\documentclass[../lic_malinka.tex]{subfiles}

\begin{document}
  Let $M$ be a countable $L$-structure. Recall, we define a topology on 
  the $G=\Aut(M)$:
  for any finite function $f\colon M\to M$ we have a basic open set 
  $[f]_{G} = \{g\in G\mid f\subseteq g\}$.  

  \subsection{Prototype: pure set} 

  In this section, $M=(M,=)$ is an infinite countable set (with no structure
  beyond equality).
  
  \begin{remark} 
	\label{remark:cojugate-classes}
	If $f_1,f_2\in \Aut(M)$, then $f_1$ and $f_2$ are conjugate if and only 
    if for each $n\in \bN\cup \{\aleph_0\}$, $f_1$ and $f_2$ have the same 
    number of orbits of size $n$.
  \end{remark}

  \begin{proof}
	It is easy to see.
  \end{proof}

  \begin{theorem}
	Let $\sigma\in \Aut(M)$ be an automorphism with no infinite orbit and with
	infinitely many orbits of size $n$ for every $n>0$. Then the conjugacy
	class of $\sigma$ is comeagre in $\Aut(M)$.
  \end{theorem}

  \begin{proof}
	We will show that the conjugacy class of $\sigma$ is an intersection of countably
	many comeagre sets. 

	Let $A_n = \{\alpha\in \Aut(M)\mid \alpha\text{ has infinitely many orbits of size }n\}$.
	This set is comeagre for every $n>0$. Indeed, we can represent this set
	as an intersection of countable family of open dense sets. Let $B_{n,k}$
	be the set of all finite functions $\beta\colon M\to M$ that consist
	of exactly $k$ distinct $n$-cycles. Then:
	\begin{align*}
	  A_n &= \{\alpha\in\ \Aut(M) \mid \alpha\text{ has infinitely many orbits of size }n\} \\
		  &= \bigcap_{k=1}^{\infty} \{\alpha\in \Aut(M)\mid \alpha\text{ has at least }k\text{ orbits of size }n\} \\ 
		  &= \bigcap_{k=1}^{\infty} \bigcup_{\beta\in B_{n,k}} [\beta]_{\Aut(M)},
	\end{align*}
	where indeed, $\bigcup_{\beta\in B_{n,k}} [\beta]_{\Aut(M)}$ is dense in
	$\Aut(M)$: take any finite $\gamma\colon M\to M$ such that $[\gamma]_{\Aut(M)}$
	is nonempty. Then also 
	$\bigcup_{\beta\in B_{n,k}} [\beta]_{\Aut(M)}\cap[\gamma]_{\Aut(M)}\neq\emptyset$,
	one can easily construct a permutation that extends $\gamma$ and has at least
	$k$ many $n$-cycles.

	Now we see that $A = \bigcap_{n=1}^{\infty} A_n$ is a comeagre set consisting
	of all functions that have infinitely many $n$-cycles for each $n$. The only
	thing left to show is that the set of functions with no infinite cycle is 
	also comeagre. Indeed, for $m\in M$ let 
	$B_m = \{\alpha\in\Aut(M)\mid m\text{ has finite orbit in }\alpha\}$. This 
	is an open dense set. It is a union over basic open sets generated by finite
	permutations with $m$ in their domain. Denseness is also easy to see.

	Finally, by the Remark \ref{remark:cojugate-classes}, we can say that 
	$$\sigma^{\Aut(M)}=\bigcap_{n=1}^\infty A_n \cap \bigcap_{m\in M} B_m,$$
	which concludes the proof.
  \end{proof}

  \subsection{More general structures}

  \begin{fact} 
	\label{fact:conjugacy}
    Suppose $M$ is an arbitrary structure and $f_1,f_2\in \Aut(M)$. 
    Then $f_1$ and $f_2$ are conjugate if and only if $(M,f_1)\cong
    (M,f_2)$ as structures with one additional unary function that is
    an automorphism.  
  \end{fact}

  \begin{proof}
    Suppose that $f_1 = g^{-1}f_2g$ for some $g\in \Aut(M)$. 
	Then $g$ is the isomorphism between $(M,f_1)$ and $(M,f_2)$. 
	On the other hand if 
    $g\colon (M, f_1)\to (M, f_2)$ is an isomorphism, then 
    $g\circ f_1 = f_2 \circ g$ which exactly means that $f_1, f_2$ conjugate.
  \end{proof} 

  \begin{theorem}
	\label{theorem:generic_aut_general}
	Let $\cC$ be a Fraïssé class of finitely generated $L$-structures. 
	Let $\cD$ be the class of structures from $\cC$ with additional unary 
	function symbol interpreted
	as an automorphism of the structure. If $\cC$ has the weak Hrushovski 
	property, $\cD$ is a Fraïssé class and $(\Pi, \sigma) = \Flim(\cD)$, then
	$\Aut(\cC)$ has a comeagre conjugacy class and $\sigma$ is in this class
	(i.e. $\sigma$ is a generic automorphism).
  \end{theorem}

  Before we get to the proof, it is important to mention that an isomorphism 
  between two finitely
  generated structures is uniquely given by a map from generators of one structure
  to the other. This allow us to treat a finite function as an isomorphism
  of finitely generated structures (if it yields one) and \textit{vice versa}. 

  \begin{proof}
	Let $\Gamma = \Flim(\cC)$ and $(\Pi, \sigma) = \Flim(\cD)$. First, by the Theorem
	\ref{theorem:isomorphic_fr_lims}, we may assume without the loss of generality
	that $\Pi = \Gamma$. Let $G = \Aut(\Gamma)$,
	i.e. $G$ is the automorphism group of $\Gamma$.
	We will construct a winning strategy for the second player in the Banach-Mazur game
	(see \ref{definition:banach-mazur-game})
	on the topological space $G$ with $A$ being $\sigma$'s conjugacy class. 
	By the Banach-Mazur theorem (see \ref{theorem:banach_mazur_thm}) this will prove 
	that this class is comeagre.

	Recall, $G$ has a basis consisting of 
	sets $\{g\in G\mid g\upharpoonright_A = g_0\upharpoonright_A\}$ for some
	finite set $A\subseteq \Gamma$ and some automorphism $g_0\in G$. In other
	words, a basic open set is a set of all extensions of some partial
	automorphism $g_0$ of finitely generated substructures of $\Gamma$. 
	By $B_{g}\subseteq G$ we denote a basic 
	open subset given by a partial isomorphism $g$. Again, Note that $B_g$
	is nonempty because of ultrahomogeneity of $\Gamma$.

	With the use of Corollary \ref{corollary:banach-mazur-basis} we can consider 
	only games where both players choose partial isomorphisms. Namely,
	player \textit{I} picks functions $f_0, f_1,\ldots$ and player \textit{II}
	chooses $g_0, g_1,\ldots$ such that
	$f_0\subseteq g_0\subseteq f_1\subseteq g_1\subseteq\ldots$, which identify
	the corresponding basic open subsets $B_{f_0}\supseteq B_{g_0}\supseteq\ldots$.
	
	Our goal is to choose $g_i$ in such a manner that 
	$\bigcap^{\infty}_{i=0}B_{g_i} = \{g\}$ and $(\Gamma, g)$ is ultrahomogeneous
	with age $\cD$. By the Fraïssé theorem (see \ref{theorem:fraisse_thm}) it will follow
	that $(\Gamma, \sigma)\cong (\Gamma, g)$, thus by the Fact \ref{fact:conjugacy}
	we have that $\sigma$ and $g$ conjugate.

	Fix a bijection $\gamma\colon\bN\times\bN\to\bN$ such that for any
	$n, m\in\bN$ we have $\gamma(n, m) \ge n$. This bijection naturally induces
	a well ordering on $\bN\times\bN$. This will prove useful later, as the
	main ingredient of the proof will be a bookkeeping argument.

	For technical reasons, let $g_{-1} = \emptyset$ and
	$X_{-1} = \emptyset$. Enumerate the elements of the Fraïssé limit 
	$\Gamma = \{v_0, v_1, \ldots\}$.
	Suppose that player \textit{I} in the $n$-th move chooses a partial
	automorphism $f_n$. We will construct a partial automorphism 
	$g_n\supseteq f_n$ together with a finitely generated substructure
	$\Gamma_n \subseteq \Gamma$ and a set $X_n\subseteq\bN^2$ 
	such that the following properties hold:

	\begin{enumerate}[label=(\roman*)]
	  \item $g_n$ is a partial automorphism of $\Gamma$ and an automorphism of
		finitely generated substructure $\Gamma_n$,
	  \item $g_n(v_n)$ and $g_n^{-1}(v_n)$ are defined.
	  \end{enumerate}

	  Before we give the third point, suppose recursively that $g_{n-1}$ already
	  satisfy all those properties. Let us enumerate 
	  $\{\langle (A_{n,k}, \alpha_{n, k}), (B_{n,k}, \beta_{n,k}), f_{n, k}\rangle\}_{k\in\bN}$
	  all pairs of finitely generated structures with automorphisms such
	  that the first substructure embed into the second by inclusion, i.e.
	  $(A_{n,k}, \alpha_{n,k})\subseteq (B_{n,k}, \beta_{n,k})$, and $f_{n,k}$
	  is an embedding of $(A_{n,k}, \alpha_{n,k})$ in the $(\FrGr_{n-1}, g_{n-1})$.
	  We allow $A_{n,k}$ to be empty. Although $g_{n-1}$ is a finite function,
	  we may treat it as a partial automorphism as we have said before.

	  \begin{enumerate}[resume, label=(\roman*)]
	  \item 
		Let 
		$(i, j) = \min\{(\{0, 1, \ldots, n\} \times \bN) \setminus X_{n-1}\}$ (with the
		order induced by $\gamma$). Then $X_n = X_{n-1}\cup\{(i,j)\}$ and
		$(B_{i,j}, \beta_{i,j})$ embeds in $(\FrGr_n, g_n)$ so that this diagram
		commutes:

		\begin{center}
		  \begin{tikzcd}
									  & (\Gamma_n, g_n) & \\
			(B_{i,j}, \beta_{i,j}) \arrow[ur, dashed, "\hat{f}_{i,j}"] &   & (\FrGr_{n-1}, g_{n-1}) \arrow[ul, dashed, "\subseteq"'] \\
									  & (A_{i,j}, \alpha_{i,j}) \arrow[ur, "f_{i,j}"'] \arrow[ul, "\subseteq"]
		  \end{tikzcd}
		\end{center}
	\end{enumerate}

	First, we will satisfy the item (iii). Namely, we will construct $\Gamma'_n, g'_n$
	such that $g_{n-1}\subseteq g'_n$, $\Gamma_{n-1}\subseteq\Gamma'_n$,
	$g'_n$ gives an automorphism of $\Gamma'_n$
	and $f_{i,j}$ extends to an embedding of
	$(B_{i,j}, \beta_{i,j})$ to $(\Gamma'_n, g'_n)$. But this can be easily
	done by the fact, that $\cD$ has the amalgamation property. 


	It is important to note that $g'_n$ should be a finite function and once
	again, as it is an automorphism of a finitely generated structure, we may
	think it is simply a map from one generators of $\Gamma'_n$ to the 
	others.	By the weak ultrahomogeneity of $\Gamma$, we may assume that 
	$\Gamma'_n\subseteq \Gamma$.

	Now, by the WHP of $\cC$ we can extend $\langle\Gamma'_n\cup\{v_n\}\rangle$ together
	with its partial isomorphism $g'_n$ to a finitely generated structure $\Gamma_n$ 
	together with its
	automorphism $g_n\supseteq g'_n$ and (again by weak ultrahomogeneity)
	without the loss of generality we
	may assume that $\Gamma_n\subseteq\Gamma$. This way we've constructed $g_n$
	that has all desired properties.

	Now we need to see that $g = \bigcap^{\infty}_{n=0}g_n$ is indeed an automorphism
	of $\Gamma$ such that $(\Gamma, g)$ has the age $\cD$ and is weakly ultrahomogeneous.
	It is of course an automorphism of $\Gamma$ as it is defined for every $v\in\Gamma$
	and is an union of an increasing chain of automorphisms of finitely generated
	substructures.

	Take any $(B, \beta)\in\cD$. Then, there are
	$i, j$ such that $(B, \beta) = (B_{i, j}, \beta_{i,j})$ and $A_{i,j}=\emptyset$.
	By the bookkeeping there was $n$ such that 
	$(i, j) = \min\{\{0,1,\ldots n\}\times\bN\setminus X_{n}\}$.
	This means that $(B, \beta)$ embeds into $(\Gamma_n, g_n)$, hence it embeds
	into $(\Gamma, g)$. Thus, $\cD$ is a subclass of the age of $(\Gamma, g)$.
	The other inclusion is obvious.	Hence, the age of $(\Gamma, g)$ is $\cH$. 

	It is also weakly ultrahomogeneous. Having $(A,\alpha)\subseteq(B,\beta)$,
	and an embedding $f\colon(A,\alpha)\to(\Gamma,g)$, we may find $n\in\bN$
	such that $(i,j) = \min\{\{0,1,\ldots n-1\}\times X_{n-1}\}$ and
	$(A,\alpha) = (A_{i,j},\alpha_{i,j}), (B,\beta)=(B_{i,j},\beta_{i,j})$ and
	$f = f_{i,j}$. This means that there is a compatible embedding of $(B,\beta)$ into
	$(\Gamma_n, g_n)$, which means we can also embed it into $(\Gamma, g)$.

	Hence, $(\Gamma,g)\cong(\Gamma,\sigma)$.
	By this we know that $g$ and $\sigma$ are conjugate in $G$, thus player \textit{II}
	have a winning strategy in the Banach-Mazur game with $A=\sigma^G$,
	thus $\sigma^G$ is comeagre in $G$ and $\sigma$ is a generic automorphism.
  \end{proof}

  \begin{theorem}
	\label{theorem:key-theorem}
	Let $\cC$ be a Fraïssé class of finitely generated $L$-structures with WHP
	and canonical amalgamation. Let $\cD$ be the class of structures from $\cC$ with 
	additional unary function symbol interpreted as an automorphism of the structure.
	Then $\cD$ is a Fraïssé class and for $(\Pi, \sigma) = \Flim(\cD)$ 
	we have that the conjugacy class of $\sigma$ is comeagre in $\Aut(\cC)$.
  \end{theorem}

  \begin{proof}
	It follows trivially from Corollary \ref{corollary:whp+canonical-iso}
	and the above Theorem \ref{theorem:generic_aut_general}.
  \end{proof}

  \subsection{Properties of the generic automorphism}

  This key theorem yields some corollaries and we present one of them below.

  Let $\cC$ be a Fraïssé class of finitely generated $L$-structures with
  weak Hrushovski property and canonical amalgamation. 
  Let $\cD$ be the Fraïssé class (by the Theorem \ref{theorem:key-theorem}
  of the structures of $\cC$ with additional automorphism of the structure).
  Let $\Gamma = \Flim(\cC)$. 

  \begin{corollary}
	\label{corollary:fixed_points}
	Let $\sigma$ be the generic automorphism of $\Gamma$. Then the set
	of fixed points of $\sigma$ is isomorphic to $\Gamma$.
  \end{corollary}

  \begin{proof}
	Let $S = \{x\in \Gamma\mid \sigma(x) = x\}$. It is a substructure of $\Gamma$,
	as it is closed under operations. For any $n$-tuple $a_1,\ldots,a_n\in S$
	and $n$-ary function symbol $f$ we have that 
	$f(a_1,\ldots a_n) = f(\sigma(a_1),\ldots, \sigma(a_n)) = \sigma(f(a_1,\ldots,a_n)$,
	which exactly says that $f(a_1,\ldots,a_n)$ is a fixed point. Also, constant
	symbols have to be fixed under $\sigma$.

	It is obvious that the
	age of $S$ is $\cC$, as every structure from $\cC$ with identity embeds
	into $\Gamma$, and hence into $S$. Also it is weakly ultrahomogeneous.
	Take $A\subseteq B$ with embedding $f\colon A\to S$. It can be thought
	as embedding of $(A, \id_A)$ into $(\Gamma,\sigma)$ and thus by its weak
	ultrahomogeneity we have $\hat{f}\colon (B,\id_B)\to (\Gamma,\sigma)$ and
	hence $\hat{f}$ is also and embedding of $B$ into $S$ such that the
	following diagram commutes:

	\begin{center}
	  \begin{tikzcd}
		A \ar[r, "f"] \ar[d, "\subseteq"] & S \\
		B \ar[ru, "\hat{f}"]
	  \end{tikzcd}
	\end{center}
	% First we need to show that it is an infinite substructure of $\Gamma$. 
	% By the theorem \ref{theorem:generic_aut_general}
	% we know that $(\Gamma, \sigma)$ is the Fraïssé limit of $\cD$, thus we
	% can embed finite $L$-structures of any size with identity as an 
	% automorphism of the	structure into $(\Gamma, \sigma)$. Thus $S$ has to be
	% infinite. Also, the same argument shows that the age of the structure is
	% exactly $\cC$. It is weakly ultrahomogeneous, also by the fact that
	% $(\Gamma, \sigma)$ is in $\cD$.
  \end{proof}
\end{document}