\Delta$ is fixed, we write $\beta_{i+1}$ to denote that it lies
in the restricted range of values.

  2. (2)

Fix a Pesin block $\Lambda:=\Lambda^{(\underline{\chi},\tau)}_{\ell}$, and let
$x\in\Lambda$. Then the $i+1$-th $(n,\epsilon,\Lambda)$-pre-tube at $x$ is
defined by

$\check{T}_{n,\epsilon}(x):=\bigcup\left\\{P_{n,\epsilon}(y):y\in
B_{\xi_{i}(x)}(x,e^{-n\chi_{i}\beta_{i+1}})\cap\Lambda\right\\}.$

Given $\Delta>0$, the $i+1$-th $(\Delta,\beta_{i+1},n,\epsilon)$\- tube at $x$
is defined by

$T_{n,\epsilon}(x):=B^{T}(x,e^{-\Delta n})\cap\\{y:|\pi_{i}(y)-\pi_{i}(x)|\leq
e^{-n\chi_{i}\beta_{i+1}+\sqrt{\epsilon}n}\\}.$

To ease notation we refer to $(\Delta,\beta_{i+1},n,\epsilon)$-tubes as
$(n,\epsilon)$-tubes or tubes.

Remark: Note, $\theta$ is well-defined and depends only on $\underline{\chi}$,
by [KdlLPW01, Appendix A]. In addition, note that we have
$\max\\{\chi_{i+1}\beta_{i+1},\chi_{i}\beta_{i+1}(1-\theta)\\}<\Delta<\chi_{i}\beta_{i+1}$.

###  4.1. Tubular covers and differentiation

######  Lemma 4.2 (Tubular covers).

For $\mu$-a.e $x$, for every $A\subseteq\xi_{i+1}(x)$, $A$ can be covered by
tubes in a Besicovitch manner, with a multiplicity bounded by
$\widetilde{C}_{d}$, where $\widetilde{C}_{d}$ is a constant depending on the
manifold and the partitions $\xi_{i},\xi_{i+1}$.

###### Proof.

We first cover $A_{i}:=\pi_{i+1}^{-1}[A]$ by a specific cover of transverse
balls: Take $x_{1}\in A_{i}$. Given $\\{x_{1},\ldots,x_{n}\\}$, take
$x_{n+1}\in A\setminus\bigcup_{j\leq n}B^{T}(x_{j},\frac{1}{2}e^{-\Delta n})$.
The process continues as long as $A_{i}$ is not covered. However, $\forall
j\neq j^{\prime}$, $B^{T}(x_{j},\frac{1}{6}e^{-\Delta n})\cap
B^{T}(x_{j^{\prime}},\frac{1}{6}e^{-\Delta n})$, therefore the process stops
at a finite time $N$, as the quotient space $\xi_{i+1}/\xi_{i}$ admits a
metric with finite volume. Let $\mathrm{Vol}_{T}$ be the volume on the
quotient space, then the cover $\mathcal{C}_{T}$ has multiplicity bounded by
$\frac{\max\mathrm{Vol}_{T}(B^{T}(\cdot,\frac{3}{2}e^{-\Delta
n}))}{\min\mathrm{Vol}_{T}(B^{T}(\cdot,\frac{1}{6}e^{-\Delta
n}))}\leq\frac{\max\mathrm{Vol}_{T}(B^{T}(\cdot,\frac{9}{2}e^{-\Delta
n}))}{\min\mathrm{Vol}_{T}(B^{T}(\cdot,\frac{1}{6}e^{-\Delta
n}))}\equiv\widetilde{C}_{i+1}$.

For every $B\in\mathcal{C}_{T}$, cover $A\cap B$ in a Besicovitch cover by
tubes, with multiplicity bounded by $C_{d}$. Denote this cover by
$\mathcal{C}^{B}$. This is possible since each tube extends across $B$, and it
is enough to cover $\pi_{i}[B\cap A]$ in $\xi_{i}$ by balls.

We claim that the cover
$\mathcal{C}:=\bigcup\\{\mathcal{C}^{B}:B\in\mathcal{C}_{T}\\}$ has
multiplicity bounded by $\widetilde{C}_{i+1}\cdot
C_{d}\equiv\widetilde{C}_{d}$. For every $B\in\mathcal{C}_{T}$,
$\bigcup\mathcal{C}^{B}\subseteq\frac{3}{2}B$. However, the number of
$B^{\prime}\in\mathcal{C}^{T}$ s.t
$\frac{3}{2}B\cap\frac{3}{2}B^{\prime}\neq\varnothing$ is bounded by
$\frac{\mathrm{Vol}_{T}(\frac{9}{2}B)}{\min\mathrm{Vol}(\frac{1}{6}B^{\prime})}\leq\widetilde{C}_{i+1}$.
Inside each $B^{\prime}$, the cover has multiplicity bounded by $C_{d}$, and
we’re done. ∎

######  Corollary 4.3 (Tubular differentiation).

Let $A\in\mathcal{B}$, then for all $\epsilon>0$, $\mu$-a.e $x\in A$:

$\lim_{\epsilon\to
0}\limsup_{n\to\infty}\frac{-1}{n}\log\frac{\mu_{\xi_{i+1}(x)}(T_{n,\epsilon}(x)\cap
A)}{\mu_{\xi_{i+1}(x)}(T_{n,\epsilon}(x))}=0.$

###### Proof.

The proof is similar to [BORH, Lemma 2.3], where Lemma 4.2 is used in place of
[BORH, Lemma 2.2]. ∎

###  4.2. Tubular dimension

######  Proposition 4.4.

Let $\epsilon>0$ small, and let $\Lambda=\Lambda^{(\underline{\chi},\tau)}$ be
a Pesin block with $0<\tau<\epsilon$. Then for all $n$ large enough, for
$\mu$-a.e $x\in\Lambda$,

$T_{n,\epsilon}(x)\supseteq\check{T}_{n,\epsilon}(x).$

###### Proof.

For all $\epsilon>0$, for every $y\in\xi_{i}(x)\cap\Lambda$,
$P_{n,\epsilon}(y)\subseteq B^{T}(x,e^{-n\chi\Delta})$ for all $n$ large
enough w.r.t $\Lambda$. Therefore, we wish to show that $\forall z\in
P_{n,\epsilon}(y)$, $|\pi_{i}(z)-\pi_{i}(x)|\leq
e^{-n\chi_{i}\beta_{i+1}+\sqrt{\epsilon}n}$.

Note, given $y_{1},y_{2}\in B_{\xi_{i}(x)}(x,e^{-n\beta_{i+1}\chi_{i}})$, and
given $z_{1}\in P_{n,\epsilon}(y_{1})$ and $z_{2}\in P_{n,\epsilon}(y_{2})$,
the disposition $|\pi_{i}(z_{1})-\pi_{i}(z_{2})|$ is bounded by:

$\displaystyle|\pi_{i}(z_{1})-\pi_{i}(z_{2})|\leq
2e^{-n\beta_{i+1}\chi_{i}}+2\cdot(e^{-n\Delta}\cdot\sin\sphericalangle(E_{i+1}(y_{1}),E_{i+1}(y_{1}))+e^{-n\beta_{i+1}\chi_{i}+n2\epsilon}).$

The angle is bounded by $\sphericalangle(E_{i+1}(y_{1}),E_{i+1}(y_{1}))\leq
C_{\mathrm{H\ddot{o}l}}(\Lambda)\cdot e^{-n\beta_{i+1}\chi_{i}\theta}$. Since
$\beta_{i+1}<\frac{\Delta}{\chi_{i}(1-\theta)}$, we have

$e^{-n\Delta}\cdot e^{-n\beta_{i+1}\chi_{i}\theta}\leq
e^{-n\beta_{i+1}\chi_{i}}.$

Thus, for all $n$ large enough,
$T_{n,\epsilon}(x)\supseteq\check{T}_{n,\epsilon}(x)$. ∎

Remark: Proposition 4.4 is the key place where we need to use the choice of
parameters $\Delta$, and consequently $\beta_{i+1}$. They are necessary to
compensate for the fact that the $E_{i+1}(\cdot)$ distribution is merely
Hölder continuous on Pesin blocks and not Lipschitz. The key is to “shorten” a
Bowen ball enough into a puck, so the Hölder regularity is sufficient. The
lack of conformality (i.e $\chi_{i}>\chi_{i+1}$) is what allows us to make
sure that the tubes remain still “elongated” in shape, with a uniform
exponential gap on their dimensions.

######  Definition 4.5 (Tubular dimension).

The $i+1$-th tubular dimension at a point $x$ is defined by

$\rho^{T}_{i+1}(x)=\rho^{T}(x,\Delta,\beta_{i+1}):=\lim_{\epsilon\to
0}\limsup_{n\to\infty}\frac{-1}{n}\log\mu_{\xi_{i+1}(x)}(T_{n,\epsilon}(x)).$

The $i+1$-th lower tubular dimension at a point $x$ is defined by

$\underline{\rho}^{T}_{i+1}(x)=\underline{\rho}^{T}(x,\Delta,\beta_{i+1}):=\lim_{\epsilon\to
0}\liminf_{n\to\infty}\frac{-1}{n}\log\mu_{\xi_{i+1}(x)}(T_{n,\epsilon}(x)).$

Remark: To understand the role of the scaling parameters for the tubular
dimension, allow us illustrate an example. Assume that $f$ is a linear map
with orthogonal eigen-directions. Then inside each $W^{2}$ leaf, we can
consider a tube $T$ around a point $x$, with sides parallel to $E_{1}$ and
$E_{2}$, with dimensions $e^{-n\chi_{1}}$ and $e^{-\Delta n}$, respectively.
The window parameter $\Delta$ can vary between $\chi_{2}$ and $\chi_{1}$,222In
this example $\beta$ is fixed with the value $1$, hence
$\frac{\Delta}{\chi_{1}}<1<\frac{\Delta}{\chi_{2}}\Rightarrow\chi_{2}<\Delta<\chi_{1}$.
yielding a Bowen ball and a square ($\|\cdot\|_{\infty}$-ball), respectively.
For both extreme values, the measure of $T$ is known by either the local
entropy estimates, or by the point-wise dimension estimates. The idea behind
tubular dimension is to be able to estimate the measure of the tube for the
rest of the range of possible values of $\Delta$. In generality, the mere
Hölder continuity of the $E_{2}$ direction restricts further the range of
possible dimensions whose measure we can estimate. Moreover, when considering
higher order intermediate foliations, the action of the differential on the
$E_{i}$ direction, $i>1$, is no longer conformal, and so we need to saturate
pucks into tubes.

######  Theorem 4.6.

For every $1\leq i\leq u-1$, for $\mu$-a.e $x$,

$\underline{\rho}^{T}_{i+1}(x)=\rho^{T}_{i+1}(x)=\Delta\cdot\frac{h_{i+1}-h_{i}}{\chi_{i+1}}+\beta_{i+1}\chi_{i}d_{i}.$

###### Proof.

We begin by showing
$\rho_{i+1}^{T}\leq\Delta\cdot(h_{i+1}-h_{i})+\beta_{i+1}\chi_{i}d_{i}$
$\mu$-a.e. By Theorem 3.3 (recall also the succeeding remark), for all
$\delta>0$, for $\mu$-a.e $x$, there exists $\epsilon_{\delta}>0$ s.t for all
$\epsilon\in(0,\epsilon_{\delta})$, for all $n$ large enough w.r.t $\epsilon$,
$\mu_{\xi_{i+1}(x)}(P_{n,\epsilon}(x))\geq e^{-n\beta_{i+1}h_{i}}\cdot
e^{-n\Delta\frac{h_{i+1}-h_{i}}{\chi_{i+1}}}e^{-\delta n}$. Let
$\epsilon\in(0,\delta)$, and let $\tau\in(0,\epsilon)$, $\ell\in\mathbb{N}$,
and $n_{\epsilon}$ s.t $\mu(\Omega_{\delta})\geq 1-\epsilon$, where

$\displaystyle\Omega_{\delta}:=\\{x\in\Lambda^{(\underline{\chi},\tau)}_{\ell}:\forall
n\geq n_{\epsilon},\text{ }$
$\displaystyle\mu_{\xi_{i+1}(x)}(P_{n,\epsilon}(x))\geq
e^{-n\beta_{i+1}h_{i}}\cdot
e^{-n\Delta\frac{h_{i+1}-h_{i}}{\chi_{i+1}}}e^{-\delta n},$
$\displaystyle\mu_{\xi_{i}(x)}(P_{n,\epsilon}(x))=e^{-nh_{i}\beta_{i+1}\pm\delta
n}\\}.$

Let $x$ be a $\mu_{\xi_{i}(x)}$-density point of $\Omega_{\delta}$, then there
exists $n_{x}\geq n_{\epsilon}$ s.t for $n\geq n_{x}$,

$\frac{\mu_{\xi_{i}(x)}(B_{\xi_{i}(x)}(x,e^{-\chi_{i}\beta_{i+1}n})\cap\Omega_{\delta})}{\mu_{\xi_{i}(x)}(B_{\xi_{i}(x)}(x,e^{-\chi_{i}\beta_{i+1}n}))}\geq
1-\delta.$

Moreover, for $\mu$-a.e $x\in\Omega_{\delta}$ exists $n_{x}^{\prime}\geq
n_{x}$ s.t $\mu_{\xi_{i}(x)}(B_{\xi_{i}(x)}(x,e^{-\chi_{i}\beta_{i+1}n}))\geq
e^{-n\chi_{i}\beta_{i+1}d_{i}-\delta n}$.

We may cover $\Omega_{\delta}\cap B_{\xi_{i}(x)}(x,e^{-\chi_{i}\beta_{i+1}n})$
by Lemma 3.1, and there must be at least
$\frac{(1-\delta)e^{-n\chi_{i}\beta_{i+1}d_{i}-\delta
n}}{e^{-nh_{i}\beta_{i+1}+\delta n}}$-many elements in this cover. Thus, by
Proposition 4.4,

$\displaystyle\mu_{\xi_{i+1}(x)}(T_{n,\epsilon}(x))\geq$
$\displaystyle\frac{1}{C_{d}}\cdot\frac{(1-\delta)e^{-n\chi_{i}\beta_{i+1}d_{i}-\delta
n}}{e^{-nh_{i}\beta_{i+1}+\delta n}}\cdot e^{-n\beta_{i+1}h_{i}}\cdot
e^{-n\Delta\frac{h_{i+1}-h_{i}}{\chi_{i+1}}}e^{-\delta n}$ $\displaystyle\geq$
$\displaystyle
e^{-n(\Delta\cdot\frac{h_{i+1}-h_{i}}{\chi_{i+1}}+\beta_{i+1}\chi_{i}d_{i})}e^{-4\delta
n}.$

Since $\delta>0$ was arbitrarily small, this concludes the upper bound of
$\rho_{i+1}^{T}$.

We continue to bound from below the lower tubular dimension: Let
$\delta\in(0,1)$, and let $h\geq 0$ s.t
$\mu([\underline{\rho}_{i+1}^{T}=\rho\pm\frac{\delta}{3}])>0$. There exists
$\epsilon\in(0,\delta^{3})$ s.t $\mu(A)>0$ where

$\displaystyle A:=\\{x\in\Lambda^{(\underline{\chi},\epsilon^{2})}:\text{ }$
$\displaystyle\liminf\frac{-1}{n}\log\mu_{\xi_{i+1}(x)}(T_{n,\epsilon}(x))=\rho\pm\frac{2\delta}{3},$
$\displaystyle\forall n\geq n_{0},\text{ }$
$\displaystyle\mu_{\xi_{i+1}(x)}(B^{T}(x,e^{-\Delta n+\delta^{2}n}))\leq
e^{-\Delta\frac{h_{i+1}-h_{i}}{\chi_{i+1}}n+\delta n}$
$\displaystyle\mu_{\xi_{i}(x)}(B_{\xi_{i}(x)}(x,e^{-n\beta_{i+1}\chi_{i}-\Delta^{2}n}))\leq
e^{-\beta_{i+1}\chi_{i}d_{i}n+\delta n}\\}.$

Let $A_{n}:=\\{x\in
A:\frac{-1}{n}\log\mu_{\xi_{i+1}(x)}(T_{n,\epsilon}(x))=\rho\pm\delta\\}$,
then by the Borel-Cantelli lemma, there exist infinitely many $n$ s.t
$\mu(A_{n})\geq e^{-\epsilon n}$.

Let $x_{n}$ s.t $\mu_{\xi_{i}(x_{n})}(A_{n})\geq e^{-\epsilon n}$. Cover
$A_{n}\cap\xi_{i}(x_{n})$ by tubes as in Lemma 4.2. Then the number of tubes
is bigger than $e^{-n\epsilon}\cdot e^{\beta_{i+1}\chi_{i}d_{i}n-\delta n}$.
On the other hand, the union of the tubes is contained in the transverse ball
$B^{T}(x,e^{-\Delta n+\delta^{2}n})$, therefore,

$e^{-\Delta\frac{h_{i+1}-h_{i}}{\chi_{i+1}}n+\delta n}\geq
e^{-n\epsilon}\cdot\frac{1}{\widetilde{C}_{d}}e^{\beta_{i+1}\chi_{i}d_{i}n-\delta
n}\cdot e^{-\rho n-\delta n}.$

Since $\delta>0$ was arbitrary small, $\epsilon<\delta$, and $n$ can be
arbitrarily large, we are done. ∎

Remark: For the case of $i=0$, one can think of a tube as the collection of
pucks centered inside a ball in $\xi_{0}(x)$. However, $\xi_{0}(x)=\\{x\\}$,
therefore the tube is simply a puck and the tubular dimension is the
intermediate entropy; And in this case we refer to Lemma 3.4. Indeed, when
substituting $h_{0}=d_{0}=0$ in the formula in Theorem 4.6, we get the formula
from Lemma 3.4.

##  5\. Leaf-wise asymptotic local product structure

The notion of tubes and tubular dimension were designed so we can compute the
measure of an object which admits a nice geometric description, and which
satisfies the property that when intersected with a transverse ball, the
intersection is a ball. This allows us to employ all three estimates
simultaneously: the measure of balls (the dimension of the conditional
measures), the measure of transverse balls (the transverse dimension), and
finally the measure of tubes (tubular dimension). The beauty of these
quantities is that they satisfy a multiplicative relationship, which allows us
to conclude the asymptotic local product structure of conditional measures.

######  Theorem 5.1 (Leaf-wise asymptotic local product structure).

For all $1\leq i\leq u-1$, for all $\Delta>0$, for all $\delta>0$, for all
$\epsilon>0$ small enough, there exists a set $K_{\delta}$ s.t
$\mu(K_{\delta})\geq 1-\delta$ and for $\mu$-a.e $x\in K_{\delta}$, for all
$n\in\mathbb{N}$ large enough,

  1. (1)

$\frac{\mu_{\xi_{i+1}(x)}(B_{\xi_{i+1}(x)}(x,e^{-n\Delta-n\epsilon})\cap
K_{\delta})}{\mu_{\xi_{i+1}(x)}(B_{\xi_{i+1}(x)}(x,e^{-n\Delta-n\epsilon}))}\geq
1-\delta$,

  2. (2)

there exist a cover of $B_{\xi_{i+1}(x)}(x,e^{-n\Delta-n\epsilon})\cap
K_{\delta}$, $\mathcal{T}_{n,\epsilon}$, by $i+1$-th $(n,\epsilon)$-tubes
centered at $K_{\delta}$, and a cover $\mathcal{B}^{T}_{n,\epsilon}$ by
transvese balls $B^{T}(\cdot,e^{-n\chi_{i}\beta_{i+1}-n\epsilon})$ centered at
$K_{\delta}$, both with multiplicity bounded by $\widetilde{C}_{d}$,

  3. (3)

set $N^{\mathcal{T}}_{n}:=\\#\mathcal{T}_{n,\epsilon}$,
$N^{\mathcal{B}}_{n}:=\\#\mathcal{B}^{T}_{n,\epsilon}$, and
$N_{n}:=\\#\\{B\in\mathcal{T}_{n,\epsilon}\vee\mathcal{B}^{T}_{n,\epsilon}:B\cap
K_{\delta}\neq\varnothing\\}$, then

$\frac{1}{n}\left|\log\frac{N_{n}^{\mathcal{T}}\cdot
N_{n}^{\mathcal{B}}}{N_{n}}\right|\leq\delta.$ (4)

Note that the definition of $\beta_{i+1}$ (Definition 4.1) guarantees that
$\Delta=\chi_{i}\beta_{i+1}\cdot(1-\theta)<\chi_{i}\beta_{i+1}$ thus the tubes
and the transverse balls are much “thinner” than the dimensions of the ball
$B_{\xi_{i+1}(x)}(x,e^{-n\Delta-n\epsilon})$.

###### Proof.

Let $\delta\in(0,1)$ small, and let $\epsilon\in(0,\delta^{8})$ small enough
so for every $\epsilon\in(0,\epsilon_{\delta})$ there exists
$n_{\epsilon}\in\mathbb{N}$ s.t $\mu(K_{\delta})\geq 1-\delta$, where

$\displaystyle
K_{\delta}:=\Big{\\{}x\in\Lambda^{(\underline{\chi},\epsilon^{3})}_{\ell_{\delta}}:\forall
n\geq n_{\epsilon},\text{ }$
$\displaystyle\mu_{\xi_{i+1}(x)}(B^{T}(x,e^{-n\chi_{i}\beta_{i+1}-n\epsilon}))=e^{-n((\chi_{i}\beta_{i+1}+\epsilon)\frac{h_{i+1}-h_{i}}{\chi_{i+1}}\pm\delta^{2})},$
$\displaystyle\mu_{\xi_{i+1}(x)}(B_{\xi_{i+1}(x)}(x,e^{-n\Delta\pm
n\epsilon}))=e^{-n(\Delta+\epsilon)d_{i+1}\pm\delta^{2}n},$
$\displaystyle\mu_{\xi_{i+1}(x)}(B^{T}(x,e^{-n\Delta+n\epsilon}))=e^{-n((\Delta-\epsilon)\frac{h_{i+1}-h_{i}}{\chi_{i+1}}\pm\delta^{2})},$
$\displaystyle\mu_{\xi_{i}(x)}(T_{n,\epsilon}(x))=e^{-n(\rho_{i+1}^{T}\pm\delta^{2})},$
$\displaystyle\mu_{\xi_{i}(x)}(B_{\xi_{i+1}(x)}(x,e^{-n\chi_{i}\beta_{i+1}+\sqrt{\epsilon}n}))=e^{-n(\chi_{i}\beta_{i+1}d_{i+1}\pm\delta^{2})}\Big{\\}}.$

This is possible by Theorem 4.6 and by [LY85b].

Let $x$ be a $\mu_{\xi_{i+1}(x)}$-Lebesgue density point of $K_{\delta}$ and a
$\mu_{\xi_{i+1}(x)}\circ\pi_{i}^{-1}$-Lebesgue density point of
$\xi_{i}[K_{\delta}]$. Let $n_{x}\geq n_{\epsilon}$ s.t for all $n\geq n_{x}$,

$\displaystyle\frac{\mu_{\xi_{i+1}(x)}(B_{\xi_{i+1}(x)}(x,e^{-n\Delta-n\epsilon})\cap
K_{\delta})}{\mu_{\xi_{i+1}(x)}(B_{\xi_{i+1}(x)}(x,e^{-n\Delta-n\epsilon}))}\geq$
$\displaystyle 1-\delta,$
$\displaystyle\frac{\mu_{\xi_{i+1}(x)}(B^{T}(x,e^{-n\Delta+n\epsilon})\cap\xi_{i}[K_{\delta}])}{\mu_{\xi_{i+1}(x)}(B^{T}(x,e^{-n\Delta+n\epsilon}))}\geq$
$\displaystyle 1-\delta.$

By Lemma 4.2, $K_{\delta}\cap B_{\xi_{i+1}(x)}(x,e^{-n\Delta-n\epsilon})$ can
be covered by a cover of $(n,\epsilon)$-tubes centered at $K_{\delta}\cap
B_{\xi_{i+1}(x)}(x,e^{-n\Delta-n\epsilon})$, $\mathcal{T}_{n,\epsilon}$, and
by a cover of $B^{T}(\cdot,e^{-n(\chi_{i}\beta_{i+1}+\epsilon)})$ transverse
balls centered at $K_{\delta}\cap B_{\xi_{i+1}(x)}(x,e^{-n\Delta-n\epsilon})$,
$\mathcal{B}^{T}_{n,\epsilon}$, both with multiplicity bounded by
$\widetilde{C}_{d}$.

In order to bound $N_{n}^{\mathcal{T}}$ from above, note that
$\bigcup\mathcal{T}_{n,\epsilon}\subseteq
B_{\xi_{i+1}(x)}(x,e^{-n\Delta+n\epsilon})$, and so for all $n$ large enough
w.r.t $\delta$,

$N_{n}^{\mathcal{T}}\leq\widetilde{C}_{d}\frac{e^{-n(\Delta+\epsilon)d_{i+1}+\delta^{2}n}}{e^{-n(\rho_{i+1}^{T}-\delta^{2})}}\leq
e^{n(\rho_{i+1}^{T}-\Delta d_{i+1}+3\delta^{2})}.$ (5)

Similarly, since $\bigcup\mathcal{T}_{n,\epsilon}\supseteq
B_{\xi_{i+1}(x)}(x,e^{-n\Delta-n\epsilon})\cap K_{\delta}$,

$N_{n}^{\mathcal{T}}\geq\frac{(1-\delta)e^{-n(\Delta-\epsilon)d_{i+1}-\delta^{2}n}}{e^{-n(\rho_{i+1}^{T}+\delta^{2})}}\geq
e^{n(\rho_{i+1}^{T}-\Delta d_{i+1}-3\delta^{2})}.$

In order to bound $N_{n}^{\mathcal{B}}$ from above, note that
$\bigcup\mathcal{B}^{T}_{n,\epsilon}\subseteq
B^{T}(x,e^{-n\Delta+n\epsilon})$, and so for all $n$ large enough,

$N_{n}^{\mathcal{B}}\leq\widetilde{C}_{d}\frac{e^{-n((\Delta-\epsilon)\frac{h_{i+1}-h_{i}}{\chi_{i+1}}+\delta^{2})}}{e^{-n((\chi_{i}\beta_{i+1}+\epsilon)\frac{h_{i+1}-h_{i}}{\chi_{i+1}}-\delta^{2})}}\leq
e^{n((\chi_{i}\beta_{i+1}-\Delta)\frac{h_{i+1}-h_{i}}{\chi_{i+1}}+3\delta^{2})}.$
(6)

Similarly, since $\bigcup\mathcal{B}^{T}_{n,\epsilon}\supseteq
B^{T}(x,e^{-n\Delta-n\epsilon})\cap\xi_{i}[K_{\delta}]$,

$N_{n}^{\mathcal{B}}\geq\frac{(1-\delta)e^{-n((\Delta-\epsilon)\frac{h_{i+1}-h_{i}}{\chi_{i+1}}-\delta^{2})}}{e^{-n((\chi_{i}\beta_{i+1}+\epsilon)\frac{h_{i+1}-h_{i}}{\chi_{i+1}}+\delta^{2})}}\geq
e^{n((\chi_{i}\beta_{i+1}-\Delta)\frac{h_{i+1}-h_{i}}{\chi_{i+1}}-3\delta^{2})}.$

Finally, notice that

$N_{n}\leq N_{n}^{\mathcal{T}}\cdot N_{n}^{\mathcal{B}},$

since $N_{n}$ is bounded by the cardinality of the cover which is the
refinement of $\mathcal{T}_{n,\epsilon}$ and $\mathcal{B}^{T}_{n,\epsilon}$.
The lower bound on $N_{n}$ is given by the observation that for every element
$B\in\mathcal{T}_{n,\epsilon}\vee\mathcal{B}^{T}_{n,\epsilon}$ s.t $\exists
x_{B}\in B\cap K_{\delta}$, $B\subseteq
B_{\xi_{i+1}(x)}(x_{B},e^{-n\chi_{i}\beta_{i+1}+\sqrt{\epsilon}n})$, and so

$\mu_{\xi_{i+1}(x)}(B)\leq e^{-n(\chi_{i}\beta_{i+1}d_{i+1}-\delta^{2})}.$

Therefore,

$N_{n}\geq\frac{(1-\delta)e^{-n(\Delta-\epsilon)d_{i+1}-\delta^{2}n}}{e^{-n(\chi_{i}\beta_{i+1}d_{i+1}-\delta^{2})}}\geq
e^{n(\chi_{i}\beta_{i+1}d_{i+1}-\Delta d_{i+1}-3\delta^{2})}.$ (7)

By Theorem 4.6,
$\rho_{i+1}^{T}=\chi_{i}\beta_{i+1}d_{i}+\Delta\frac{h_{i+1}-h_{i}}{\chi_{i+1}}$,
therefore by (5),(6), and (6),

$N_{n}\geq e^{-9\delta^{2}n}\cdot N_{n}^{\mathcal{T}}\cdot
N_{n}^{\mathcal{B}}.$

Therefore, for $\delta>0$ small enough, we are done. ∎

##  6\. Entropy gap and volume growth

In §5 we gave an application of the tubular dimension which was introduced in
§4. In this section we present a second application of the tubular dimension.
We bound the difference between any two consecutive conditional entropies
$h_{i+1}-h_{i}$ by the volume growth of disks of dimension corresponding to
the Oseledec space of $\chi_{i+1}$, $E_{i+1}$.

######  Definition 6.1 (Newhouse [New88]).

Let $1<r\in(\mathbb{R}\setminus\mathbb{N})\cup\\{\infty\\}$, and assume
$f\in\mathrm{Diff}^{r}(M)$. Let $1\leq k\leq d$ be an integer, then the
$(k,r)$-uniform volume growth is defined by

$\mathcal{V}_{k}^{r}(f):=\limsup_{n\to\infty}\frac{1}{n}\log\sup_{\eta}\mathrm{Vol}_{k}(f^{n}\circ\eta[B_{\mathbb{R}^{k}}(0,1)]),$

where the supremum is taken over all diffeomorphisms
$\eta:B_{\mathbb{R}^{k}}(0,1)\to M$ s.t $\|\eta\|_{C^{1+\\{r\\}}}\leq 1$,
where $\\{r\\}:=r-\lfloor r\rfloor$. Such disks are called standard $k$-disks.

######  Proposition 6.2.

For every Pesin block $\Lambda^{(\underline{\chi},\tau)}$ and a measurable set
$A$ s.t $\mu(A\cap\Lambda^{(\underline{\chi},\tau)})>0$, for $\mu$-a.e
$x\in\Lambda^{(\underline{\chi},\tau)}\cap A$, $\exists\epsilon\in(0,\tau)$
s.t for all $n$ large enough, there exist at least

$e^{n(d_{i+1}-d_{i})(\chi_{i}\beta_{i+1}-\Delta)-7d\tau n}\text{-many}$

disjoint transverse balls $B^{T}(\cdot,e^{-n\chi_{i}\beta_{i+1}+3n\tau})$, s.t
their intersection with $T_{n,\epsilon}(x)$ contains a
$(\beta_{i+1}\chi_{i},\frac{\beta_{i+1}}{1-\frac{\theta}{2}},n,\epsilon)$-tube
centered at a point in $\Lambda^{(\underline{\chi},\tau)}\cap A$.

###### Proof.

Let $n_{\tau}\in\mathbb{N}$ s.t $\mu(K_{\tau})\geq
e^{-\frac{\tau}{2}}\mu(\Lambda^{(\underline{\chi},\tau)})\cap A$, where

$K_{\tau}:=\\{x\in\Lambda^{(\underline{\chi},\tau)}\cap A:\forall n\geq
n_{\tau},\mu_{\xi_{i+1}(x)}(B_{\xi_{i+1}(x)}(x,e^{-\chi_{i}\beta_{i+1}n}))\leq
e^{-nd_{i+1}\chi_{i}\beta+\tau n}\\}.$

By Lemma 4.3, for $\mu$-a.e $x\in K_{\tau}$, there exist
$\epsilon_{x}\in(0,\tau^{2})$ s.t $\forall\epsilon\in(0,\epsilon_{x}]$
$\exists n_{x}(\epsilon)\in\mathbb{N}$ s.t $\forall n\geq n_{x}$,

$\frac{\mu_{\xi_{i+1}(x)}(T_{n(1+\sqrt{\epsilon}),\epsilon}(x)\cap
K_{\tau})}{\mu_{\xi_{i+1}(x)}(T_{n(1+\sqrt{\epsilon}),\epsilon}(x))}\geq
e^{-\tau n}.$

Let $\epsilon\in(0,\tau)$ and $n_{\tau}^{\prime}\geq n_{\tau}$ s.t
$\mu(A^{\prime})\geq\mu(K_{\tau})e^{-\frac{\tau}{2}}\geq\mu(A\cap\Lambda^{(\underline{\chi},\tau)})e^{-\tau}$,
where

$A^{\prime}:=\\{x\in K_{\tau}:\epsilon_{x}\geq\epsilon,n_{x}(\epsilon)\leq
n_{\tau}\\}.$ (8)

Let $x\in A^{\prime}$. Cover $K_{\tau}\cap
T_{n(1+\sqrt{\epsilon}),\epsilon}(x)$ with Besicovitch cover of transverse
balls of the form $B^{T}(\cdot,e^{-n\chi_{i}\beta_{i+1}+n\tau})$,
$\mathcal{B}^{T}$.

Notice that for every $B^{T}\in\mathcal{B}^{T}$, if $y\in K_{\tau}\cap
T_{n(1+\sqrt{\epsilon}),\epsilon}(x)\cap B^{T}$, then
$T_{n(1+\sqrt{\epsilon}),\epsilon}(x)\cap B^{T}\subseteq
B_{\xi_{i+1}(x)}(y,e^{-\chi_{i}\beta_{i+1}n})$, and so
$\mu_{\xi_{i+1}(x)}(T_{n(1+\sqrt{\epsilon}),\epsilon}(x)\cap B^{T})\leq
e^{-nd_{i}\chi_{i}\beta+\tau n}$.

By Theorem 4.6, we may also assume that $\epsilon>0$ is sufficiently small and
$n$ is large enough so
$\mu_{\xi_{i+1}(x)}(T_{n(1+\sqrt{\epsilon}),\epsilon}(x))\geq
e^{-n\rho^{T}_{i+1}-n\tau}$. Then, since by Theorem 4.6
$\rho^{T}_{i+1}=n\beta_{i+1}\chi_{i}d_{i}+\Delta(d_{i+1}-d_{i})$, we have

$\\#\mathcal{B}^{T}\geq\frac{1}{C_{d}}\cdot\frac{e^{-n\rho^{T}_{i+1}-n\tau}\cdot
e^{-n\tau}}{e^{-nd_{i+1}\chi_{i}\beta_{i+1}+\tau n}}\geq
e^{n(d_{i+1}-d_{i})(\chi_{i}\beta-\Delta)-4\tau n}.$

Finally, notice that for all $y\in K_{\tau}\cap
T_{n(1+\sqrt{\epsilon}),\epsilon}(x)\cap B^{T}$, for all $n$ large enough
w.r.t $\Lambda^{(\underline{\chi},\tau)}$,

$T_{\beta_{i+1}\chi_{i},\frac{\beta_{i+1}}{1-\frac{\theta}{2}},n,\epsilon}(y)\subseteq
T_{n,\epsilon}(x)\cap B^{T}(x_{B},e^{-n\chi_{i}\beta_{i+1}+3n\tau}),$

where $B^{T}=B^{T}(x_{B},e^{-n\chi_{i}\beta_{i+1}+n\tau})$. In order to bound
from below the number of disjoint elements
$B^{T}(\cdot,e^{-n\chi_{i}\beta+3\tau n})$ which we can find in
$\mathcal{B}^{T}$, we wish to bound their multiplicity in the cover
$\mathcal{B}^{T}$. We refer to the computation in the tubular covers lemma,
Lemma 4.2, to recall that the multiplicity for “inflated” transverse balls is
bounded by the volume estimates on the quotient space $\xi_{i+1}/\xi_{i}$, and
so, $\forall B^{T}\in\mathcal{B}^{T}$,

$\\#\\{(B^{T})^{\prime}\in\mathcal{B}^{T}:(e^{2\tau
n}(B^{T})^{\prime})\cap(e^{2\tau
n}B^{T})\neq\varnothing\\}\leq\widetilde{C}_{d}e^{3d\tau n}.$

Thus in total, for all $n$ large enough, the statement follows. ∎

Remark: Allow us to explain the idea behind the following theorem. Imagine
that the distribution $E_{i+1}$ were Lipschitz continuous, and so $\theta=1$
(recall Definition 4.1). In this case, a tube could be of unbounded
eccentricity, for example $B^{T}(x,e^{-\epsilon
n})\cap\\{y:|\pi_{i}(y)-\pi_{i}(x)|\leq e^{-n\chi_{i+1}}\\}$, with a measure
corresponding to its section by $\xi_{i}$-conditionals-
$e^{-nd_{i}\chi_{i+1}+O(\epsilon n)}$. We could then divide the tube into
$\sim e^{n(h_{i+1}-h_{i})}$-many balls of radius $e^{-n\chi_{i+1}}$ which
contain a good Pesin point, in the spirit of Proposition 6.2. In that case, we
can consider a disk which passes through the Bowen ball of the good point in
each one of those balls, as they all lie in one tube we can guarantee that the
disk does not “wiggle” too much.

Starting with a tube which is long in the transverse metric and thicker than
$e^{-n\chi_{i+1}}$ in the $\xi_{i}$-direction means that the disk which passes
through the points may “wiggle” much, as it may have to go up and down a
distance larger than $e^{-n\chi_{i+1}}$ over a distance of order
$e^{-n\chi_{i+1}}$. Unfortunately, the distribution $E_{i+1}$ is merely Hölder
continuous, and $\theta$ is bounded away from $1$.

The idea of the theorem below, is to use the fact that tubes have some non-
zero exponential eccentricity, in order to intermediately bridge between their
eccentricity and the desired eccentricity; while in each step controlling the
“wiggling” of the disk. This process relies heavily on the geometric measure
theoretic properties of tubes, as a differentiation basis and their geometric
description.

######  Theorem 6.3 (Entropy gap).

Let $1<r\in(\mathbb{R}\setminus\mathbb{N})\cup\\{\infty\\}$, and assume
$f\in\mathrm{Diff}^{r}(M)$. Let $\mu$ be an $f$-invariant ergodic measure with
$u$ distinct positive Lyapunov exponents, and let $k_{i+1}$, $0\leq i\leq
u-1$, be the dimension of the Oseledec subspace of the $i+1$-th largest
exponent. Then,

$h_{i+1}-h_{i}\leq\mathcal{V}_{k_{i+1}}^{r}(f).$

###### Proof.

The case of $i=0$: In that case $h_{0}=0$, and we consider $\xi_{1}(x)$ of a
$\mu$-typical point $x$. For every $\delta\in(0,\frac{1}{2})$, there exist
$\epsilon\in(0,\delta^{2})$, a Pesin block $\Lambda^{(\underline{\chi},\tau)}$
with $\tau\in(0,\epsilon^{2})$, an integer $n_{0}\in\mathbb{N}$, and a set
$K_{\delta}$ s.t $\mu(K_{\delta})\geq 1-\delta$ where

$\displaystyle K_{\delta}:=\\{x\in\Lambda^{(\underline{\chi},\tau)}:\forall
n\geq n_{0},\mu_{\xi_{1}(x)}(B_{\xi_{1}(x)}(x,e^{-(\chi_{1}+\epsilon)n}))\leq
e^{-nh_{1}+\delta n}\\}.$

We can assume w.l.o.g that $x$ is a $\mu_{\xi_{1}(x)}$-density point of
$K_{\delta}$, and cover $B_{\xi_{1}(x)}(x,e^{-\epsilon n})\cap K_{\delta}$
with a Besicovitch cover by balls
$B_{\xi_{1}(x)}(\cdot,e^{-(\chi_{1}+\epsilon)n})$. Note, by [Rue79, Theorem
6.1], $W^{1}(x)$ is a standard $k_{1}$-disk. Assume further that
$\mu_{\xi_{1}(x)}(B_{\xi_{1}(x)}(x,e^{-\epsilon n}))\geq e^{-nd\epsilon}$ for
all $n\geq n_{1}\geq n_{0}$. Notice that for all $y\in
K_{\delta}\cap\xi_{1}(x)$,
$\mathrm{Vol}_{k_{1}}(f^{n}[B_{\xi_{1}(x)}(x,e^{-(\chi_{1}+\epsilon)n})])\geq
e^{-2\delta n\cdot k_{1}}$ then

$\displaystyle\frac{1}{C_{d}}\frac{(1-\delta)\cdot e^{-n\epsilon
d}}{e^{-nh_{1}+\delta n}}\cdot e^{-2\delta
nk_{1}}\leq\mathrm{Vol}_{k+1}(f^{n}[B_{\xi_{1}(x)}(x,e^{-\epsilon n})]).$

Therefore, since $\delta>0$ can be arbitrarily small and $0<\epsilon<\delta$,
we get $h_{1}\leq\mathcal{V}_{k_{1}}^{r}(f)$.

The case of $i\geq 1$:

This case adds a significant difficulty, which is the lack of conformality for
the action of $f$ in the $\xi_{i}$-direction- i.e different expanding factors.
To address this, we use the geometric properties of tubes. The heart of the
proof is to find a tube inside which we can control the number of smaller
tubes around Pesin regular points, and continuing to go into smaller and
smaller scale, in a way which allows us to find a standard $k_{i+1}$-disk with
a bounded norm where we control its volume growth quantitatively. The proof
follows a recursive argument.

Step 1: We define the following starting values:

  1. (1)

Let $\theta>0$ smaller than the Hölder exponent of the $E_{i+1}(\cdot)$ on
Pesin blocks of $\mu$ (see [KdlLPW01]), and small enough so
$(1-\theta)\chi_{i}>\chi_{i+1}$.

  2. (2)

Let $\kappa>0$ small.

  3. (3)

$\Delta_{0}:=(1-\frac{\theta}{2})\kappa$.

  4. (4)

$\beta^{(0)}:=\frac{\Delta_{0}}{(1-\frac{\theta}{2})\chi_{i}}\in(\frac{\Delta_{0}}{\chi_{i}},\frac{\Delta_{0}}{\chi_{i}(1-\theta)})$.

Let $0<\tau\ll(\frac{\kappa^{2}}{7d})^{3}$ small, let
$\Lambda^{(\underline{\chi},\tau)}$ be a Pesin block with a positive measure.

Step 2: We define the recursive values: Let $T^{j}_{n,\epsilon}(x)$ be a
$(\Delta_{j},\beta^{(j)},n,\epsilon)$-tube, where

  1. (1)

$\Delta_{j+1}:=\beta^{(j)}\chi_{i}$,

  2. (2)

$\beta^{(j)}:=\frac{\Delta_{j}}{(1-\frac{\theta}{2})\chi_{i}}$.

Then it follows that for all $j\in\mathbb{N}$,

$\Delta_{j}:=\frac{\kappa}{(1-\frac{\theta}{2})^{j-1}}\text{,
}\beta^{(j)}:=\frac{\kappa}{(1-\frac{\theta}{2})^{j}\chi_{i}}.$ (9)

Set

$N=N(\underline{\chi},\theta,\kappa):=\min\\{j\in\mathbb{N}:\Delta_{j}\geq\chi_{i+1}\\}.$
(10)

Step 3: Let $x\in\Lambda^{(\underline{\chi},\tau)}=:A_{0}$,
$n_{0}\in\mathbb{N}$, and $\epsilon_{0}>0$ given by Proposition 6.2. Define

$L_{x}^{(0)}:=\exp_{x}^{W^{i+1}(x)}[E_{i+1}(x)]\cap
B^{T}(x,e^{-\Delta_{0}n-\epsilon_{0}n}).$

By the construction of the tube $T_{n,\epsilon_{0}}^{0}(x)$,
$L_{x}^{(0)}\subseteq T_{n,\epsilon_{0}}^{0}(x)$. By [Rue79, Theorem 6.1],
$W^{i+1}(x)$ is a $C^{r}$ manifold, and so is the exponential map of it, hence
$L_{x}^{(0)}$ is a $C^{r}$ disk. A standard result regarding the regularity of
local unstable leaves guarantees that
$\sup_{y\in\xi_{i+1}(x)}\|\exp_{y}^{W^{i+1}(x)}\|_{C^{1+\\{r\\}}}<\infty$.

Step 4: Let $1\leq j\leq N$, and $A_{j}$, and define
$A_{j+1}:=(A_{j})^{\prime}$ (recall (8)), $\epsilon_{j+1}\in(0,\epsilon_{j})$,
and $n_{j+1}\geq n_{j}$ given by Proposition 6.2.

Assume that $L^{(j)}_{x}$ is a standard $k_{i+1}$-disk in
$T^{j}_{n,\epsilon_{j}}(x)$. We continue to modify $L_{x}^{(j)}$ in patches,
in a way which keeps its $C^{1+\\{r\\}}$-norm small. Consider all disjoint
transverse balls given by Proposition 6.2, $\mathcal{B}^{T}$. By construction,
each one contains a strictly smaller transverse ball with the same center, and
radius $e^{-n\chi_{i}\beta^{(j)}+\tau n}$, s.t the small transverse ball
contains $x\in A_{j}$. For each such transverse ball $B^{T}$, we get for all
$n\geq n_{j+1}$,

$A_{j}\ni y_{B}\in(e^{-2\tau n}B^{T})\cap
T_{n(1+\sqrt{\epsilon}_{j+1}),\epsilon_{j+1}}^{j}(x)\Subset B^{T}\cap
T_{n,\epsilon_{j+1}}^{j}(x).$

The patching is done over the segment $L_{x}^{(j)}\cap B^{T}$, where we modify
the disk by a “bump” so it contains

$\exp_{y_{B}}^{W^{i+1}(x)}[B_{E_{i+1}(y_{B})}(0,e^{-\chi_{i}\beta^{(j)}n-\epsilon_{j+1}n})].$

Note, $\chi_{i}\beta^{(j)}=\Delta_{j+1}$, and
$\sphericalangle(E_{i+1}(y_{B}),E_{i+1}(x))\leq
e^{-(\Delta_{j}+\sqrt{\epsilon}_{j+1})n\theta}$. The dimensions of the sets
$(e^{-2\tau n}B^{T})\cap T_{n(1+\sqrt{\epsilon}_{j+1}),\epsilon_{j+1}}^{j}$
and $B^{T}\cap T_{n,\epsilon_{j+1}}^{j}(x)$ guarantee that this can be done
without causing the $C^{1+\\{r\\}}$-norm of the disk inside the box to exceed
$4\sup_{y\in\xi_{i+1}(x)}\|\exp_{y}^{W^{i+1}(x)}\|_{C^{1+\\{r\\}}}$.333In each
box $B^{T}\cap T_{n,\epsilon_{j+1}}^{j}(x)$, the modified $L_{y_{B}}^{(j)}$
can be thought of as a graph of a function over $L_{x}^{(j)}$, with derivative
bounded by $2$; If $\eta:B_{\mathbb{R}^{k}}(0,1)\to M$ is the disk, we can
write $\eta=\exp_{y_{B}}^{W^{i+1}(x)}\circ\widetilde{\eta}$, then
$\|\eta\|_{C^{1+\\{r\\}}}\leq\|d_{\cdot}\widetilde{\eta}\|_{\mathrm{Lip}}^{1+\\{r\\}}\cdot\|\exp_{y_{B}}^{W^{i+1}(x)}\|_{C^{1+\\{r\\}}}\leq
4\|\exp_{y_{B}}^{W^{i+1}(x)}\|_{C^{1+\\{r\\}}}$. Call the modified standard
$k_{i+1}$-disk $L_{y_{B}}^{(j+1)}$.

Step 5: We continue this way to modify all $L_{x}^{(j)}$ over all segments
$\\{L_{y_{B}}^{(j+1)}\\}_{B\in\mathcal{B}^{T}}$, and call the total modified
disk $L_{x}^{(j+1)}$. By the main statement of Proposition 6.2, at each step
$j$, we divide each tube of order $j$, into at least
$e^{n(h_{i+1}-h_{i})(\frac{\chi_{i}\beta^{(j)}}{\chi_{i+1}}-\frac{\Delta_{j}}{\chi_{i+1}})-7d\tau
n}$-many tubes of order $j+1$. Therefore, together with (9), the total number
of tubes of order $N$ for all $n\geq n_{N}$, for all $x\in A_{N}$, is at least

$\displaystyle\exp$
$\displaystyle\left(n(h_{i+1}-h_{i})\sum_{j=0}^{N-1}(\frac{\chi_{i}}{\chi_{i+1}}\frac{\kappa}{(1-\frac{\theta}{2})^{j}\chi_{i}}-\frac{1}{\chi_{i+1}}\frac{\kappa}{(1-\frac{\theta}{2})^{j-1}})-7Nd\tau
n\right)$ $\displaystyle=$
$\displaystyle\exp\left(n(h_{i+1}-h_{i})\frac{1}{\chi_{i+1}}(\frac{\kappa}{(1-\frac{\theta}{2})^{N-1}}-\kappa(1-\frac{\theta}{2}))-7Nd\tau
n\right)$ $\displaystyle=$
$\displaystyle\exp\left(n(h_{i+1}-h_{i})\frac{1}{\chi_{i+1}}(\Delta_{N}-\kappa(1-\frac{\theta}{2}))-7Nd\tau
n\right)$ $\displaystyle\text{(}\because\text{\eqref{theNvaluation})}\geq$
$\displaystyle\exp\left(n(h_{i+1}-h_{i})(1-\kappa\frac{1-\frac{\theta}{2}}{\chi_{i+1}})-7Nd\tau
n\right)$ $\displaystyle\geq$
$\displaystyle\exp\left(n(h_{i+1}-h_{i})-\sqrt{\kappa}n\right).$ (11)

The last inequality holds for all $\tau>0$ small enough w.r.t $\kappa$,
$\theta$, and $\chi_{i+1}$, which determine the value of $N$; and for all
$\kappa>0$ small enough w.r.t $(h_{j})_{j\leq u}$ and $\underline{\chi}$.

Step 6: Also notice that for every tube $T^{N}_{n,\epsilon_{N}}$ (since
$\Delta_{N}<\chi_{i+1}(1-\frac{\theta}{2})$),

$\mathrm{Vol}_{k_{i+1}}\Big{(}f^{n(1-\frac{\theta}{2})}\Big{[}L_{x}^{(N)}\cap
T_{n,\epsilon_{N}}^{(N)}(y_{B})\Big{]}\Big{)}\geq e^{-4\tau
n(1-\frac{\theta}{2})k_{i+1}}\geq e^{-4\tau nk_{i+1}},$

thus, by (6),

$\mathrm{Vol}_{k_{i+1}}\left(f^{n(1-\frac{\theta}{2})}[L_{x}^{(N)}]\right)\geq
e^{n(h_{i+1}-h_{i})-\sqrt{\kappa}n}\cdot e^{-4\tau nk_{i+1}}\geq
e^{n(h_{i+1}-h_{i})-2\sqrt{\kappa}n}.$ (12)

Step 7: Finally,
$L_{x}^{(0)}\subseteq\exp_{x}^{W^{i+1}(x)}[B_{E_{i+1}(x)}(0,e^{-\Delta_{0}n})]$,
and let $\widetilde{\eta}_{n}:B_{E_{i+1}(x)}(0,e^{-\Delta_{0}n})\to
L_{x}^{(N)}$ be the representing function of the modified disk $L_{x}^{(N)}$
over $B_{E_{i+1}(x)}(0,e^{-\Delta_{0}n})$. Let $b:B_{E_{i+1}(x)}(0,1)\to
B_{E_{i+1}(x)}(0,e^{-\Delta_{0}n})$, $t\mapsto te^{-\Delta_{0}n}$, and set
$\eta_{n}:=\widetilde{\eta}_{n}\circ b:B_{E_{i+1}(x)}(0,1)\to L_{x}^{(N)}$,
then $\|\eta\|_{C^{1+\\{r\\}}}\leq e^{-\Delta_{0}n}\cdot
4\sup_{y\in\xi_{i+1}(x)}\|\exp_{y}^{W^{i+1}(x)}\|_{C^{1+\\{r\\}}}\leq 1$, for
all $n$ large enough w.r.t $\kappa$.

Therefore,

$h_{i+1}-h_{i}-2\sqrt{\kappa}\leq\frac{1}{1-\frac{\theta}{2}}\mathcal{V}^{r}_{k_{i+1}}(f).$

Since $\kappa>0$ can be arbitrarily small,

$h_{i+1}-h_{i}\leq\frac{1}{1-\frac{\theta}{2}}\mathcal{V}^{r}_{k_{i+1}}(f).$

Since $\theta>0$ can be arbitrarily small, we are done. ∎

Remark: If $f\in\mathrm{Diff}^{2+\gamma}(M)$, then the unstable leaves
$W^{i+1}$ are embedded $C^{2+\gamma}$ leaves, and so the exponential map
$\exp_{x}^{W^{i+1}(x)}$ is $C^{2+\gamma}$. By the same computation in 3, and
Step $7$, the disk $L_{x}^{(N)}$ will in fact have a $C^{2}$-norm bounded by
$1$.

######  Lemma 6.4.

Assume that $f\in\mathrm{Diff}^{r+\gamma}(M)$, $r\in\mathbb{N}$ and
$\gamma>0$, and let $\mu$ be an ergodic $f$-invariant Borel probability with
$u$ distinct positive Lyapunov exponents. Then for all $i\leq u$,

$h_{i}\leq\mathcal{V}_{\ell_{i}}^{r+\gamma}(f).$

###### Proof.

Let $i\leq u$, and let $\tau\in(0,\delta^{2})$, $\epsilon\in(0,\delta^{2})$,
and $n_{\delta}$ s.t $e^{-n_{\delta}\epsilon^{2}}\ll\frac{1}{\ell_{\delta}}$
and $\mu(K_{\delta})>0$ where

$\displaystyle
K_{\delta}:=\\{x\in\Lambda^{(\underline{\chi},\tau)}_{\frac{1}{\ell_{\delta}}}:\forall
n\geq n_{0},\mu_{\xi_{i}(x)}(B_{\xi_{i}(x)}(x,n,e^{-\epsilon
n}))=e^{-nh_{i}\pm\delta n}\\},$ (13)

and $\Lambda^{(\underline{\chi},\tau)}_{\frac{1}{\ell_{\delta}}}$ is a Pesin
block. Let $x\in K_{\delta}$ which is a $\mu_{\xi_{i}(x)}$-Lebesgue density
point of $K_{\delta}$. Assume further that for all $n\geq
n_{\delta}^{\prime}\geq n_{\delta}$,
$\mu_{\xi_{i}(x)}(B_{\xi_{i}(x)}(x,e^{-\epsilon n}))\geq e^{n\epsilon(d+1)}$,
and so for all $n$ large enough

$\displaystyle\mu_{\xi_{i}(x)}(K_{\delta}\cap B_{\xi_{i}(x)}(x,e^{-\epsilon
n}))\geq e^{-\delta}e^{-n\epsilon(d+1)}.$ (14)

Given $n\geq n_{\delta}^{\prime}$, cover $K_{\delta}\cap
B_{\xi_{i}(x)}(x,e^{-\epsilon n})$ by a cover $\mathcal{C}_{n}$ of sets of the
form
$f^{-n}[B_{f^{n}[W_{i}(x)]}(f^{n}(\cdot),e^{-2\epsilon n})]$, with
multiplicity bounded by a constant $C_{k}$ which depends only on the dimension
of $M$. This is possible by the Besicovitch covering lemma.

Note that $\forall B\in\mathcal{C}_{n}$, $B\subseteq B(y_{B},n,e^{-\epsilon
n})\cap\xi_{i}(x)$, where $y_{B}\in K_{\delta}$. Therefore for all $n$ large
enough, together with (14) and (13) we get,

$\displaystyle\\#\mathcal{C}_{n}\geq\frac{e^{-\delta}e^{-n\epsilon(d+1)}}{e^{-nh_{i}+n\delta}}\geq
e^{nh_{i}-2\delta n}.$

It follows that for all $n$ large enough,

$\displaystyle\mathrm{Vol}_{\ell_{i}}(f^{n}[B_{\xi_{i}(x)}(x,e^{-\epsilon
n})])\geq\frac{1}{C_{k}}\cdot e^{nh_{i}-2\delta
n}\cdot\frac{1}{2}e^{-2\epsilon nk}\geq e^{nh_{i}-3\delta n}.$

By [Rue79], $W_{i}(x)$ is $C^{r+\gamma}$-manifold, therefore in total,

$h_{i}-3\delta\leq\mathcal{V}_{\ell_{i}}^{r+\gamma}(f).$

Since $\delta>0$ was arbitrary, the statement follows. ∎

######  Corollary 6.5.

Let $f\in\mathrm{Diff}^{\infty}(M)$ and let $\mu$ be an ergodic measure of
maximal entropy with $u$ distinct positive Lyapunov exponents,
$\chi_{1}>\ldots>\chi_{u}>0$. Let $\ell_{i}$ be the dimension of the Oseledec
subspace corresponding to $\chi_{1},\ldots,\chi_{i}$, and assume that for all
$0\leq i\leq u-1$, for $\gamma>0$,

$\mathcal{V}^{1+\gamma}_{\ell_{i+1}-\ell_{i}}(f)\leq\mathcal{V}^{1+\gamma}_{\ell_{i+1}}(f)-\mathcal{V}^{1+\gamma}_{\ell_{i}}(f).$

Then, for all $i\leq u$,

$h_{i}=\mathcal{V}^{1+\gamma}_{\ell_{i}}(f).$

###### Proof.

By [Yom87], $h_{\mathrm{top}}(f)\geq\mathcal{V}^{1+\gamma}_{u}(f)$. By
[LY85a], $h_{u}=h_{\mu}(f)$. Therefore, since $\mu$ is a measure of maximal
entropy,

  1. (1)

$h_{u}\geq\mathcal{V}^{1+\gamma}_{u}(f)$.

By Theorem 6.3 and by our assumption, for all $i\leq u-1$,

  1. (2)

$h_{i+1}-h_{i}\leq\mathcal{V}^{1+\gamma}_{\ell_{i+1}-\ell_{i}}(f)\leq\mathcal{V}^{1+\gamma}_{\ell_{i+1}}(f)-\mathcal{V}^{1+\gamma}_{\ell_{i}}(f)$.

Therefore, combining $(1)$ and $(2)$, we get that

$h_{u-1}=h_{u}-(h_{u}-h_{u-1})\geq\mathcal{V}^{1+\gamma}_{u}(f)-(\mathcal{V}^{1+\gamma}_{\ell_{u}}(f)-\mathcal{V}^{1+\gamma}_{\ell_{u-1}}(f))=\mathcal{V}^{1+\gamma}_{\ell_{u-1}}(f).$

Continuing by induction, we get for all $i\leq u$,

  1. (3)

$h_{i}\geq\mathcal{V}^{1+\gamma}_{\ell_{i}}(f)$.

However, by Lemma 6.4, for all $i\leq u$,

  1. (4)

$h_{i}\leq\mathcal{V}^{1+\gamma}_{\ell_{i}}(f)$.

Thus, putting together (3) and (4), we conclude

$\forall i\leq u\text{, }h_{i}=\mathcal{V}^{1+\gamma}_{\ell_{i}}(f).$

∎

Remarks:

  1. (1)

In a follow up paper, we show that for holomorphic endomorphisms of
$\mathbb{C}\mathbb{P}^{k}$, $k\geq 1$, the assumptions of Corollary 6.5 hold,
and as a consequence we provide a formula for the Hausdorff dimension of the
measure of maximal entropy. This gives an answer to a question of Fornæss and
Sibony in their list of fundamental open problems in higher dimensional
complex analysis and complex dynamics ([FsS01, Question 2.17]), and proves the
Binder-DeMarco conjecture ([BD03, Conjecture 1.3]).

  2. (2)

It follows that under the assumptions of Corollary 6.5, the measure of maximal
entropy not only maximizes the total entropy over all invariant measures, but
also maximizes each separate conditional entropy over all invariant measures
with the same index (recall Lemma 6.4), maximizes each entropy gap over all
invariant measures with the same index (recall Theorem 6.3), and moreover it
attains the maximal possible bounds given by the volume growth on the
manifold.

## Acknowledgements

I would like to thank the Eberly College of Science in the Pennsylvania State
University for excellent working conditions. I wish to deeply thank Prof.
Federico Rodriguez-Hertz for many enlightening and useful discussions, from
which I have learned a lot.

## References

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S. Ben Ovadia, Department of Mathematics, Pennsylvania State University, State
College, Pennsylvania 16801, United States. E-mail address:
<EMAIL_ADDRESS>