\documentclass[trans]{beamer} %\documentclass{beamer} \usepackage{beamerthemesplit, verbatim} \useoutertheme{infolines} \input{nielsenmacros} \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}