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\def\SSS{{\openo S}}

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\title{
JORDAN MANIFOLDS
}

\author{L. L. STACH\'O \ and \ W. WERNER }

\date{03/09/2009, Iasi}



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\usetheme{Madrid}



\begin{document}



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\frame[<+->]{

%\begin{block}{} \centerline{\textcolor{blue}{The concept of reaction paths (RP)}} \end{block}

\frametitle{MOTIVATION: THE UNIT BALL OF A JB*-TRIPLE}

$Z$ complex Banach space

\

ASSUMPTION: \ \ ${\rm Ball}_Z(\hskip-7mm\vtop{\hbox{$\phantom{{\Kis cent}}0$}\hbox{{\Kis center}$^\uparrow$}},
\vtop{\hbox{$1$}\hbox{$^\uparrow${\Kis radius}}}
\hskip-7mm)$ \ \ is {\bf symmetric}

\

\

Automatic algebraic structure: \ $\{ x,a,y\}$ \ 3-variable product

\

(J1) \ $\{ x,a,y\}$ {\it symmetric bilinear} in $x,y$, {\it conjugate linear} in $a$

\

(J2) \ $\underbrace{i D(c)}_{\hbox{$\delta$}} : x\mapsto i\{ c,c,x\}$ \
\vtop{\hbox{{\it derivation} of $\{.,.,.\} \phantom{_{\hbox{$\downarrow$}}}$}
\hbox{\hskip-2cm $\phantom{^{\hbox{$\uparrow$}}}\delta\{x,a,y\}=\{\delta x,a,y\}+\{ x,\delta a,y\}+\{ x,a,\delta y\}$ } }

\

(J3) \ $\Vert \{ a,a,a\} \Vert = \Vert a \Vert^3$, \ \
$\Vert \{ x,a,y\} \Vert \le \Vert x\Vert \Vert a\Vert \Vert y\Vert $

\

(J4) \ $D(c) \in {\rm Her}_+(Z)$ \qquad
$\big[ \ \Vert \exp(\zeta D(c) \Vert \le 1$ \ for \ ${\rm Re}\,\zeta \le 0 \ \big]$

}


\frame[<+->]{

\frametitle{COMPLETE HOLOMORPHIC VECTOR FIELDS}
\hbox{
\vtop{
\hbox{Henceforth \ $Z$ JB*-triple}
\hbox{$B(Z) := {\rm Ball}_Z(0,1)$}}
\ \
\vtop{
\hbox{$\Vert .\Vert,\ \{ .,.,.\}, \ D(.)$}
\hbox{$D(a,b):z\!\mapsto\! \{ a,b,z\}$, $Q(z): a\!\mapsto\! \{ z,a,z\}$}}}

\

\

$v:Z\to Z$ holomorphic vector function

$V := v(z){\partial\over\partial z}$ \ vector field: \quad
$f \mapsto \lim_{t\downarrow 0}  \, {1\over t} \big[ f\big( z+tv(z)\big) - f\big( z\big) \big]$

\

\

$\exp (t V) : Z \ni z \mapsto z_t = \big[ \hbox{solution of ${d\over dt} z_t = v(z_t), \ z_0 =z$ (if defined)}\big]$

\

$V$ \ is {\it complete} in $S \subset Z$ \ \ if \ \
$\exp(tV) S \subset S$ \ for all \ $t\in\RR$

\

\

${\rm aut} B(Z)$ \vtop{\hbox{$= \big[ \hbox{ compl. hol, v-fields on $B(Z)$} \big] =$}
\hbox{$= \big\{ [ a - Q(z)a  + \delta z ] {\partial\over\partial z} : \
a \in Z,\ \delta\in {\rm Der}(Z) \big\}$}}


}


\frame[<+->]{

\frametitle{TRIPOTENTS}

$e\in Z$ \ is {\it tripotent} \ \ if \ \ $e = \{ e,e,e\}$

\

Example: \ $Z$ C*-algebra \quad
\vtop{
\hbox{$\{ x,a,y\} = {1\over 2} x a^* y + {1\over 2} y a^* x$}
\hbox{$e \in {\rm Tri}(Z)$ \ iff \ $e e^* e =e$ partial isometry}}

\

$Z_\lambda(e):=\{ z:\ D(e)z=\lambda z\}$, \quad $Z_1^\eps(e) := \{ z:\ Q(e)z=\eps e\}$

\

{\bf Peirce rules:}
\hbox{
\vtop{
\hbox{$Z = Z_0(e) \oplus Z_{1/2}(e) \oplus Z_1(e)$}
\hbox{$\big\{ Z_\xi(e),Z_\alpha(e),Z_\eta(e)\big\} \subset Z_{\xi-\alpha+\eta}(e)$}
\hbox{$\{ Z_0(e),Z_1(e),Z\} = \{ Z_1(e),Z_0(e),Z\} =0$}}
\quad
\vtop{
\hbox{$Z_1(e)=Z_1^+(e) \oplus Z_1^-(e)$}
\hbox{$\big\{ Z_1^\eps(e),Z_1^\varphi(e),Z_1^\psi(e)\big\} \subset Z_1^{\eps\varphi\psi}(e)$}}}

\

\

Finite dimensions: \
\vtop{
\hbox{${\rm Tri}(Z) \leftrightarrow
\big\{ \hbox{closed faces of $\overline{B(Z)}$} \big\}$}
\hbox{\hskip9mm $e \leftrightarrow  e+ \big[ Z_0(e) \cap \overline{B(Z)} \big]$}}

}


\frame[<+->]{

\frametitle{TANGENT SPACES IN \ $\partial B(Z)$}

${\cal X}$ Banach space, \ \ ${\cal S} \subset {\cal X}$


${\rm T}_p ({\cal S}) := \big\{ \left. {d\over dt}\right\vert_{t=0} p_t :\
\big[ t(\in\RR)\mapsto p_t \big] \ \hbox{smooth curve in ${\cal S}$, $p_0=p$} \big\}$

\

$e\in {\rm Tri}(Z)$, \ $x\in B\big( Z_0(e) \big)$ \hskip1.5cm $\big[ e+B(Z_0(e))$ face of $\partial B(Z) \big]$

\

{\bf THM.} \ ${\rm T}_{e+x} \partial B(Z) = \vtop{
\hbox{$Z_0(e) \oplus Z_{1/2}(e) \oplus Z_1^-(e) =$}
\hbox{$= {\rm range}\big[ 1 - Q(e+x) \big]$}}$

\

{\bf Proof.} $v:= {d\over dt}\big\vert_{t=0} p_t$

1) \ $P_1^+(e) ={1\over 2} Q(e)+ {1\over 2} Q(e)^2: Z\to Z_1^+(e)$ \ contractive projection,

$\ \ \ $ $P_1^+(e) v=0$. \ \
Reason: \ $\Vert P_1^+(e) p_t \Vert \le 1$, \ $\Rightarrow$
${d\over dt}\big\vert_{t=0}\Vert e + P_1^+(e)v  \Vert =0$

2) \ $a= a_0+a_{1/2}+a_1^-$, \ \
$p_t := \exp\big[ t \big( a-Q(z)a\big){\partial\over \partial z}\big] (e+x)$

$\ \ \ $ $v={d\over dt}\big\vert_{t=0} p_t = a - Q(e+x)a$,

$\ \ \ $ $v_0 = a_0 - \{ x,a_0,x \} = [1 - Q(x)] a_0$, \ \
$\Vert Q(x)\Vert = \Vert x\Vert^2$ \ $\big[ 1-Q(x)$ inv.$\big]$

$\ \ \ $ $v_1^- = a_1^- \{ e, a_1^-,e\} =2 a_1^-$

$\ \ \ $ $v_{1/2} = a_{1/2} - 2\{ x,a_{1/2},e\} = [ 1 - 2 Q(x,e) ] a_{1/2}$


$\ \ \ \ \ \ $  on $Z_{1/2}(e)$, $Q(x,e)^2 = D(x)$
with norm $\le {1\over 4} \Vert x\Vert^2$ \ $\big[ 1 - 2 Q(x,e)$ inv.$\big]$

}


\frame[<+->]{

\frametitle{TANGENT SPACES OF \ ${\rm Tri}(Z)$}

$e_t \in {\rm Tri}(Z)$, \quad $e_0=e$ \hskip 2cm $\{ e_t,e_t,e_t \} = e_t$

\


$2 \big\{ {d\over dt} e_t, e_t, e_t \big\} + \big\{ e_t,{d\over dt} e_t, e_t\big\} = {d\over dt} e_t$

\

$v := {d\over dt}\big\vert_{t=0} e_t$, \quad $v=v_0+v_{1/2}+v_1^-$

\

$2 \{ e,e,v\} - \{ e,v,e\} =v \iff
0=v_0 \ \big[ v_{1/2}-0 = v_{1/2}, \ 2 v_1^- - v_1^- = v_1^- \big]$

\

{\bf THM.} \ ${\rm T}_e \big( {\rm Tri}(Z)\big) = Z_{1/2}(e) \oplus Z_1^-(e)$

\

{\bf Proof.} \ $v = v_{1/2}+v_1^- \in Z_{1/2}(e) \oplus Z_1^-(e)$

\

$\delta:=D(a,b)-D(b,a) \in {\rm Der}(Z)$, $\Rightarrow$
$\exp(t\delta) \in {\rm Aut}(Z)$, \
$\delta e = {d\over dt}\big\vert_{t=0} \exp(t\delta)e \in {\rm T}_e \big( {\rm Tri}(Z)\big)$

\

$a:=  a_{1/2}+ a_1^-$, \ $b:=e$ \ \ $\Rightarrow$ \
$\delta e = {1\over 2} a_{1/2} + 2 a_1^-$ arb. $\in Z_{1/2}(e) \oplus Z_1^-(e)$

}


\frame[<+->]{

\frametitle{DIRECT SUBMANIFOLDS}

${\cal S}$ {\it direct complex submanifold} in ${\cal X}$:

\


$\ \ \ \ \ $ $\forall\ p\!\in\! {\cal S} \quad \exists\ {\cal U}_p, {\cal V}_p$ nbh. of $p$ resp. $0$ \ \
$\exists\ {\cal X} \!=\! {\cal X}_p^{(0)} \!\oplus\! {\cal X}_p^{(1)}$

$\ \ \ \ \ $ $\exists\ \Phi_p: {\cal V}_p \!\leftrightarrow\! {\cal U}_p \
\hbox{bihol.} \qquad {\cal U}_p \cap {\cal S} = \Phi_p \big( {\cal V}_p \cap {\cal X}_p^{(0)}\big) $

\

\

\it If \quad \rm
\vtop{
\hbox{$\forall\ p\!\in\! {\cal S} \quad
\exists\ {\cal X} \!=\! {\cal X}_p^{(0)} \!\oplus\! {\cal X}_p^{(1)}$ \quad
$\exists\ \Psi_p: [$nbh. of $0] \to {\cal X} \
\hbox{hol.}$}
\hbox{$\Psi_p(0)=p,\ \Psi_p^\prime(0)$ invertible, \
$\Psi_p(x) \in {\cal S} \!\iff\! x\in {\cal X}_p^{(0)}$}}

\

\it then \quad\rm
\vtop{
\hbox{${\cal S}$ \ is a complex direct submanifold in ${\cal X}$}
\hbox{with local charts by restrictions of the maps $\Psi_p$.}}

}


\frame[<+->]{

\frametitle{${\rm Tri}(Z)$ AS DIRECT SUBMANIFOLD}

$e\in {\rm Tri}(Z)$, \quad $P_\lambda(e): Z\to Z_\lambda(e)$ Peirce projection

$$K(e,a) := D\big( 2 \!\! \underbrace{a_{1/2}}_{P_{1/2}(e)a} + \
2^{-1} \!\!\! \underbrace{a_1^-}_{P_1^-(e)a} , e \big) -
D\big( e , 2 a_{1/2} + 2^{-1} a_1^- \big)$$

$K(e,a) \in {\rm Der}(Z)$, $\Rightarrow$ $\exp[t K(e,a)] \in {\rm Aut}(Z)$ $(t\in\RR)$

\

$p_t := \exp[t K(e,a)] e \in {\rm Tri}(Z)$

\

${d\over dt}\big\vert_{t=0} p_t = K(e,a)e = a_{1/2}+a_1^-$

\

{\bf THM.} ${\rm Tri}(Z)$ direct submanifold in $Z$ with local charts

$X_e: Z_{1/2}(e)\oplus Z_1^-(e) \ni a \mapsto \big[ \exp K(e,a) \big] e$

\

{\bf Proof.} $\Psi_e : Z \ni a \mapsto \big[ \exp K(e,a) \big]  e + a_0 + a1^+ )$
\qquad $\Psi_e^\prime(0)={\rm id}$


}



\frame[<+->]{

\frametitle{ALGEBRAIC CONNECTION ON ${\rm Tri}(Z)$}

$X \equiv X(e) {\partial\over \partial e} , Y \equiv  Y(e) {\partial\over \partial e}$ \quad
smooth vector fields on ${\rm Tri}(Z)$

 \

%\hskip2cm
$X(e),Y(e) \in {\rm T}_e \big({\rm Tri}(Z)\big) = Z_{1/2}(e)\oplus Z_1^-(e)$

$$\nabla_X Y := ( P_{1/2}(e) + P_1^-(e) ) \big[ Y_{X(e)}^\prime (e) - X_{Y(e)}^\prime (e) \big]
{\partial\over\partial e}
$$

\

{\bf THM.} $\nabla$ \it is affine and torsionfree \rm

\

$(X,Y) \mapsto \nabla_X Y$ real bilinear

$\nabla_X \varphi Y (e) = \varphi_{X(e)}^\prime (e) Y (e) + \varphi(e) \nabla_X Y (e)$

$\nabla_X \big[ Y_1,Y_2\big] = \big[ \nabla_X Y_1,Y_2\big] + \big[ Y_1,\nabla_X Y_2\big]$

}


\frame[<+->]{

\frametitle{GEODESICS OF $\nabla$}

$\gamma: I\to {\rm Tri}(Z)$ \quad smooth curve, $0\in I$ open interval

\

$\gamma$ is {\it geodesic} of $\nabla$: \qquad
$\nabla_{\dot\gamma(t)} \dot\gamma(t) =0$

$$\underbrace{P_\gamma(t)}_{\phantom{int^{\int}}P_1^-(\gamma(t))+P_{1/2}(\gamma(t))}
\hskip-10mm \Big( {d^2 \over dt^2} \gamma(t) \Big) = 0$$

\

Ricatti type \quad $\big[$%False solution in Chu-Isidro,
solution not correct in \  Math. Z. 243 (2000) 741-754$\big]$

\

Solution with EXPONENTIAL INTEGRALS

\

$F:\RR\to {\rm Der}(Z)$ \quad
%$-\kern-5pt-\kern-5pt\longrightarrow$
$\sim\kern-2.1pt\sim\kern-2.1pt\sim\kern-2.1pt\sim\kern-2.1pt\sim\kern-2.1pt\sim\kern-2.1pt\sim\kern-7pt >$
\quad $\strut^{{\bf L}}F : \RR\to {\rm Aut}(Z)$

$\displaystyle{ {d\over dt}\, \strut^{{\bf L}} F(t) = \big[\, \strut^{{\bf L}} F(t) \big] F(t) }$,
\quad $\strut^{{\bf L}}F(0) = {\rm Id}$
\qquad left mult. primitive funct.

}


\frame[<+->]{

\frametitle{FORMULA FOR THE GEODESICS}

{\bf THM.} Any smooth curve \
$\gamma : \RR \to {\rm Tri}(Z)$, $\gamma(0)=e$ \
has the form
$$\gamma(t) = \strut^{{\bf L}} K\big( e, a(t) \big) e$$
with suitable \ $a: \RR\to Z_{1/2}(e) \oplus Z_1^-(e)$.
\quad
$\gamma$ \ is a geodesic of $\nabla$ \ iff
$$a(t) = v_0 + \exp\big( 3t D(e,v_0) \big) u_0 \qquad
 \ u_0 \!\in\! Z_{1/2}(e), v_0 \!\in\! Z_1^-(e)$$
For any $c \in {\rm T}_e (Z) = Z_{1/2}(e) \oplus Z_1^-(e)$ there is a unique \
$\nabla$-geodesic $\gamma_{e,c}$ \ with $\gamma_{e,c}(0)=e$, $\dot\gamma_{e,c}(0)=c$.

\

\

{\bf COR.} \ $t\mapsto \exp[ t K( e, c) ] e$, \ $c\in Z_{1/2}(e)\oplus Z_1^-(e)$ \ \ is $\nabla$-geodesic

$\iff$ \quad
$\{ c_1^-, e, c_{1/2} \} = \{ e, c_1^-, c_{1/2} \} =0$


}



\frame[<+->]{

\frametitle{METRIC ASPECTS}

$[0,1] \ni t\mapsto \gamma(t)$ \ {\it metric stationary curve} in ${\rm Tri}(Z)$:

$${\partial\over \partial s}\big\vert_{s=0} {\rm Length} \big[t\mapsto \widetilde{\gamma}(t,s)\big] =0$$
for pert. $\widetilde{\gamma}:[0,1]\times I\to{\rm Tri}(Z)$ \
with $\gamma(t)=\widetilde{\gamma}(t,s)$ \ if\ $s=0$ or $t=0,1$

\

{\bf THM.} $\gamma(t) :=\strut^{{\bf L}} K\big( e, a(t) \big) e$ metr. stat. curve $\Longrightarrow$
$$\int_0^1 \Big[ \xi(t) \underbrace{\delta\big( a(t), K(e,a(t)) u\big)}_{
\delta(x,y):={d\over dt}\big\vert_{t=0+} \Vert x + t y\Vert} +
\xi^\prime(t) \delta\big( a(t),u \big) \Big] \ dt =0$$
if $\xi(0)=\xi(1)=0$ and $u\in Z_{1/2}(e)\oplus Z_1^+(e)$

\

{\bf CONJ.} \ Geodesics of $\nabla$ are metr. stat. curves in ${\rm Tri}(Z)$

}


\frame[<+->]{

\frametitle{EQUIVALENCE OF TRIPOTENTS}

$e \sim f$: \ $D(e)=D(f)$

\

{\bf LEM.} \ $e,f \in {\rm Tri}(Z)$. \qquad T.f.a.e.:

\hskip6mm $1) \ e\sim f$ \quad

\hskip6mm $2) \ \{ e,e,f\} =f$ , \ $\{f,f,e\}=e$ \quad

\hskip6mm $3) \ Z_1(e)=Z_1(f)$ \quad

\hskip6mm $4) \ Q(e)Z=Q(f)Z$

%\

%{\bf REM.} \ $Q(w)Z = Q(e)Z$ for some $e\in{\rm Tri}(Z)$ $\iff$ $w \in {\rm Reg}(Z)$

%\

%\hskip1.3cm $w\in Z$ is {\it regular}\ : \quad
%$0 \not\in {\rm Sp}\Big( D(w)\big\vert \hbox{JB$^*$Span} \, \{ w\} \Big)$

\

\

{\bf THM.} \qquad $\gamma_{e,c}^\nabla (t) \sim \exp[ t K( e, c) ] e$

\

\

{\bf Problem.} \ Natural diffgeo str. of \ \  ${\rm Tri}^\sim (Z) := \big\{$ equiv. classes of $\sim\big\}$


}


\frame[<+->]{

\frametitle{GRASSMANIAN REPRESENTATION OF ${\rm Tri}^\sim(Z)$}

{\bf Kaup, 2001.}

$w\in Z$ is {\it regular}\ : \quad
$0 \not\in {\rm Sp}\Big( D(w)\big\vert \hbox{JB$^*$Span} \, \{ w\} \Big)$

\

$Q(w)Z = Q(e)Z$ for some $e\in{\rm Tri}(Z)$ \ $\iff$ \ $w \in {\rm Reg}(Z)$

$$\PP := \big\{ J_w :\ w\in{\rm Reg}(Z) \big\}
\qquad
J_w := Q(w)z$$


$J_w$ \ inner ideal: \quad $\{ J_w,Z,J_w \} \subset J_w$, \qquad
$\exp D(x,a)J_w \in \PP$

\

{\bf THM.} \ $\PP = \big\{$closed complemented inner ideals of $Z\big\}$ is a

\hskip14mm submanifold in \ \ ${\rm Grassm}(Z)$ \ \ with atlas

\hskip14mm $Z_{1/2}(e) \ni u \mapsto \exp D(e,u) J_e, \quad e\in{\rm Tri}(Z)$

\

{\bf REM.} \ $\PP = \{ J_e: \ e\in {\rm Tri}(Z)\}$, \quad $J_e = J_f \iff e\sim f$


}



\frame[<+->]{

\frametitle{SUPPORT TRIPOTENTS}

$w\in{\rm Reg}(Z)$, \qquad $\Omega_w :={\rm Sp} D(w) \subset \RR_+$, \ \ $\inf \Omega_w >0$

Functional calculus: \ \
\vtop{
\hbox{$F_w :{\cal C}(\Omega_w) \leftrightarrow \hbox{JB$^*$Span} \{ w\}$, \quad
$F_w w = {\rm Id}_{\Omega_w}$}
\hbox{$\{ F_w(\varphi),F_w(\alpha),F_w(\psi)\} = F_w(\varphi\overline{\alpha}\psi)$}}

\

{\bf DEF.} \ $s(w) := F_w(1_{\Omega_w})$ \ \ support tripotent of $w$

\

$\{ s(w),s(w),w \} = \{ s(w),w,e(w)\}$, \quad $J_w = J_{s(w)} = Z_1\big( s(w)\big)$

\

{\bf THM.} \ $Z$  JC$^*$-triple and $e\in M$ \ $\Rightarrow$

$s_t := s\big( [\exp t D(u,e)] e \big)$ \quad is the solution of

\

$(ST) \qquad {d\over dt} s_t = P_{1/2}(s_t) \{ u,e,s_t \}, \qquad s_0 =e$


\

{\bf CONJ.} \ Not only for JC$^*$-triples


}

\frame[<+->]{


\frametitle{OPERATOR REPRESENTATION OF ${\rm Tri}^\sim(Z)$}

For short: \ $M:={\rm Tri}(Z)$, \ \
$\MM := {\rm Tri}^\sim(Z) = \{ \hskip-15mm
\underbrace{{\bf e}^\sim}_{\hskip15mm\{ f\in{\rm Tri}(Z) :\ f\sim e\}}
\hskip-15mm :\ e\in M \}$


\

$\DD := \{ i D(e): \ e\in M \} \subset {\rm Der}(Z)$ \hskip8mm {\Kis single algebraic objects}

$$\vtop{
\hbox{\hskip9mm$M$}
\hbox{\hskip5mm $\swarrow$ $\downarrow$ $\searrow$}
\hbox{$\DD \leftrightarrow \MM \leftrightarrow \PP$}}
\hskip15mm
\vtop{
\hbox{\hskip14.6mm$e$}
\hbox{\hskip9.5mm $\swarrow$ $\downarrow$ $\searrow$}
\hbox{$iD(e) \leftrightarrow {\bf e}^\sim \leftrightarrow \, J_e$}}
$$

\

\

{\bf THM.} \ $\Vert D(e)-D(f)\Vert < {1\over 66}$ $\Longrightarrow$
$\exists\ f^\prime \in {\bf f}^\sim \quad \Vert e-f^\prime\Vert \le 16 \Vert D(e)-D(f)\Vert$

\

$\big[$quotient topology $M/\sim\big]$
\vtop{
\hbox{$= \big[$top. by bias
$\delta({\bf e}^\sim,{\bf f}^\sim):= \inf_{e\in{\bf e}^\sim, f\in{\bf f}^\sim} \Vert e-f\Vert \big]$}
\hbox{$= \big[$Hausdorff dist. top. on $\MM\big]$}}

\

${\bf e}^\sim \mapsto iD(e)$ \ is bilipschitzian \ $\MM \leftrightarrow \DD$

}


\frame[<+->]{

\frametitle{PROBLEMS}

$$\vtop{
\hbox{\hskip9mm$M$}
\hbox{\hskip5mm $\swarrow$ $\downarrow$ $\searrow$}
\hbox{$\DD \leftrightarrow \MM \leftrightarrow \PP$}}
\hskip15mm
\vtop{
\hbox{\hskip18mm$e$}
\hbox{\hskip10.4mm $\swarrow$ \ \ $\downarrow$ \ \ $\searrow$}
\hbox{$iD(e) \leftrightarrow \underbrace{{\bf e}^\sim}_{M \cap J_e} \leftrightarrow \, J_e$}}
$$

\

Natural {\it complex} structure on $\PP$

\

$\sim\kern-2.1pt\sim\kern-2.1pt\sim\kern-2.1pt\sim\kern-2.1pt\sim\kern-2.1pt\sim\kern-2.1pt\sim\kern-7pt >$
\quad complex str. on \ \ $\MM,\DD$

\

Algebr. simple charts on $\PP$ \ with \ "big" objects

\

$J_e \cap J_f \ne \emptyset,J_e,J_f$ \ \ may happen in \ $\PP$

\

${\bf e}^\sim={\bf f}^\sim$ or ${\bf e}^\sim\cap {\bf f}^\sim =\emptyset$ \ \ in \ $\MM$

\

{\it DESCRIBE HOLOMORPHY ON} \ $\MM$, $\DD$ \ in terms of that of $Z$

}



\frame[<+->]{

\frametitle{$\DD$ \ AS DIRECT SUBMANIFOLD}

$\pi_{k\ell}(e): {\cal L}(Z)\ni L \mapsto  P_{k/2}(e) L P_{\ell/2}(e)$ \qquad
$\displaystyle{\Pi_m(e) := \hskip-2mm \sum_{\vert k-\ell\vert=m} \hskip-2mm \pi_{k\ell}(e)}$

\

{\bf LEM.} \ $L\in{\rm Der}(Z)$ \ with \ $Le \in Z_{1/2}(e)$ \ $\Longrightarrow$

$\displaystyle{D(e)_\#^2 L = [D(e), [ D(e),L]] = \sum_{m=0}^2 \left({m\over 2}\right)^2 \Pi_m(e) L =
{1\over 4} K(e,Le) }$

\

{\bf THM.} \ 1) ${\rm Der}(Z) = \DDD_1(e) \oplus \DDD_0(e)$ \qquad $\DDD_m(e) := \Pi_m(e){\rm Der}(Z)$

2) \ $K(e,\, . \, ): \ Z_{1/2}(e) \leftrightarrow \DDD_1(e)$ %$ = \big\{ K(e,u):\ u\in Z_{1/2}(e) \big\}$


3) \ $\Psi_e : K(e,u)\oplus L \mapsto
\big( \exp K(e,u) \big) \big( iD(e)+L\big) \big( \exp K(e,u)\big)^{-1}$

$\ \ \ $ real bianalytic between some neighborhoods of $0$ resp. $e$ in ${\rm Der}(Z)$

$\ \ \ $ with \ $\Psi_e\big( K(e,u)\oplus L \big) \in \DD \iff L=0$

\

{\bf COR.} $ \big\{ \big[ Z_{1/2}(e) \ni u \mapsto i D\big( \exp K(e,u) \big) \big] : \ e\in M\big\}$ \ \
atlas for $\DD$

$\ \ \ \ \ \ \ \,$ $ \big\{ \big[ Z_{1/2}(e) \ni u \mapsto i \big( \exp K(e,u) \big)^\sim \big] : \ e\in M\big\}$ \ \
atlas for $\MM$

}



\frame[<+->]{

\frametitle{THE TANGENT MANIFOLD OF \ $\MM$}

${\rm T}_{iD(e)} \DD = \Delta_1(e) =
\big\{ K(e,u):\ u\in Z_{1/2}(e) \big\} \approx Z_{1/2}(e)$

\

$t\mapsto {\bf e}^\sim_t$  \ \ tangent curve to $\MM$, \quad\qquad
$\displaystyle{L_t := {d\over dt} iD(f,f)}_{f\in {\bf e}^\sim_t}$

\

$u(t,f) := \Big[ u\in Z_{1/2}(f) : \ K(f,u) = L_t \Big]$, \qquad $f\in {\bf e}^\sim_t$

\

$\displaystyle{X(f) := {\partial\over\partial t} u(t,f)}$

\

{\bf LEM.} If \ \ $e\sim f \in {\bf e}^\sim_t$ \ \ then \ \

\

(*) \ \ $D\big( X(e),e\big) = D\big( X(f),f\big)$ \ and \ $D\big( e,X(e)\big) = D\big( f,X(f)\big)$

\

{\bf DEF.} \ $X : M \to P_{1/2}{\rm T}M$ \ equivariant: \quad (*) holds whenever $e\sim f\in M$.

\

$\ \ $ $X \approx Y$: \quad
$D\big( X(e),e\big) = D\big( Y(e),e\big)$ \ and \ $D\big( e,X(e)\big) = D\big( e,Y(e)\big)$


}


\frame[<+->]{

\frametitle{LIE STRUCTURE OF EQUIVAR. FIELDS}

{\bf THM.} \ 1)
$X: M\to P_{1/2}{\rm T}M$ is complete equivariant field in $M$ \ $\Longrightarrow$

\hskip1.8cm $\exp tX$ maps equi. classes of $\sim$ onto equi. classes of $\sim$

\

2) $\big[ X,Y \big]_* := P_{1/2} \big[ X,Y \big] = \nabla_X Y - \nabla_Y X$ \ \
Lie alg. for equivar. fields

\hskip5mm $X\approx \widehat{X}$ and $Y\approx \widehat{Y}$ $\Rightarrow$ \
$\big[ X,Y \big]_* \approx \big[ \widehat{X},\widehat{Y} \big]_*$

\

3) ${\rm T}\MM$ can be identified with ${\rm T}\DD$ by \ \
$[t\mapsto {\bf e}^\sim_t] \mapsto {d\over dt}\big\vert_{t=0} D({\bf e}^\sim_t)$

\hskip5mm With this identification, \
$\{ M\buildrel{equivar}\over\longrightarrow P_{1/2}{\rm T} M\} \leftrightarrow {\rm T}\MM$
\ \ is given by

\hskip10mm $\Lambda(X) := \Big[ \MM \ni {\bf e}^\sim \mapsto 2 D\big( e,X(e)\big) - 2 D\big( X(e),e\big) \Big]$


}


\frame[<+->]{

\frametitle{DISJOINTIFICATION \ OF \ $\PP$}

$\displaystyle{\SSS := \bigcup_{e\in M} \{ e\} \times Z_1(e) \ =
\big\{ (e,x):\ e\in M, \ x\in J_e \big\} \ \subset Z\times Z}$

\

$Z^-(e) := Z_{1/2}(e) \oplus Z_1^-(e)$, \quad $Z^+(e) := Z_0(e)\oplus Z^+(e)$, \quad
$P^\pm(e): Z\to Z^\pm(e)$

\

{\bf THM.} \ $\SSS$ \ real anal. dir. subm. of $Z^2$, \quad ${\rm T}_{(e,x)} \SSS = Z_1^-(e) \times Z_1(e)$

\

For \ $(e,x)\in\SSS$ \ and \ $(v,y)\in Z^-(e)\times Z$ \ \ t.f.a.e.

1) $(v,y)\in {\rm T}_{(e,x)} \SSS$, \quad
2) $P_0(e)y=0$ and $P_{1/2}(e)y = 2 \{ P_{1/2} v, e, x\}$,

3) $y = \{ v,e,x\} + \{ e,v,y \} + \{ e,e,y\}$

\


{\bf Proof.} \ Investigate the map

$\Phi_{(e,x)}(w,z) := $

$\Big( \
[ \exp K( e ,P^-(e)w)] [ e+ P^+(e)w ] \ , \
[\exp K(e, P^-(e)w) ] (x+z) \
\Big)$

}


\frame[<+->]{

\frametitle{TANGENT VECT0R FIELDS TO \ ${\Twelveopen S}\hskip-2.5mm{\Twelveopen S}$}

{\bf COR.} \ \
${\rm T}_{(e,x)} \SSS \supset
\Big\{ \Big( P_{1/2}(e) D(a,b)e, \ D(a,b)x \Big): \ (a,b)\in Z\!\times\! Z \Big\}$

\

If \ \ $V,Y$ \ \ smooth v.fields on \ $Z$ \ and \ $V$ tangent to $M$. \ \ T.f.a.e.

\

$\ $ 1) $\forall\ (e,x)\in\SSS \quad [\exp tY]x \in Z_1\big( [\exp tV]e \big)$ \qquad
for \ $\vert t\vert$ \ small

\

$\ $ 2) $Y(x) = \{ \{ V(e),e,x \} + \{ e,V(e), x\} + \{ e,e,Y(x)\}$, \qquad $(e,x)\in\SSS$

\

If $V$ complete in $M$  \ and \ $Y$ complete in $Z$ \quad $\Rightarrow$ \ 1)+2) is equivalent to

\

$\ $ 3) $[\exp(tY)] Z_1(e) = Z_1\big( [\exp tV]e \big)$, \quad $(e,x)\in\SSS,\ t\in\RR$


}


\frame[<+->]{

\frametitle{COORDINATES VIA  \ $(ST)$}

$C_u^{(e)} : M \ni f \mapsto P_{1/2}(f) D(u,e) f$ \quad $e\in M,\ u\in Z_{1/2}(e))$

\

\

{\bf REM.} \ Solution of $(ST)$: \ $t\mapsto [\exp tC_u^{(e)}]e \in M$

%$\{ s(w),s(w),w \} = \{ s(w),w,e(w)\}$, \quad $J_w = J_{s(w)} = Z_1\big( s(w)\big)$

\

\

{\bf DEF.} \ \ $Y_e(u) := [\exp C_u^{(e)}]e$, \qquad $e\in M,\ u\in Z_{1/2}(e)$

\

\

{\bf THM.} \ $C_u^{(e)}$ \ \ equivar. complete v.field on $M$,

\

\hskip14mm  $\exists\ e\mapsto W_e$ $0$-nbh. in $Z_{1/2}(e)$ \quad $Y_e \vert W_e$ real-bianal.


$$Y_e(0) = e, \quad [\exp D(u,e)] J_e = J_{Y_e(u)} \qquad\qquad (u\in W_e)$$


}



\frame[<+->]{

\frametitle{COMPLEX STRUCTURES}

$\widetilde{Y}^{(e)} (u) := \Big[ \big[\exp C_u^{(e)}\big]e \Big]^\sim =
Y_e(u)^\sim {\phantom{\Big]^{\int}}}^{\kern-10mm{{\rm equiv.class}\atop\downarrow}} \ \ \in\MM$

\

\

$\overline{Y}^{(e)} (u) := Z_{1/2} \Big[ \big[\exp C_u^{(e)}\big]e \Big] =
J_{Y_e(u)} \hskip6mm \in \PP$

\

$\widehat{Y}^{(e)} (u) := i D\big( \big[ \exp C_u^{(e)} \big] e \big)
\hskip24mm \in\DD$

\

{\bf THM.} \ $\big\{ \widetilde{Y}^{(e)} :\ e\in M\big\}$, \
$\big\{ \overline{Y}^{(e)} :\ e\in M\big\}$, \
$\big\{ \widehat{Y}^{(e)} :\ e\in M\big\}$

\

\hskip14mm holomorphic atlases for \ $\MM,\PP,\DD$

$$\vtop{
\hbox{\hskip5mm$Z_{1/2}(e)$}
\hbox{\hskip5mm$\swarrow \ \downarrow \ \searrow$}
\hbox{$\DD\leftrightarrow \MM  \leftrightarrow\PP$}}
\qquad
\vtop{
\hbox{\hskip23.5mm$u$}
\hbox{\hskip14mm$\swarrow \hskip6mm \downarrow \hskip6mm \searrow$}
\hbox{$\widehat{Y}^{(e)}(u)\leftrightarrow \widetilde{Y}^{(e)}(u)  \leftrightarrow \overline{Y}^{(e)}(u)$}}
$$


}



\frame[<+->]{

\frametitle{HOLOMORPHIC MAPS ON \ $\MM$, $\PP$}

{\bf THM.} \ ${\openo U}$ open $\subset \MM$, \
$U:= %\bigcup{\openo U} :=
\{ e\in M:\ \exists\ \widetilde{f}\in \MM \ \ e\in\widetilde{f}\}$,

$\Phi:{\openo U}\to B$ map to compl. Banach space .


T.f.a.e. $\phantom{ \Big]^{\int} }$

\

\hskip5mm 1) \ $\Phi$ is holomorphic

\

\hskip5mm 2) $\forall\ e \in U \ \quad \exists\ V \hbox{open}\subset M \ \quad \exists\ \phi:V\to B$ holomorphic \
s.\hskip-0.3mm th.

\

\hskip 20mm $\phi(f) = \Phi(f^\sim) \qquad \hbox{whenever} \quad f \in U\cap V$

\

\

{\bf REM.} \ $\Psi: [$nbh. of $J_e] \to B$ \ is holomorphic $\iff$

\

\hskip13.6mm $\Phi\big( \underbrace{J\cap M}_{\in \MM} \big) := \Psi(J)$ \ \ is holomorphic


}



\end{document}


\frame[<+->]{

\frametitle{NUMERICAL TREATMENT}

\begin{itemize}

\item[]
It follows
$$a_{8+n} =
{\rm Re}\!\!\! \!\!\!\!\!\!\! \sum_{k=\in\{1,2,3,5\}} \!\!\!\!\!\!\!\!
\rho_k^n \alpha_k, \quad
b_{8+n} =
{\rm Re}\!\!\!  \!\!\!\!\!\!\! \sum_{k=\in\{1,2,3,5\}} \!\!\!\!\!\!\!\!
\rho_k^n \beta_k, \quad
c_{8+n} =
{\rm Re}\!\!\! \!\!\!\!\!\!\!  \sum_{k=\in\{1,2,3,5\}} \!\!\!\!\!\!\!\!
\rho_k^n \gamma_k$$

\item[]
\begin{eqnarray*}
& &\alpha_1 \approx \ \ 0.0001015421, \quad
\alpha_2 \approx 3375.0749184657,\\
& &\beta_1 \approx   -0.0000538719,\quad
\beta_2 \approx 3552.5887804916,\\
& &\gamma_1\approx   -0.0000206576\quad
\gamma_2\approx 4039.314501374481984826,
\end{eqnarray*}

\item[]
etc.

\item[]
\begin{eqnarray*}
& &\alpha_5 \approx    2\cdot(-0.0380308729 +  1.4388617102 i) ,\\
& &\beta_5 \approx \ \  2\cdot(2.6921768571   -1.4860158672 i) ,\\
& &\gamma_5\approx \ \  2\cdot(2.34766856938 +  0.9004836949 i) .
\end{eqnarray*}

\end{itemize}

}


\frame[<+->]{

\frametitle{GENERATOR FUNCTIONS}

\begin{itemize}

\item[]
$v_n=\sum_{p=1}^K v_p A_p$, \qquad $n>K$

$\phantom{xxxxx}$


\item[]
$g(t) := \sum_{n=1}^\infty t^n v_n$ \qquad\qquad scalar$\to$vector

$\phantom{xxxxx}$

\item[] Classical Fibonacci: \quad $f_n=f_{n-1}+f_{n-2}$, $f_0=0$, $f_1=1$

\item[]
$\phantom{xxxxx}
g(t) = \sum_{n=1}^\infty t^n f_n$ \quad scalar$\to$scalar

\item[]
$\phantom{xxxxx}
g(t)-t g(t)-t^2g(t) = \sum_{n=2}^\infty (\underbrace{f_n-f_{n-1}-f_{n-2}}_0)t^n + f_0 + (f_1-f_0)t$

\item[]
$\phantom{xxxxx}
g(t) = t/(1-t-t^2) = \alpha_1 (t-\rho_1)^{-1} + \alpha_2 (t-\rho_2)^{-1}$

\item[]
$\phantom{xxxxx}
g(t) = \sum_{n=0}^\infty \big( \underbrace{\beta_1 \rho_1^n + \beta_1 \rho_1^n}_{f_n} \big) t^n$


\end{itemize}

}


\frame[<+->]{

\frametitle{GENERATOR FUNCTION VIA PHASE SPACE}

\begin{itemize}

\item[]
$g(t):= v_1 + t v_2 + t^2 v_3 + \cdots =
\sum_{n=0}^\infty t^n v_{n+1}$

\item[]
\begin{eqnarray*}
g(t)\!\!\!\! &-&\!\!\!\! \Big[ t g(t)A_1 +\cdots + t^K g(t) A_K +
\sum_{n=K+1}^\infty t^n b \Big] =\\
%&=& \sum_{n=0}^\infty t^n v_{n+1} -
%\sum_{n=1}^\infty t^n v_n A_1 -
%\sum_{n=2}^\infty \! t^n v_{n-1} A_2 - \cdots -
%\sum_{n=K}^\infty \! t^n v_{n+1-K} A_K - \!\!\!\!
%\sum_{n=K}^\infty \!\! t^n b =\\
%&=& v_1 + \sum_{n=1}^{K-1} t^n \big[ v_{n+1} -
%( v_n A_1 + v_{n-1} A_2 +\cdots+ v_1 A_n)\big] +\\
%& &\hskip5mm + \!\!
%\sum_{n=K}^\infty t^n
%\big[ v_{n+1} -
%(v_n A_1 + v_{n-1} A_2 + \cdots + v_{n+1-K} A_K +b)\big] =\\
&=& v_1 + \sum_{n=1}^{K-1} t^n \Big[ v_{n+1} - \sum_{r=1}^n v_{n+1-r} A_r \Big]
\end{eqnarray*}

\item[]
$$g(t) =  \Big[ {t^K \over 1 - t} b + \sum_{n=0}^{K-1} t^n b_n \Big]
\Big[ I - \sum_{p=1}^{K-1} t^p A_p \big]^{-1}$$

\item[]
$\phantom{xxxxxxxxxxxxxxxxxx}
b_n := v_{n+1} - \!\!\! \sum_{1\le r\le n} v_{n-r} A_r$


\end{itemize}

}

\def\eps{\varepsilon}

\frame[<+->]{

\frametitle{THEOREM 2}

\it
\begin{itemize}

\item[]
$f:(-\eps,\eps)\to\CC^d$ \qquad rational function

\item[]
$f(t) = u(t) U(t)^{-1}$ \qquad
$u:\RR\to \CC^d$, \ $U:\RR\to\CC^{d\times d}$

\item[]
$\Longrightarrow$

$\phantom{xxxxx}$

$f =\big[$gen. funct. of
$v_1,v_2,\ldots\in\CC^d$ with affine recursion prop.$\big]$

\item[]
$n\ge K$, \quad $\Rightarrow$ \ \
Taylor coefficients of \ $f$

$\phantom{xxxxx}$

$n!^{-1} d^n / dt^n \vert_{t=0} f(t) =$

$\phantom{xxxxx}$

\item[]
$=v^{(0)} +
\sum_{q=1}^N
\sum_{p=0}^{\min\{ n-K,m_q\}}
\left( {n-K \atop p} \right) \lambda_q^n v^{(p,q)}$.

\end{itemize}

}


\end{document}
