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\title[Coincidences of iterates]{
Nielsen coincidence theory of iterates}
\subtitle{(preliminary report)}
\author[Staecker]{P. Christopher Staecker (with Philip Heath)}
\institute[Fairfield U.]{Fairfield University, Fairfield CT}
\date[]{AIMS, Dresden 2010}

\newcommand{\aside}{\textcolor{red}}

\begin{document}
\frame{\titlepage}

\frame{
\frametitle{Nielsen theories}

Nielsen fixed point theory studies
\[ \Fix(f) = \{ x \mid f(x) = x \} \]
in a homotopy-invariant way.
\pause
\vfill
Generalizes to:
\pause
\vfill
{\bf Coincidence theory:}
\[ \Coin(f,g) = \{ x \mid  f(x) = g(x) \} \]
\pause

{\bf Periodic points theory:}
\[ f^n(x) = x \]
Can study points with minimal period $n$, or points with any period $n$.
}

\frame{
\frametitle{Why not?}
Let's try both:
\[ f^n(x) = g^n(x) \]

\pause
\vfill

Or even:
\[ f^n(x) = g^m(x) \]
\vfill

\pause
For now, let's just stick with $n=m$. 
}

\frame{
\frametitle{$NP_n$ and $N\Phi_n$}
As in the Nielsen periodic point theory, we want Nielsen
numbers $NP_n(f,g)$ and $N\Phi_n(f,g)$ which are meant to satisfy
something like:
\begin{eqnarray*}
N\Phi_n(f,g) &\le& \min \#\{ x \mid f^n(x) = g^n(x) \} \\
\pause
NP_n(f,g) &\le& \min \#\{ x \mid f^n(x) = g^n(x) \text{ but }
f^d(x) \neq f^d(x) \text{ for } d \mid n\} 
\end{eqnarray*}
}

\frame{

Call the set $\{x, f(x), f^2(x), \dots\}$ the \emph{trajectory of $x$
under $f$}. Then we are finding points $x$ for which the
trajectories under $f$ and $g$ meet at various iteration levels.

\vfill
\pause
\begin{itemize}
\item $NP_n(f,g)$ should measure how many points have intersecting
trajectories iterate $n$ but not at any iterate $k \mid n$.
\pause\vfill
\item $N\Phi_n(f,g)$ should measure how many points have intersecting
trajectories at iterate $n$.
\end{itemize}
}

\frame{
\frametitle{Setting}
We consider compact manifolds without boundary.
\pause
\vfill
Unlike in coincidence theory, we must require $f$ and $g$ to be
selfmaps (so that we can iterate).

}

\frame{
\frametitle{Periodicity and commutativity}
To mimic the Nielsen periodic points theory,
we want some kind of periodicity: like
\pause
\[ f^k(x) = g^k(x) \quad \Rightarrow \quad f^{n}(x) = g^{n}(x) \qquad \text{
  for } k \mid n\]
\pause
This is not automatic.
\pause
\vfill

If $f(x) = g(x)$, we get 
\[ f^2(x) = f(g(x)) \text{ and } g(f(x)) = g^2(x) \]

\pause
So we're going to need commutativity: $f\circ g = g \circ f$. (We'll
be able to loosen this a bit.)
}

\frame{ 
\frametitle{Ingredients}
The Nielsen periodic points theory has 3 main ingredients:
\begin{itemize}
\pause
\item The fixed point index
\pause
\item Reidemeister classes and boosts
\pause
\item Reidemeister orbits
\end{itemize}
}

\frame{
\frametitle{The index}
We will use the coincidence index. 
\vfill
\pause
Coincidence points split into \emph{coincidence classes}: $x,y \in
\Coin(f,g)$ are in the same class when there is some path $\gamma$
connecting them with $f(\gamma) \simeq g(\gamma)$.
\vfill
\pause
A coincidence class $C \subset \Coin(f^n,g^n)$ is \emph{essential} when
$\ind(f^n,g^n,C)$ is nonzero.
}

\frame{
\frametitle{Reidemeister classes}

Every coincidence class has an associated \emph{Reidemeister class}
$[\alpha] \in \Reid(f^n,g^n)$, where 
\[ \Reid(f^n,g^n) = \pi_1(X) / \sim \]

with the relation
\[ [\alpha] \sim [\beta] \iff \exists z\in \pi_1(X) \quad \beta = f^n_*(z) \alpha
g^{-n}_*(z). \]
}

\frame{
\frametitle{The boost}
There is a \emph{boost} function from one iteration level to another: for $k
\mid n$, we have
\[ \iota_{k,n}: \Reid(f^k,g^k) \to \Reid(f^n,g^n) \]
given by \pause
\[ \iota_{k,n}([\alpha]) = [ f_*^{n-k}(\alpha)\,
f_*^{n-2k}(g_*^k(\alpha))\, \dots f_*^k(g_*^{n-2k}(\alpha)) \,
g_*^{n-k}(\alpha) ] \]

\pause
\vfill
This is well-defined on Reidemeister classes. \pause

(Absolutely needs commutativity of
$f_*$ and $g_*$!)
}

\frame{
\frametitle{$NP_n$}
We say that a class $[\alpha] \in \Reid(f^n,g^n)$ is
\emph{irreducible} if it is not in the image of any boost. 
\pause
\vfill
Define:
\[ NP_n(f,g) = \# \text{ of essential irreducible classes of $(f^n,
  g^n)$}. \]

\pause
\vfill
Note: no orbits! We have $f^n(x) = g^n(x) \Rightarrow
f^{2n}(x) = g^{2n}(x)$.
\pause

Orbits require $f^n(x) = f^{2n}(x)$, which is something different
entirely.
}

\frame{
\frametitle{$NP_n$}
$NP_n$ as defined above satisfies:
\begin{itemize}
\pause 
\item $NP_n(f,g)$ is homotopy invariant for both $f$ and $g$.
\pause
\item $NP_n(f,g) \le \# \{ x \mid f^n(x) = g^n(x), \quad f^{n/d}(x)
  \neq g^{n/d}(x) \}$
\pause
\begin{item} $NP_n(f,\id) = NP_n(f)$ when $X$ is essentially toral
\end{item}
\end{itemize}

\vfill
\pause
$NP_n$ does \emph{not} actually require $f\circ g = g \circ f$, but
only $f_*\circ g_* = g_* \circ f_*$. 
\vfill
\pause
This is a nicer requirement because it is preserved by homotopy.
}

\frame{
\frametitle{The number of coincidences}
From periodic points theory we have
\[ \sum_{k \mid n} NP_k(f) \le \min_{h \htp f} \#\Fix(f^n) \]
\vfill
\pause
This is not true simply replacing ``$\Fix$'' with ``$\Coin$''. 
\pause
\vfill
In fact
\[ \min_{h \htp f, l \htp \id} \#\Coin(f^n,g^n) \] 
isn't even a generalization.
}

\frame{
\frametitle{One, rather than both}
``On removing coincidences of two maps when only one, rather 
than both, of them may be deformed by a homotopy'': 
\[ \min_{h \htp f} \#\Fix(h) = \min_{h \htp f, l \htp \id}
\#\Coin(h,l) \]
when the spaces are manifolds. (Brooks, 1971)
\vfill
\pause
This does not hold for iterates. 
\vfill
\pause That is, for $n \neq 1$ it is
possible for 
\[ \min_{h \htp f} \#\Fix(h^n) \neq \min_{h \htp f, l \htp \id}
\#\Coin(h^n,l^n) \]
\vfill
\pause
(Let $f: S^1 \to S^1$ be $f(z)=\bar z$, $g$ a small rotation.)
}

\frame{
\frametitle{Union of coincidences}
The proper set of coincidences to look at is:
\[ \bigcup_{k \mid n} \Coin(f^k,g^k) \]
\pause
This gives you $\Fix(f^n)$ when $g=\id$.

\vfill
\pause
We can prove
\[ \sum_{k\mid n} NP_k(f,g) \le \# \bigcup_{k \mid n}
\Coin(f^k,g^k) \]
\vfill
\pause
(Still unsure whether \[\min\#\Fix(f^n) = \min \#\bigcup_{k\mid n}
\Coin(f^k,g^k)\] for $g \htp \id$. \pause I don't think so.)
}

\frame{
\frametitle{Two Nielsen numbers}
Nielsen periodic point theory has two basic invariants:
\begin{align*}
NP_n(f) &\le \text{number of periodic points of \emph{least} period $n$} \\
N\Phi_n(f) & \le \text{number of periodic points of period $n$}
\end{align*}

\vfill
\pause
The definition of the second one is a bit tricky- the trick is to make
it a homotopy invariant of $f$ (not $f^n$). 
\vfill
}

\frame{
\frametitle{$N\Phi_n$}
A set $\mathcal G$ of coincidence classes is a set of {\bf
  $n$-representatives} if every essential class of $(f^k,g^k)$ for $k
\mid n$ reduces to something in $\mathcal G$.

\vfill
\pause

The minimal size of a set of $n$-representatives is called
$N\Phi_n(f,g)$.

\vfill
\pause
Easy to see that 
\[ \sum_{k \mid n} NP_k(f,g) \le N\Phi_n(f,g) \]
}

\frame{
\frametitle{What we all want}
Several theorems we would like to have about the sum of the $NP_k$:
\vfill
\pause
If $f,g$ are essentially reducible, then
\[ N\Phi_n(f,g) = \sum_{k \mid n} NP_k(f,g). \]
\vfill
\pause
With a few more conditions (Jiang, essentially reducible to the gcd,
$N(f^n,g^n) \neq 0$),
\[ N\Phi_n(f,g) = \sum_{k\mid n} NP_k(f,g) = N(f^n,g^n) \]
\vfill \pause
and
\[ NP_n(f,g) = \sum_{\tau \subset \mathbf p(n)} (-1)^{\#\tau}
N(f^{n:\tau},g^{n:\tau}) \]
}

\frame{
\frametitle{Essential reduction to the gcd}
Some of those will be easy, some are difficult. Some of these conditions
are more complicated for coincidences.
\vfill\pause
With great effort, we have shown (using new methods)
\begin{thm}
If $f,g:S^1 \to S^1$ have degrees $a$ and $b$, then $f,g$ essentially
reduce to the gcd if and only if $\gcd(a,b) = 1$.
\end{thm}

\vfill\pause
Hopefully something like this is true for tori and solvmanifolds, but our argument for
circles doesn't generalize.

\vfill\pause
(Injective boosts (``essential torality'') works fine for coincidences for
circles and tori.)
}

\frame{
\frametitle{Circles}
On circles (and tori), all maps are essentially reducible, so we get
\[ N\Phi_n(f,g) = \sum_{k \mid n}NP_k(f,g) \]
\vfill
\pause
But not all maps on the circle are essentially reducible to the gcd,
so we may expect
\[ \sum_{k \mid n} NP_k(f,g) \neq N(f^n,g^n) \]
even when $N(f,g) \neq 0$.
}

\frame{
\frametitle{An example}
Let $f,g:S^1 \to S^1$ have degrees 0 and 2. Then the Reidemeister
classes are:
\[ \Reid(f,g) \cong \Z_2 \quad \Reid(f^2,g^2) \cong \Z_4 \quad
\Reid(f^3,g^3) \cong \Z_8 \quad \Reid(f^6,g^6) \cong \Z_{64}. \]
\vfill
\pause
 The boosts are [multiplication by]
\[ \iota_{1,2} = 2, \quad \iota_{1,3} = 4, \quad \iota_{1,6} = 32,
\quad \iota_{2,6} = 16, \quad \iota_{3,6} = 8 \]
\[ (\iota_{2,6} = 0^4 + 0^2 2^2 + 2^4 = 16 ) \]
\vfill \pause
So $[16] \in \Reid(f^6,g^6)$ reduces to levels 3 and 2, but not 1. 
}

\frame{
\begin{align*} \Reid(f,g) \cong \Z_2 \quad \Reid(f^2,g^2) \cong \Z_4 \quad
\Reid(f^3,g^3) \cong \Z_8 \quad \Reid(f^6,g^6) \cong \Z_{64}. \\
 \iota_{1,2} = 2, \quad \iota_{1,3} = 4, \quad \iota_{1,6} = 32,
\quad \iota_{2,6} = 16, \quad \iota_{3,6} = 8 
\end{align*}

For this example, we can compute
\begin{align*}
NP_1(f,g) &= 2, \\ NP_2(f,g) &= 4-2 = 2, \\ NP_3(f,g) &= 8 - 2 = 6, \\
NP_6(f,g) &= 64 - 8 = 56 \end{align*}
}

\frame{
So we get
\[ \sum_{k\mid 6} NP_k(f,g) = 66 \quad \text{but} \quad 
 N(f^6,g^6) = 64. \]
\vfill
\pause
So 
\[ \sum_{k\mid 6} NP_k(f,g) \neq N(f^6,g^6). \]
\vfill\pause
And the M\"obius inversion fails too:
\begin{align*} \sum_{\tau \subset \mathbf p(6)}
(-1)^{\#\tau}N(f^{6:\tau},g^{6:\tau}) &= N(f,g) - N(f^2,g^2) -
N(f^3,f^3) + N(f^6,g^6) \\&= 2 - 4 - 8 + 64 = 54 \end{align*}
\vfill\pause so 
\[ 56 = NP_6(f,g) \neq \sum_{\tau \subset \mathbf p(6)}
(-1)^{\#\tau}N(f^{6:\tau},g^{6:\tau}) \]
\pause\vfill
So essential reducibility to the gcd is necessary for these, even on the circle.
}

\frame{
\frametitle{It's over!}
Thanks!
}



\end{document}