% slides completed February 27, 2012
%
% for GTDS Dynamical Systems Seminar
%

\documentclass{beamer}

% \usetheme{Antibes}
% \usetheme{Berkeley}
% \usetheme{Berlin}
% \usetheme{Boadilla}
% \usetheme{CambridgeUS}
% \usetheme{Copenhagen}
 \usetheme{Darmstadt}
% \usetheme{default}
%% \usetheme{Dresden}
% \usetheme{Frankfurt}
% \usetheme{Goettingen}
% \usetheme{Hannover}
% \usetheme{Ilmenau}
% \usetheme{JuanLesPins}
% \usetheme{Luebeck}
% \usetheme{Madrid}
% \usetheme{Malmoe}
% \usetheme{Marburg}
% \usetheme{Montpellier}
% \usetheme{PaloAlto}
% \usetheme{Pittsburgh}
% \usetheme{Rochester}
% \usetheme{Singapore}
% \usetheme{Szeged}
% \usetheme{Warsaw}
 
% \usecolortheme{crane}
 % \beamertemplatesolidbackgroundcolor{craneorange!25}
 
 \parskip=4pt
 
\input{top.sty}
 
 
\date{\small February 27, 2012}
   
\title{Self-intersections of foliation cycles}


\author{Steven Hurder}

\institute[UIC] {University of Illinois at Chicago\\www.math.uic.edu/$\sim$hurder\\ \quad \\
{\small Geometry, Topology and Dynamics Seminar}}

 
\begin{document}

\frame{\titlepage} % # 1




 \section{Introduction}
 
  
\frame % 2
{
  \frametitle{Realizing homology classes}

New, \emph{improved} version of result with Yoshihito Mitsumatsu, 

``The intersection product of transverse invariant measures'',   {\it Indiana Univ. Math. J.},  1991.
  
 \pause
 
 \medskip
 
 Let $M$ be closed $m$-manifold.
 
 $C$ a closed, oriented $n$ manifold for $0 < n < m$.
 
 $\iota \colon C \to M$   defines class $[\iota C] \in H_n(M, \mZ)$.
 
 {\bf Problem:}   
Which homology classes   $x \in H_n(M ; \mZ)$ can be realized by a geometric cycle $\iota \colon C \to M$?

\pause

Thom [1953,1954] solved this problem (up to torsion) in terms of bordism classes. 

     \vfill   

}

\frame % 2
{
  \frametitle{}

 {\bf Problem:}  Which homology classes   $[C] \in H_n(M ; \mR)$ can be realized by a geometric cycle $\iota \colon C \to M$?

\pause

\medskip

 {\bf Example:} Realize class $x = \alpha \cdot e_1 + \beta \cdot e_2 \in H_1(\mT^2, \mR)$ 
 
For  $\alpha, \beta \in \mR$ construct foliation of $\mT^2$ with slope $\lambda = \beta/\alpha$

 
  \begin{center}
\includegraphics[width=0.8\textwidth]{pix/torus_linear.png}
\end{center}


   \vfill   

}

  
\frame % 2
{
  \frametitle{}

  {\bf Another Example:}  

 Lamination $\Lambda$ embedded in surface $\Sigma_g$ (carried by train track)
 
 Leaf $L$ defines asymptotic class   $[L] \in H_1(\Sigma_g; \mR) \cong H_1(\mT^{2g}, \mR)$
 
 Non-trivial if  branched cover of algebraic Anosov map of $\mT^2$.  

  \begin{center}
\includegraphics[width=0.5\textwidth]{pix/TrainTrack.png}
\end{center}

    \vfill   

}

  
\frame % 2
{
  \frametitle{Asymptotic cycles}

  $L$ a complete Riemannian $n$-manifold is ``closed at infinity'' 
   if there is $x \in L$ and sequence $R_{\ell} \to \infty$ so that 
   
   $$\rho(L) = \lim_{\ell \to \infty} ~ \frac{| \partial B(x, R_{\ell}) | }{| B(x, R_{\ell})|} = 0$$

$B(x, R) = \{y \in L \mid d_L(x,y) \leq R)$

$\partial B(x, R) =  \{y \in L \mid R -1 \leq d_L(x,y) \leq R)$

$| X |$ denotes Riemannian volume of $X \subset L$.

 

    \vfill   

}
  
\frame % 2
{
  \frametitle{Asymptotic cycles}


{\bf Theorem:}  Let $L$ be oriented, then $\cC = \{ F_{\ell} = B(x, R_{\ell}) \mid \ell = 1,2, \ldots \}$ defines an \emph{asymptotic fundamental class} for the bounded $n$-forms on $L$. 

$$\langle [\cC], \psi \rangle ~ = ~ \lim_{i \to \infty}  ~ \frac{1}{| F_{\ell_i} |} \cdot \int_{F_{\ell_i}} ~ \psi $$

\medskip

{\bf Theorem:} Let $\iota \colon L \to M$ be   embedding with bounded geometry, and assume $L$ is ``closed at infinity'' and oriented.
Then it defines \emph{asymptotic geometric cycle}, and homology class $[\iota \cC] \in H_n(M, \mR)$.

\pause

 \medskip
 
 {\bf Problem:}  Which homology classes   $x \in H_n(M ; \mR)$ can be realized by an asymptotic  geometric cycle $\iota \colon \cC \to M$?

    \vfill   

}

  
\frame % 2
{
  \frametitle{Foliated spaces}

{\bf Definition:} A \emph{foliated space of dimension $n$} is a   continuum $\Lambda$ with a partition $\F$ into leaves, such that  there exists a compact separable metric space $\fX$, and
for each $x \in \Lambda$ there is a compact subset $\fT_x \subset \fX$, an open subset $U_x \subset \Lambda$ with $x \in U_x$, and a homeomorphism defined on the closure
$\vp_x \colon \oU_x \to [-1,1]^n \times \fT_x$ such that   for each $y \in U_x$ the connected component of $\F | U_x$ containing $y$ is defined by $\vp_x^{-1}((-1,1)^n \times w_y)$ for some $w_y \in \fT_x$. 
  
\pause

\bigskip

   $\{\Lambda, \F\}$ is \emph{smoothly embedded} in $M$ if $\Lambda \subset M$, and 
for each $x \in \Lambda$, there exists a  $C^{\infty,0}$-chart for $M$, 
$\psi_x \colon \overline{W}_x \to [-1,1]^m$ about $x$ which     restricts to a foliation chart for $\Lambda$.

Codimension $q = m-n$.

     \vfill   

}

  
\frame % 2
{
  \frametitle{Foliated spaces}

{\bf Remark:} $L \subset \Lambda \subset M$ is embedded with bounded geometry. So $L$ ``closed at infinity'' yields asymptotic geometric cycle in $M$. 


 \medskip
 
 {\bf Remark:} Embedded foliated spaces $\Lambda \subset M$ arise naturally as invariant (attractors) in differentiable dynamics. Examples include:
 
 $\bullet$  Hyperbolic invariant  set for Axiom A diffeomorphism $f \colon M \to M$.

\pause

 $\bullet$ Orbit closures for diffeomorphism $f \colon M \to M$: 

 \begin{center}
\includegraphics[width=0.36\textwidth]{pix/chaos.png}
\end{center}


    \vfill   

}

  
\frame % 2
{
 \frametitle{Transverse invariant measures}

  A  \emph{transverse invariant measure} $\mu$ for a foliated space $(\Lambda, \F)$ is a family of finite Borel measures $\{\mu_{\alpha} \mid \alpha \in \cA \}$ defined on a family of transversals  $\fT_{\alpha} \subset \Lambda$ to $\F$, such that for each holonomy map  $h_{\beta, \alpha}$ from an open subset of $\fT_{\alpha}$ to an open subset of $\fT_{\beta}$ and Borel subset $E \subset {\rm Domain}(h_{\beta, \alpha})$ then, 
  $$\mu_{\beta}(h_{\beta, \alpha}(E)) = \mu_{\alpha}(E)$$

\medskip

{\bf Theorem:} [Ruelle-Sullivan, Plante 1976] $L \subset \Lambda$ leaf which is ``closed at infinity'' yields transverse invariant measure $\mu$ for $(\Lambda, \F)$. 


    \vfill   

}


\frame % 2
{
  \frametitle{Atomics}

{\bf Definition:} A measure $\mu$ has \emph{atoms} if there exists $x \in \fT_{\alpha}$ such that $\mu_{\alpha}(\{x\}) > 0$. 

  $\cC = \{ F_{\ell} = B(x, R_{\ell}) \subset L \mid \ell = 1,2, \ldots \}$ defines $\mu$, then an atom for $\mu$ corresponds to a compact leaf $L_0 \subset \Lambda$, which is a type of Poincar\'{e}-Bendixson limit cycle for $\F$.

 \begin{center}
\includegraphics[width=0.5\textwidth]{pix/torus_asymp.png}
\end{center}

    \vfill   

}

  
\frame % 2
{
  \frametitle{Intersections of cycles}

  {\bf Example:} Let $M = \mS^2 \times \mS^2$, then obtain classes 
  $$x = [\mS^2 \times \{y_0\}] \in H_2(M; \mZ) ~ , ~ y = [ \{x_0\} \times \mS^2] \in H_2(M; \mZ)$$
Their intersection product $x \cap y = [1] \in H_0(M, \mZ)$

\medskip

 $x \in H_n(M, \mZ)$ \& $y \in H_{q}(M, \mZ)$   represented by geometric cycles 
$$\iota_x \colon C_x \to M \quad , \quad \iota_y \colon C_y \to M$$

Then $x \cap y \in H_0(M; \mZ)$ can be calculated via counting ``signed'' points of intersection.


\medskip

     \vfill   

}

  
\frame % 2
{
  \frametitle{Self-intersections of cycles}

  For $x \in H_n(M, \mZ)$ how to calculate $x \cap x \in H_{m-2q}(M; \mZ)$?
  
  {\bf Step 1:} Represent $x$ by geometric cycle $\iota_x \colon C_x \to M$
  
  {\bf Step 2:} Choose perturbation $\iota_x' \colon C_x \to M$
  
{\bf Step 3:} Count intersection homology classes $\iota_x(C_x) \cap \iota_x'(C_x)$ 

\pause

\medskip

Alternate approach: Construct  closed form $\omega$ on $M$ of degree $q = m-n$  which is Poincar\'{e} dual to $[\iota_x C_x]$
$$[\iota_x C_x] \in H_n(M ; \mZ) \to  H_n(M ;  \mR) \cong H^{q}(M ; \mR) \cong H_{deR}^{q}(M)$$

Then $([\iota_x C_x] \cap [\iota_x C_x])^* = [\omega \wedge \omega] \in H_{deR}^{2q}(M) \cong H_{m-2q}(M ; \mR)$. 
    \vfill   

}

  
\frame % 2
{
  \frametitle{Main Theorem}

  {\bf Theorem:} Let $\F$ be a $C^{\infty,0}$-foliation of   a foliated space $\Lambda \subset M$   embedded in a closed oriented manifold $M$, such that the leaves of $\F$ are oriented,   immersed $C^1$-submanifolds of $M$.   Let $\mu$ be a  transverse invariant measure  for $\F$ without atoms. Let $C_{\mu}$ be the closed foliation $n$-current associated to $\mu$. 
  Then the self-intersection product $\ds  [C_{\mu}] \cap [C_{\mu}]  \in H_{m-2q}(M ; {\mR})$ vanishes.

More precisely, for the  Poincar\'{e} dual closed $q$-form $\omega_{\mu}$ on $M$,
$$0 = [\omega_{\mu} \wedge \omega_{\mu}] \in H_{deR}^{2q}(M)$$

\pause

\medskip

{\bf Corollary:} If $x \in H_n(M ; \mR)$ can be realized by an asymptotic  geometric cycle $L \subset \Lambda \subset  M$ with no atoms, then $x \cap x = 0$.

    \vfill   

}

 
 \section{Application to dynamics}
  
\frame % 2
{
  \frametitle{Anosov diffeomorphisms}

  $f \colon M \to M$ is Anosov diffeomeorphism if there exists   $\lambda > 1$, and
  
  $\bullet$ $TM = E^- \oplus E^+$, $E^-$   dimension $n$, $E^+$   dimension $q$, $n + q = m$
  
  $\bullet$ $E^{\pm}$ are invariant under the differential $Df$, 
  
  $\bullet$ $Df | E^+$ is uniformly expanding: $\|DF(\vec{X}) \| \geq \lambda \|\vec{X}\|$,  $\vec{X} \in E^+$
  
 $\bullet$ $Df | E^-$ is uniformly contracting: $\|DF(\vec{X}) \| \leq \lambda^{-1} \|\vec{X}\|$,  $\vec{X} \in E^-$
  
\medskip

The distributions $E^+$ and $E^-$ are uniquely integrable, giving foliations $\F^{\pm}$, whose leaves are smoothly immersed submanifolds with \emph{polynomial growth rate}.

The foliations are $C^{\infty,0}$ -   continuous, with smooth leaves - but rarely smooth unless the map $f$ is algebraic. 


    \vfill   

}

  
\frame % 2
{
  \frametitle{Ruelle-Sullivan currents}

  Ruelle and Sullivan [1976] showed that   the leaves 
  $L^{\pm} \subset M$ for $\F^{\pm}$ define   closed $n$-currents:
    $$[C_{-}] \in H_{n}(M, \mR) \quad {\rm and} \quad [C_{+}] \in H_{q}(M, \mR)$$   
  with $[C_{-}] \cap  [C_{+}]  = 1$. Thus both classes are non-zero.
  
  \medskip
  The result holds more generally for Axiom A diffeomorphisms. 

\pause

\medskip

The leaves define invariant measures without atoms. 
 
Main Theorem implies  $[C_{\pm}] \cap  [C_{\pm}]  = 0$.

    \vfill   

}

  
\frame % 2
{
  \frametitle{Example}

  {\bf Theorem:} [Kleptsyn \& Kudryashov (2009)] $M = \mS^2 \times \mS^2$ admits no Axiom A diffeomorphism $f \colon M \to M$. 

Note that $H_2(M; \mZ) \cong \mZ \oplus \mZ$, so there does exists a smooth map   $f \colon M \to M$ whose action on homology is hyperbolic. 

\pause

{\it Proof:} If exists, then $0 \ne [C_{-}] = \alpha \cdot e_1 + \beta \cdot e_2 \in H_2(\mS^2 \times \mS^2 ; \mR)$
 
$[C_{-}] \cap [C_{-}] =0$ implies $\alpha = 0$, or $\beta = 0$.
 
 If $[C_{-}] = \alpha \cdot e_1$ then $(f^{\ell})_*[C_{-}]  = \lambda^{-\ell} \cdot [C_{-}] \to 0$.
 
 But $e_1$ is an integral class, so this is impossible. 
 
 Ditto for $[C_{-}] = \beta \cdot e_2$. \hfill $\Box$
 
 \pause
 
    Shiraiwa [1973] proved    using   machinery of Axiom A dynamics. 

   \vfill   

}

 
  

 \section{Sketch of Proof}
 

\frame % 2
{
  \frametitle{Two key ideas}

  {\bf  (1)}  A finite Borel measure $\mu$ on $\fT$ has no atoms iff the diagonal 
   $\Delta \subset \fT \times \fT$ has measure $0$ for $\mu \times \mu$.
   
\pause

   \medskip
   
 {\bf (2)}  For a closed $(m -2q)$-form $\psi$ on $M$,  the pairing
   $$\langle [\psi] , [C_{\mu}] \cap C_{\mu}] \rangle ~ = ~ \int_M ~ \psi \wedge \omega_{\mu} \wedge \omega_{\mu} $$
 
 reduces to calculating the mass of the diagonal in $\ds \bigcup_{\alpha \in \cA} ~ \fT_{\alpha} \times \fT_{\alpha}$ for the measure $\mu \times \mu$. 
 
 
 
    \vfill   

}

 

\frame % 2
{
  \frametitle{Constructing the normal bundle}

    Each $L \subset \Lambda$ is smoothly embedded, with normal bundle $Q_L = TL^{\perp}$.
    Together these give $Q_{\Lambda} = T\F^{\perp} \subset TM$. 
 
\pause
 \medskip
 
 Let $G_m(TM)$ be the  Grassmann bundle of $n$-dimensional subspaces of $TM$.
 Obtain $\sigma_{\Lambda} \colon \Lambda \to G_m(TM)$.
 
\pause
\medskip

 $\Lambda \subset M$ is \emph{ANR},  so there is open neighborhood $\Lambda \subset W \subset M$ and extension $\sigma_W \colon W \to G_m(TM)$.
 
 Choose smooth approximation to $\sigma_W$, obtain smooth subbundle $Q \subset TM$, $\pi_W \colon Q \to W$, transverse to $T\F$. 
 
    \vfill   

}



\frame % 2
{
  \frametitle{Thom classes}

  $\Phi_{W}$ denotes a Thom form on $\pi_W \colon Q_{W} \to W$ with support contained in the unit disk subbundle $Q_{W}^1$.   
 
 $\bullet$  $\Phi_{W}$ is closed $q$-form with fiberwise compact support
 
 $\bullet$  $\ds \int_{Q_x} ~ \Phi_{W}  = 1$ for each fiber $Q_x = \pi_W^{-1}(x)$
 
 $\bullet$ \emph{Integration over the fiber} map, for $\Omega_{\pi,c}^p(Q_{W})$   the space of   $p$-forms on $Q_{W}$ with fiberwise compact supports,   
 $$\int_{\pi} ~ \colon ~ \Omega_{\pi,c}^p(Q_{W}) \to \Omega^{p-q}(W)$$
 
  
 
    \vfill   

}



\frame % 2
{
  \frametitle{Properties of fiberwise integration  }

   
$\bullet$ For 
  $\phi \in \Omega^p(W)$   and   $\widetilde{\psi}  \in \Omega^p(Q_{W})$ with bounded uniform norm,
$$\int_{\pi} \pi_{W}^* \phi \wedge \Phi_{W} = \phi \quad , \quad d \, \int_{\pi} \widetilde{\psi} \wedge \Phi_{W} = \int_{\pi} d \, \widetilde{\psi} \wedge \Phi_{W}$$

 $\bullet$ Uniform norm estimate
 $$ \| \int_{\pi}  \widetilde{\psi} \wedge \Phi_{W} \|_W ~ \leq ~  B_{\Phi} \cdot \|\widetilde{\psi}\|_{Q_W}$$

 
    \vfill   

}



\frame % 2
{
  \frametitle{Renormalization}

    For $s > 0$,   $\nu_s \colon Q_{W} \to Q_{W}$ is the fiberwise linear map defined by multiplication by $s$. $Q_{W}^{s}$ denotes $s$-disk subbundle. 
    
    \medskip
    
  $\bullet$  $\nu_s$ maps $Q_{W}^1$ diffeomorphically to $Q_{W}^s$. 
  
    \medskip
    
 $\bullet$  Define $\ds \Phi_{W}^s = \nu_{1/s}^* (\Phi_{W})$, then  $\nu_s^*(\Phi_{W}^s) = \Phi_{W}$.   
  
    \medskip
    
   $\bullet$    $\Phi_{W}^s$ is   a  smooth form on $Q_{W}$ with support in  $Q_{W}^s$,  
    
    \medskip
    
    $\bullet$  integral of $\Phi_{W}^s$ over  each  fiber of $\pi_{W}$ equals $1$.   

    \vfill   

}



\frame % 2
{
  \frametitle{Poincar\'{e} dual classes}

   $exp_M \colon TM \to M$ is geodesic exponential map.
   
   $\exp_W^Q \colon Q \to M$ is restriction to $Q \subset TM$.
 
For compact subset $K \subset L$, there is push-forward map

$$\omega_K^s = \frac{1}{|K|} \cdot (\exp_W^Q)_*(\Phi_{W}^s) \in \Omega^q(M)$$
 
 \pause
 
 {\bf Proposition:} For each $0 < s < \e_0$, and sequence of compact sets $K_{\ell}$ which are closed at infinity, then the following limit exists, 
 $$\lim_{\ell \to \infty} ~ \omega_{K_{\ell}}^s = \omega^s$$
 Poincar\'{e} dual to $[\cC]$ where $\cC$ is closed $n$-current defined by  the $K_{\ell}$.
 
 
 
 
 
    \vfill   

}



\frame % 2
{
  \frametitle{Bounds}

   For a closed $(m -2q)$-form $\psi$ on $M$,    calculate
   $$  \int_M ~ \psi \wedge \omega_{K_{\ell}}^s  \wedge \omega_{K_{\ell}}^s $$
 
\pause

 Use exponential coordinates given by $\exp_L^Q \colon Q_L^s \to M$ to reduce 
 to an integral over normal bundle $Q_L^s \to L$ if all $K_{\ell} \subset L$.

\pause

Local embedding  $\exp_L^Q \colon Q_L^s \to M$ is \emph{recurrent}, so must estimate values of $\omega_{K_{\ell}}^s  \wedge \omega_{K_{\ell}}^s $ where image overlaps. This estimate is most delicate, and is heart of extension of original result from $C^1$-foliations, to $C^{\infty,0}$-foliated spaces. 

\pause

Assumption of ``no atoms'' implies that tame estimates exist for integrals as $\ell \to \infty$, and the limit tends to $0$. 
    \vfill   

}



\frame % 2
{
  \frametitle{Further applications}

    $\bullet$ Obstructions to existence of Axiom A diffeomorphisms
 \medskip
 
 \pause
    
    $\bullet$ Higher-order intersection products for laminations dimensions at least $2$, analogous to self-linking numbers for flows 
  [Gambaudo and Ghys, Khesin, Kotschick and Vogel]
     
     
    \vfill   

}


     
 
\end{document}


 