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\controldates{8-SEP-1997,8-SEP-1997,8-SEP-1997,8-SEP-1997}
 
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\begin{document}
\title[Prevalence of Non-Lipschitz Anosov foliations]{Prevalence of
non-Lipschitz Anosov foliations}

\author{Boris Hasselblatt}
\address{Department of Mathematics\\
Tufts University\\
Medford, MA 02155-5597}
\email{bhasselb@@tufts.edu}

\author{Amie Wilkinson}
\address{Department of Mathematics\\
Northwestern University\\
Evanston, IL 60208-2730}
\email{wilkinso@@math.nwu.edu}

\dedicatory{To the memory of Gunnar Hasselblatt, 19.8.1928--12.7.1997}
\keywords{Anosov system, hyperbolic system, invariant
  foliations, stable foliation, Anosov
splitting, horospheric foliations, holonomy, H\"older
structures, conjugacy} 
%\subjclass{Primary: 58F15; Secondary: 53C12}
\subjclass{Primary 58F15; Secondary 53C12}
%\date{\today}
\date{May 9, 1997}
\commby{Krystyna Kuperberg}
\issueinfo{3}{14}{January}{1997}
\dateposted{September 11, 1997}
\pagespan{93}{98}
\PII{S 1079-6762(97)00030-9}
\copyrightinfo{1997}{American Mathematical Society}
\begin{abstract}
We give sharp regularity results for the invariant
subbundles of hyperbolic dynamical systems and give open
dense sets of codimension one systems where this regularity
is not exceeded as well as open dense sets of symplectic,
geodesic, and codimension one systems where the analogous
regularity results of Pugh, Shub, and Wilkinson %[PSW] %\cite{PSW} 
are optimal. We 
produce {\em open sets} of symplectic Anosov diffeomorphisms
and flows with low transverse H\"older regularity of the
invariant foliations {\em almost everywhere}. Prevalence of
low regularity of conjugacies on large sets is a
corollary. We also establish a new connection between the
transverse regularity of foliations and their tangent
subbundles.
\end{abstract}
\maketitle

\section{Introduction}
A diffeomorphism $f$ of a compact Riemannian manifold $M$ is
called {\em Anosov} if the tangent bundle splits
(necessarily uniquely and continuously) into two $Df$-invariant
subbundles $TM=E^u\oplus E^s$, and there exist constants $C$
and $a<1$ such that
\[
\|Df^n(v)\|\le Ca^{n}\|v\|\,\text{ and }\,\|Df^{-n}(u)\|\le Ca^{n}\|u\|
\quad\text{ for }
v\in E^s,\ 
u\in E^u,\ 
n\in\N.
\]
Thus, for two nearby points positive or negative
iterates move apart exponentially fast.  Any hyperbolic
matrix $A\in SL(m,\Z)$ induces an Anosov diffeomorphism of
the $m$-torus ${\Bbb T}^m$; $E^u$ and $E^s$ are the
expanding and contracting root spaces of $A$. The main subject of 
this paper is the regularity of
the {\em unstable} and {\em stable} bundles $E^u$ and $E^s$,
which bears in particular on the regularity of conjugacies.

Anosov systems are structurally stable, meaning that if $f\colon M\to M$ 
is Anosov and $g$ is
$C^1$-close to $f$ then there is a unique homeomorphism
$h\colon M\to M$ close to the identity such that $h\circ f =
g\circ h$ (see, e.g., \cite[Corollary~18.2.4]{KH}).  
The conjugacy $h$ is rarely
differentiable, for if $f^n(p)=p$ and $h$ is
differentiable at $p$ then by the Chain Rule, $Dh(p)Df^n(p)=Dg^n(h(p))Dh(p)$.
Thus $Df^n(p)$, $Dg^n(h(p))$ have the same eigenvalues, which
is not a $C^1$-open condition. 

In fact these eigenvalues
also give  an
obstruction to $h$ being Lipschitz or having high H\"older
exponent [$\varphi\colon X\to Y$ is {\em $C^\alpha$ at $x\in
X$} or $\alpha$-H\"older (if $\alpha=1$ we say $\varphi$ is
Lipschitz) if there exist $C,d > 0$ such that
\begin{equation}\label{E=holder}
d_Y(\varphi(x),\varphi(x')) \leq Cd_X(x,x')^\alpha \text{
whenever }d_X(x,x')\leq d;
\end{equation}
$\varphi$ is $C^\alpha$ {\em on a set $S$} if it is
$C^\alpha$ at every $x\in S$]: While conjugacies are 
uniformly bi-$C^\beta$ for some $\beta>0$
(see, e.g., \cite[Theorem~19.1.2]{KH}), i.e., for some $C,d>0$
\[
(1/C) d_M(x,x') ^{1/\beta} \leq d_M(h(x),h(x')) \leq C
d_M(x,x')^{\beta}
\text{ for }
d_M(x,x')\le d,
\]
differing eigenvalue data will affect the H\"older exponent.
As an  illustration of this relationship,
consider the related (non-Anosov) example of the expanding map 
$f(x)=\lambda x$ on $\R$. Suppose that
$g\colon(-\epsilon,\epsilon)\to\R$, $g(0)=0$,
$0<\lambda<\mu\dfn g'(0)$ and $h\circ f=g\circ h$,
where $h$ is bi-$C^\alpha$. Then
\[
h(f^n(x_0))
=
g^n(h(x_0))
\approx
\mu^nh(x_0)
=
(\lambda^nx_0)^\alpha(h(x_0)/x_0^\alpha)
=
C(f^n(x_0))^\alpha
\]
for $\alpha\dfn\log\mu/\log\lambda<1$, i.e., $h$ is not Lipschitz. 

A similar argument gives an obstruction to Lipschitz conjugacy between
Anosov diffeomorphisms, but it is not a very sensitive one;
if $g$ is $C^1$-close to $f$
then the periodic data are not very different and thus do not
preclude a relatively high H\"older exponent. Furthermore
these data are no obstruction to higher regularity off the
countable set of periodic points (in fact, in dimension two
they are the only obstructions to smooth conjugacy
\cite{Llave}).  It was brought to our attention by John
Franks that low regularity of the Anosov splitting
(Theorem~\ref{T=main}) provides a different and effective
mechanism for showing that the H\"older exponent of {\em
many conjugacies} arbitrarily {\em close to the identity}
may be {\em very low} on a {\em very large set}:

\begin{Thm}\label{T=Franks}
For $\alpha\in(0,1)$ there is a linear symplectic Anosov
diffeomorphism $A$ and a $C^1$-neighborhood $U$ of $A$ of
symplectic diffeomorphisms such that for a $C^1$-open
$C^\infty$-dense set of $f\in U$ the conjugacy to $A$ is
almost nowhere bi-$C^\alpha$.
\end{Thm}
In his proof of ergodicity for volume-preserving Anosov
diffeomorphisms, Anosov showed that there is always an
$\alpha>0$ such that  $E^u$, $E^s$ are
uniformly $C^\alpha$; i.e., if
\begin{equation}\label{E=bigH}
H^\alpha_{C,d}\dfn\{p\in M\mid
d_B(E^u(p),E^u(q))\leq C d_M(p,q)^{\alpha}\text{ if
  }d_M(p,q)\le d\},
\end{equation}
for a smooth metric $d_B$ on subbundles, then
$M=H^\alpha_{C,d}$ for some $C,d$. He did not estimate
$\alpha$ \cite{Anosovtwo}.  Hirsch-Pugh-Shub \cite{HPS} estimated $\alpha$
via global contraction and expansion rates of $Df$,
and Hasselblatt \cite{thesis} obtained better estimates from local data and
showed that they are not exceeded for a generic symplectic
Anosov system, or the geodesic flow of a generic negatively
curved metric.  We give optimal regularity results for
subbundles and foliations and show that they are quite sharp
and extreme nonsmoothness is quite common in the manifold
and the space of Anosov systems. We
\begin{itemize}
\item sharpen the regularity results in
  \cite{thesis},
\cite{periodicbunching},
\cite{ETDSII} and extend the
  sharpness statement to a wider class of Anosov systems
  (Theorem~\ref{T=regularity}),
\item show generic sharpness of the holonomy estimates in \cite{PSW} 
   (Theorems~\ref{T=dist-fols}, \ref{T=regularity}),
\item find {\em large sets} of Anosov systems on which
  the regularity of $E^u$, $E^s$, $W^u$ {\em and} $W^s$ is
  low {\em on a large set of points} (Theorems \ref{T=main}
  and \ref{T=distrib}, Proposition~\ref{P=pert}).
\end{itemize}
Our construction is elementary; it
relies on a simple locally defined
obstruction.
Say that $S\subset M$ is {\em negligible} if
$M\smallsetminus S$ is residual and $\mu(S) = 0$ for any
ergodic invariant probability measure $\mu$ that is fully
supported, i.e., positive on  open sets.

\begin{Thm}\label{T=distrib}
For $\alpha\in(0,1)$ there is a $C^1$-open set of symplectic
Anosov flows and diffeomorphisms for which $E^u$ {\em and}
$E^s$ are $C^\alpha$ only on a negligible set.
\end{Thm}

Many homotopy classes of Anosov systems contain such open
sets. Note that if $\varphi$ is continuous and $S$ is dense,
then $\varphi$ is $C^\alpha$ on $S$ if \eqref{E=holder}
holds for all $x,x'\in S$, so these results show that no
restrictions to large sets are very regular.  

An {\em
Anosov flow} $\varphi^t\colon M\to M$ is a fixed point free
flow with $D\varphi^t$-invariant splitting $TM=E^{su}\oplus
E^{ss}\oplus\langle\dot\varphi\rangle$, where $D\varphi^t$
expands and contracts $E^{su}$ and $E^{ss}$, respectively.
Examples are suspensions (mapping torus) of Anosov
diffeomorphisms (cf. \cite[Section~0.3]{KH}) and geodesic flows
(on the unit tangent bundle) of compact Riemannian
manifolds with negative sectional curvature
\cite[\S22]{Anosov} or \cite[Theorem~17.6.2]{KH}, where the
foliations are called horospheric foliations. The {\em
weak} stable and unstable bundles $E^\ell\dfn
E^{s\ell}\oplus\langle\dot\varphi\rangle$ are tangent to
weak foliations $W^\ell$, $\ell=s,u$.

The regularity of $E^s$ and $E^u$ is closely related to that
of the foliations $W^u$, $W^s$ tangent to $E^u$, $E^s$
\cite[Theorem~8]{Anosov}, \cite{Fenichel}, which have smooth
leaves but low transverse regularity. Regularity
is measured by examining the natural locally defined homeomorphism
between two sufficiently close smooth disks transverse to
the foliation (determined by following the leaves and called
the holonomy map).  If, for any two transversals from a compact family,
the  holonomy map is H{\"o}lder with exponent $\alpha$ and
a uniformly bounded constant, then the foliation
is said to be $C^\alpha$.  Pugh-Shub-Wilkinson
\cite{PSW} showed that $W^u$ and $W^s$ are $C^\alpha$,
with the same $\alpha$ as found for the subbundles in
\cite{thesis}.  There the exponent was not shown to be
optimal.
\begin{Thm}\label{T=main}
For $\alpha\in(0,1)$ there is a $C^1$-open set of symplectic
Anosov flows and diffeomorphisms whose unstable {\em and} stable
holonomies are almost nowhere $C^\alpha$.
\end{Thm}

By \cite{thesis} there generically is a periodic point in
$M$ where $\alpha$ cannot exceed the predicted value. This
left open the possibility that a larger $\alpha$ might work
on the rest of the manifold.  Anosov \cite[p.\ 201]{Anosov}
has an example of an Anosov diffeomorphism where $E^u$ is
almost nowhere $C^{2/3+\epsilon}$, for any
$\epsilon>0$. ($E^u$ is almost nowhere $C^\alpha$ with
respect to $\mu$ if $\mu(\bigcup_{C,d}H^\alpha_{C,d}) = 0$;
see \eqref{E=bigH}.)  In our examples regularity is also low
on a large set, but stably so.

An optimal regularity estimate is interesting for several
reasons.  The regularity of the foliations is related to
that of the conjugacy to a linear system, hence
Theorem~\ref{T=Franks}. Also, $E^u$ and $E^s$ are invariants
of a smooth system that show a marked lack of smoothness
measured by the largest possible exponent $\alpha$---if they
are $C^1$ one can take $\alpha=1$ (conversely, when
$\alpha=1$ the subbundles are often $C^1$). It can be
related to dimension characteristics (see, e.g.,
\cite{SchmSieg}).  Work on stable ergodicity in partially
hyperbolic systems \cite{GPS},
\cite{amiethesis},
\cite{PS} used invariant
subbundles with $\alpha$ close to $1$.  The regularity of
horospheric foliations bears directly on that of Busemann
functions \cite{green}; smoothness characterizes locally
symmetric metrics \cite{BCGII}.

If $f^n(p)=p$ let
$\mu_f(p)<\mu_s(p)<1<\nu_s(p)<\nu_f(p)$ be the minimal and
maximal absolute values of the eigenvalues of $D_pf^n$ in
and outside the unit circle and
\[
B_{\text{per}}^u(f)\dfn\inf\Big\{
\frac{\log\mu_s(p)-\log\nu_s(p)}{\log\mu_f(p)}
\mid
f^n(p)=p\text{ for some }n\in\N\Big\}
.
\] 
E.g., if $f$ is symplectic then $\nu_s\mu_s=1$, so
$B_{\text{per}}^u(f)=2\inf_p{\log\mu_s(p)}/{\log\mu_f(p)}$
is close to 2 if and only if the contraction rates $\mu_s$
and $\mu_f$ are close together. $f$ is said to be {\em
transitive} if it has a dense orbit; this holds for Anosov
systems preserving volume (by ergodicity) and for systems
with $\codim(E^s)=1$ \cite{newhouse}.  Denote by $C^r$ the
space of $C^{\lfloor r\rfloor}$ maps whose $\lfloor
r\rfloor$th derivatives have modulus of continuity
$O(x^{r-\lfloor r\rfloor})$.

\begin{Thm}\label{T=regularity}
\begin{enumerate}
\item If $f$ is transitive Anosov and
  $B_{\text{per}}^u(f)\notin\N$ then $E^u\in
  C^{B_{\text{per}}^u(f)}$.\label{I=notinN}
\item If $f$ is transitive Anosov and
  $B_{\text{per}}^u(f)\in\N$ then $E^u\in
  C^{B_{\text{per}}^u(f)-1,O(x|\log x|)}$.\label{I=inN}
\item \ref{I=notinN} and \ref{I=inN} hold for flows.\label{I=flows} 
\item For an open dense set of symplectic diffeomorphisms
  and flows the regularity of $E^u$ and  $\W^u$ is at most
  that asserted in \ref{I=notinN} and 
  \ref{I=inN}.\label{I=symplectic}
\item For an open dense set of diffeomorphisms and flows
  with $\codim(E^s)=1$ the regularity of $E^u$ and
  $\W^u$ is at most that asserted in
  \ref{I=notinN} and \ref{I=inN}.\label{I=codim1}
\item
  \ref{I=notinN}--\ref{I=flows} hold for hyperbolic sets. 
\item Of the metrics on a compact manifold with sectional
  curvature $\le-k^2$ and injectivity radius $\ge\log2/k$
  an open dense set has horospheric foliations
  (hence structure at infinity) of at most the regularity
  claimed in \ref{I=notinN} and
  \ref{I=inN}.\label{I=horospheric}
\end{enumerate}
\end{Thm}
The results about the subbundles are due to the first
author; the others then follow from
Theorem~\ref{T=dist-fols}. Theorem~\ref{T=regularity}
encompasses some known results.  For example, an
area-preserving diffeomorphism or geodesic flow in dimension
2 has $C^{1,O(x|\log x|)}$ foliations \cite{HK}.
Transitivity can be replaced by bunching along all orbits
\cite{ETDSII}.

A foliation tangent to a H{\"o}lder subbundle may not have
H{\"o}lder holonomy maps, even when the leaves are uniformly
smooth \cite{amiethesis}. This is related to the fact that a
H{\"o}lder vector field need not be uniquely integrable;
trajectories near nonunique ones can move apart rapidly.
Surprisingly, the converse holds; H\"older regularity of
holonomy maps implies (essentially) the same regularity for
the tangent subbundles:
\begin{Thm}\label{T=dist-fols}
If the holonomies of a foliation ${\cal F}$ of a Riemannian
manifold $M$ with uniformly $C^{n+1}$ ($C^\infty$) leaves
are $C^{\alpha-}\dfn\bigcap_{\beta<\alpha}C^\beta$ then
$T{\cal F}$ is $C^{\alpha n/(n+1)-}$ ($C^{\alpha-}$).

Suppose $\lambda$ is a Borel probability measure on $M$
whose ${\cal F}$-conditionals are absolutely continuous and
such that for some family $\{D_x\}_{x\in M}$ of smooth
transversals and for $\lambda$-almost every $x$, $y$ with
$y\in {\cal F}(x)$, the ${\cal F}$-holonomy map
\(
 h\colon D_x\to D_y
\),
\(
 h(p)={\cal F}(p)\cap D_y
\)
is absolutely continuous and $C^{\alpha-}$ a.e. (with
respect to Riemannian volume on $D_x$). Then $T{\cal F}$ is
$C^{\alpha n/(n+1)-}$ ($C^{\alpha-}$) $\lambda$-a.e.
\end{Thm}

This is proved by induction on derivatives along leaves.
There might be a true difference in regularity between
holonomy and tangent distribution when the leaves are not
$C^\infty$ but this question remains open.  Note that
Theorem~\ref{T=main} now follows from
Theorem~\ref{T=distrib} because the foliations are
absolutely continuous: If the unstable subbundle is almost
nowhere $C^{\alpha-\epsilon}$ the invariant set where the
holonomies are $C^\alpha$ cannot have full measure by
Theorem~\ref{T=dist-fols}, hence is a null set by
ergodicity.

Our techniques need failure of bunching everywhere, not just
at a periodic point:
\begin{Def}\label{D=spread}
  $f$ is called {\em$\alpha$-u-spread\/} if there are
  $E^{fs}\subseteq E^{ms}\subsetneq E^s$ and constants
  $\mu_f$, $\mu_s$, $\nu$, with $\nu\mu_f^\alpha<\mu_s$, such that
  $\|Df^n\rest{E^{fs}}\|< %\const
\text{cst.}\mu_f^n$,
  $\|Df^{-n}(v)\|< %\const
\text{cst.}\mu_s^{-n}\|v\|$ for $v\in
  E^s\smallsetminus E^{ms}$, and
  $\|Df^{-n}\rest{E^u(x)}\|> %\const
\text{cst.}\nu^{-n}$ for all $x$.
\end{Def}
This means the Mather spectrum has annuli in $\{|z|<\mu_f\}$
and $\{\mu_s<|z|<1\}$ and an annulus overlapping
$\{1<|z|<\nu\}$, and is an open condition \cite{Pesin}. For
example, $A\dfn \pmatrix
B&0\\0&B^{\lfloor2/\alpha\rfloor+1}\endpmatrix$, where
$B=\pmatrix 2&1\\1&1\endpmatrix$, induces a symplectic
Anosov automorphism of ${\Bbb T}^4$. Both $A$ and $A^{-1}$
are $\alpha$-u-spread.  We obtain Theorem~\ref{T=distrib}
from
\begin{Prop}\label{P=pert}
In a small neighborhood of an $\alpha$-u-spread
symplectic or codimension one Anosov diffeomorphism or flow
the systems whose unstable subbundle is
$C^\alpha$ only on a negligible set are $C^1$-open $C^\infty$-dense.

If a metric of negative curvature on a compact manifold has
$\alpha$-u-spread geodesic flow, then in a small neighborhood
of this metric there is a $C^3$-open $C^\infty$-dense set of
metrics whose horospheric subbundles are $C^\alpha$ on a
negligible set only.
\end{Prop}
This gives Theorem~\ref{T=distrib} for $E^u$ or $E^s$ and
the intersection of open sets is open. Suspending these
examples proves the result for flows.  No $\alpha$-u-spread
geodesic flow is known to us. Perturbations of
volume-preserving (e.g., linear) codimension one flows and
diffeomorphisms have $C^1$ subbundles \cite[p.\ 11]{Anosov},
hence are not examples.

To define the obstruction assume $\codim(E^s)=1$ for simplicity and at 
$x\in M$ choose coordinates which split as $\R\times\R^s$,
with $\R\times\{0\}$ corresponding to $W^u(x)$ and
$\{0\}\times \R^s$ to $W^s(x)$.  Accordingly, the
differential of $f$ at $y\in W^s(x)$ is
\begin{equation*}%\label{Df}
  D_yf=\pmatrix a&0\\B&C\endpmatrix
\end{equation*}
because $W^s(x)$ is preserved by $f$.  Represent $E^u(y)$ as
the graph of a linear map $D\colon\R\to\R^s$ or  the
image of $ \pmatrix
1\\D\endpmatrix\colon\R\to\R\times\R^s$. Then $D_yf(E^u)$
is the image of
\[
D_yf\pmatrix 1\\D\endpmatrix = \pmatrix
a&0\\B&C\endpmatrix\pmatrix 1\\D\endpmatrix = \pmatrix
a\\B+CD\endpmatrix = \pmatrix 1\\(B+CD)/a\endpmatrix a,
\]
which is unchanged when we reparametrize the
preimage by $a^{-1}$, so
by invariance $E^u(f(y))$ is the graph of
$f^*D\dfn(B+CD)/a$. If $z_i\dfn f^i(y)$ then
$D(z_1)=(B(z_0)+C(z_0)D(z_0))/a(z_0)$
or $D(z_{i+1})=(B(z_i)+C(z_i)D(z_i))/a(z_i)$.  
If $f$ is $\alpha$-u-spread then
for $y\in\W_{\text{loc}}^{fs}(x)$
(the local fast stable leaf of $x$ defined by the adapted
coordinate neighborhood) $C$
is lower block triangular. Denote the upper left $k\times k$
block corresponding to the complement of $E^{ms}$ by $c$.
$B$ and $D$ are column vectors whose top $k$
entries define column vectors $b$ and $d$, and
$
  d(z_{i+1})=\big(b(z_i)+c(z_i)d(z_i)\big)/a(z_i).
$
If
\(
\xi^n_{z_0}\dfn\prod\limits_{i=0}^{n-1}c(z_{n-i-1}),\ 
\eta^n_{z_0}\dfn\prod\limits_{i=0}^{n-1}a(z_i)^{-1},\text{ and }
\Delta^n_{z_0}\dfn
-\sum\limits_{i=0}^{n-1}(\xi^{i+1}_{z_0})^{-1}b(z_i){(\eta^i_{z_0})}^{-1}
\)
then
$|(\eta^i_{z_0})^{-1}|\le %\const
\text{cst.}\nu^i$, 
$\|(\xi^i_{z_0})^{-1}\|\le %\const
\text{cst.}\mu_s^{-i}$,
$\|b(z_i)\|\le %\const
\text{cst.}\|z_i\|\le %\const
\text{cst.}\mu_f^i$, and
$\mu_f\nu\le\nu\mu_f^\alpha<\mu_s$, so the
$\Delta^n_{z_0}$ converge uniformly, and the obstruction
\begin{equation*}%\label{E=obstruction}
  O(x)\dfn\sup\{\|d(z)-\Delta^f_z\|\mid z\in\W_{\text{loc}}^{fs}(x)\}
\end{equation*}
is continuous in $x$ and $f$.

For
an $\alpha$-u-spread periodic point $x$ take $y\in
W_{\text{loc}}^{fs}(x)$ with a neighborhood $U$ such that
$U\cap\{f^i(y)\mid i\in\Z\}=\{y\}$ and $x\notin U$
\cite[Proposition~4.1]{thesis}. Let $J$ be a perturbation of
the identity on $M$ supported in $U\smallsetminus\{y\}$ such
that $D_yJ=I+\epsilon e$ in adapted coordinates at $x$,
where the only nonzero entry of $e$ is $e_{21}=1$. Then
$J\circ f$ has $x$ periodic and $y\in
W_{\text{loc}}^{fs}(x)$ not returning to $U$ with unstable
subspace \(
J\pmatrix1\\D\endpmatrix=\pmatrix1\\D\endpmatrix+(0,\epsilon,0,\dots,0)^t,
\) so $d(y)\neq\Delta^f_{y}=\Delta^{J\circ f}_{y}$
and $O\neq0$ (stably). 

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\end{document}

\endinput
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