Proof. Let $\|\cdot \|$ be an Hermitian metric on X. Suppose that there is $x\in W^s_f(p)\cap F_f^+$ . By normality, for any tangent vector $v\in T_x X$ , the sequence $(\|d_xf^n(v)\|)$ is bounded.

Let $\mathscr {C}_p$ be an homogeneous cone in $T_pX$ such that the tangent space to the unstable manifold $W_f^u(p)$ is contained in the interior of $\mathscr {C}_p$ , and the tangent space to the stable manifold $W_f^s(p)$ is contained in the interior of $T_p X\setminus \mathscr {C}_p$ . For $k\geq 0$ large enough, $d_pf^k(\mathscr {C}_p)\subset \mathrm {int}(\mathscr {C}_p)$ and $\|d_pf^k(v)\|> 2\|v\|$ for all $v\in \mathscr {C}_p\setminus \{0\}$ . The cone $\mathscr {C}_p$ can be extended to a cone field $\mathscr {C}_\bullet $ in a neighbourhood U of p such that

$$ \begin{align*} d_zf^k(\mathscr{C}_z)\subset \mathrm{int}(\mathscr{C}_{f(z)}) \quad\text{and}\quad \|d_zf^k(v)\|> 2\|v\| \quad \text{for all } v\in \mathscr{C}_z\setminus\{0\} \end{align*} $$

for all $z\in U\cap f^{-1}(U)$ .

Since $x\in W^s_f(p)$ , there exists an integer m such that $f^n(x)\in U$ for all $n\geq m$ . Choose a non-zero $v\in T_xX$ such that $d_xf^m(v)\in \mathscr {C}_{f^m(x)}\setminus \{0\}$ . Then,

$$ \begin{align*}\|d_xf^{m+kj}(v)\|> 2^k\|d_xf^m(v)\|\end{align*} $$

for all $j\geq 1$ , which contradicts the boundedness of the sequence $(\|d_xf^n(v)\|)$ .

Proof. The proof is so similar to the proof in [Reference Arosio and Lárusson4] of the analogous result for endomorphisms of an Oka–Stein manifold [Reference Arosio and Lárusson4, Theorem 1(a)] that we will only sketch it.

Given a compact subset K of X and an integer $m\geq 1$ , let $\mathscr T(m,K)\subset {\operatorname {Aut}}\,X$ be the open set of automorphisms such that every periodic point in K of X of minimal period at most m is transverse. Let $\mathscr H(m,K)\subset {\operatorname {Aut}}\,X$ be the open set of automorphisms of X such that every period point in K of minimal period at most m is hyperbolic.

We prove, in three steps as in [Reference Arosio and Lárusson4], that for all K and m, $\mathscr H(m,K)$ is dense in ${\operatorname {Aut}}\, X$ . We then exhaust X by compact sets $K_m$ and conclude that every periodic point of an automorphism in the residual subset $\bigcap \nolimits _{m\geq 1}\mathscr H(m,K_m)$ of ${\operatorname {Aut}}\, X$ is hyperbolic.

Step 1. For every compact $K\subset X$ , $\mathscr T(1,K)$ is dense in ${\operatorname {Aut}}\, X$ . This is the analogue of [Reference Arosio and Lárusson4, Proposition 1]. Let $f_0\in {\operatorname {Aut}}\, X$ . We will produce a perturbation of $f_0$ , that is, a continuous map $f:P\to {\operatorname {Aut}}\, X$ , where the parameter space P is a neighbourhood of the origin $0$ in some ${\mathbb {C}}^k$ , with $f(0)=f_0$ , such that there are arbitrarily small $t\in P$ with $f_t=f(t)\in \mathscr T(1,K)$ . Equivalently, for arbitrarily small $t\in P$ , the map $X\to X\times X$ , $x\mapsto (x,f_t(x))$ , is transverse to the diagonal $\Delta \subset X\times X$ on K. By the parametric transversality theorem, this holds if the associated map $F:X\times P\to X\times X$ , $F(x,t)=(x,f_t(x))$ , is holomorphic (or merely $C^1$ ) and transverse to $\Delta $ on $K\times P$ .

Take finitely many complete holomorphic vector fields $v_1,\ldots ,v_r$ on X that span each tangent space of X on a neighbourhood U of $f_0(K)$ . Let $\psi _\bullet ^j$ be the flow of $v_j$ and let $f_t=\psi _{t_r}^r\circ \cdots \circ \psi _{t_1}^1\circ f_0\in {\operatorname {Aut}}\,X$ , $t\in {\mathbb {C}}^r$ . Then, $(\partial /{\partial t_j})f(x,t)\vert _{(x,0)} = v_j(f_0(x))$ for all $x\in X$ , so the associated map F is transverse to the diagonal on $f_0^{-1}(U)\times \{0\}$ and hence on $K\times P$ for a sufficiently small neighbourhood P of $0$ in ${\mathbb {C}}^k$ .

Step 2. For every compact $K\subset X$ and $m\geq 1$ , $\mathscr H(m,K)$ is dense in $\mathscr T(m,K)$ . This is the analogue of [Reference Arosio and Lárusson4, Proposition 2] and the proof is verbatim the same, except with ${\operatorname {End}}\,X$ replaced by ${\operatorname {Aut}}\,X$ . The proof of [Reference Arosio and Lárusson4, Proposition 2] refers to the endomorphism case of [Reference Arosio and Lárusson3, Lemma 1]; here, we need the automorphism case of the lemma.

Step 3. For every compact $K\subset X$ and $m\geq 2$ , $\mathscr T(m,K) \cap \mathscr H(m-1,K)$ is dense in $\mathscr H(m-1,K)$ . This is the analogue of [Reference Arosio and Lárusson4, Proposition 3] and the proof is the same, except the perturbation $f:X\times ({\mathbb {C}}^r)^\ell \to X$ of the given automorphism $f_0\in \mathscr H(m-1,K)$ needs to be defined differently. As in the proof of [Reference Arosio and Lárusson4, Proposition 3], we define $f_t=f(\cdot , t_1,\ldots ,t_\ell ) = \phi _{t_1}^{V_1}\circ \cdots \circ \phi _{t_\ell }^{V_\ell }\circ f_0$ , but now, $\phi _\bullet ^{V_1},\ldots , \phi _\bullet ^{V_\ell }$ are holomorphic families of automorphisms, parametrized by ${\mathbb {C}}^r$ , of the kind $\phi _\bullet ^V$ described as follows.

In the notation of the proof of [Reference Arosio and Lárusson4, Proposition 3], were it possible to find finitely many complete holomorphic vector fields $v_1,\ldots ,v_r$ on X that span each tangent space of X on a neighbourhood of $\overline V$ and are as close to zero as we wish on $L_1\cup \cdots \cup L_{m-1}$ , then we would take $\phi _t^V=\psi _{t_r}^r\circ \cdots \circ \psi _{t_1}^1$ , with $\psi _\bullet ^j$ being the flow of $v_j$ (as in Step 1).

Without the completeness requirement, the fields $v_1,\ldots ,v_r$ can be chosen as above. Using the density property, then, we approximate $v_j$ uniformly on $\overline V\cup L_1\cup \cdots \cup L_{m-1}$ by a sum $v=u_1+\cdots +u_s$ of complete holomorphic vector fields $u_1,\ldots ,u_s$ on X with flows $\eta _\bullet ^1,\ldots , \eta _\bullet ^s$ . Then, $\eta _\tau =\eta _\tau ^s\circ \cdots \circ \eta _\tau ^1$ is an algorithm for v, meaning that $\eta _0={\operatorname {id}}_X$ and $\partial /{\partial \tau }\vert _{\tau =0}\eta _\tau (x)=v(x)$ for all $x\in X$ , and we take $\psi _{t_j}^j=\eta _{t_j}$ and $\phi _t^V= \psi _{t_r}^r\circ \cdots \circ \psi _{t_1}^1$ .

Proof. Take finitely many complete vector fields $v_1,\ldots ,v_m$ that span $T_p X$ for some $p\in X$ and hence for every p outside a proper closed analytic subset A of X. We want to produce finitely many fields that span $T_q X$ for a point q in each irreducible component of A and then induct on dimension. Let the countable set D consist of one point from each component of A.

By [Reference Winkelmann19, Corollary 8], there is an automorphism f of X with $f(x)\notin A$ for every $x\in D$ . (The corollary applies because by hypothesis, the orbits of ${\operatorname {Aut}}\,X$ are open, so there is only one orbit. Also, as for an arbitrary complex manifold, ${\operatorname {Aut}}\,X$ is completely metrizable, and hence Baire, and separable.) Then, the inverse of f takes a point in $X\setminus A$ into each component of A, so the fields $f_*^{-1}v_1,\ldots , f_*^{-1}v_m$ span $T_q X$ for a point q in each component of A.

Proof. In Buzzard’s notation, let $p_1$ and $p_2$ be saddle periodic points of an automorphism F of X and $q_0 \in W^s_F(p_1)\cap W^u_F(p_2)$ be a homoclinic or heteroclinic point. We need suitable replacements for his families $\Psi _k$ , $k\geq 1$ , of automorphisms and family $\Psi $ of diffeomorphisms of X, defined in [Reference Buzzard6, p. 501].

The total F-orbit of $q_0$ accumulates only on the union of the cycles of $p_1$ and $p_2$ , which is a finite set. Let K be the compact closure of the orbit. Let U be a Runge neighbourhood of K, consisting of finitely many coordinate balls with mutually disjoint closures, such that $q_0$ is the only point of K in the ball $U_0$ that contains it. Let $U_1 = U\setminus U_0$ . Let $V_0\subset U_0$ be a smaller ball containing $q_0$ and let B be a closed ball in ${\mathbb {C}}^n$ , $n=\dim X$ , centred at the origin, so that $\overline V_0+z\subset U_0$ for all $z\in B$ with respect to the coordinates in $U_0$ . Write $V=V_0\cup U_1$ .

Define $\Phi :V\times B\to X$ by the formula $(x,z)\mapsto x+z$ on $V_0\times B$ and by the formula $(x,z)\mapsto x$ on $U_1\times B$ . Extend $\Phi $ to a smooth family $\Psi :X\times B\to X$ of diffeomorphisms. Use the parametric Andersén–Lempert theorem [Reference Forstnerič9, Theorem 1.1] to approximate $\Phi $ locally uniformly on $V\times B$ by a sequence of smooth families $\Psi _k:X\times B\to X$ of automorphisms of X. The theorem is stated for ${\mathbb {C}}^n$ , $n\geq 2$ , but holds more generally for Stein manifolds with the density property. The Runge domains in the theorem should then be taken to be Stein (as they are here), but they need not be connected.

These families have the properties needed for Buzzard’s proof. The remainder of the proof consists of local arguments, and transversality and genericity arguments that straightforwardly extend to our more general setting.

Proof. After enlarging K, we may assume that it is holomorphically convex. Let $p\notin f(K)\cup K$ . By [Reference Varolin18, Theorem 2], there is $h\in {\operatorname {Aut}}\,X$ such that $d_{f(K)}(h,{\operatorname {id}})<\epsilon $ , $h(f(p))=p$ , and $d_{f(p)}h\circ d_pf$ is a saddle or attracting or repelling. Now take $g=h\circ f$ .

Proof of Theorem 4.1

By Remark 4.3, there is a countable dense subset $\{\tilde \psi _1, \tilde \psi _2, \ldots \}$ in ${\operatorname {Aut}}\,X$ such that each $\tilde \psi _j$ admits a saddle fixed point $\eta (\tilde \psi _j)$ . After passing to a suitable subsequence of $(\tilde \psi _j)$ , which we call $(\psi _j)$ , we can construct by induction compact subsets $H_j\subset X$ and numbers $\gamma _j>0$ such that:

1. (1) $\eta (\psi _j)\in H_j^\circ $ ;

2. (2) the open set $B_j:=\{h\in {\operatorname {Aut}}\,X: d_{H_j}(\psi _j,h)<\gamma _j\}$ is disjoint from $B_1,\ldots ,B_{j-1}$ ;

3. (3) $A:=\bigcup \nolimits _{j\geq 1}B_j$ is dense in ${\operatorname {Aut}}\,X$ since it contains $\{\tilde \psi _1, \tilde \psi _2, \ldots \}$ ;

4. (4) every automorphism $h\in B_j$ has a unique saddle fixed point $\eta (h)$ in the neighbourhood of $\eta (\psi _j)$ provided by Lemma 2.1, which we can assume to be contained in $H_j^\circ $ .

Let $\{U_n:n\geq 1\}$ be a countable basis for the topology of X consisting of relatively compact open sets, and define the sets

$$ \begin{align*} S_n:=\{f\in A: W^s_f(\eta(f))\cap U_n\neq \varnothing\}\end{align*} $$

and

$$ \begin{align*} T_n:=\{f\in A: W^u_f(\eta(f))\cap U_n\neq \varnothing\},\end{align*} $$

which are open by Lemma 2.3. Hence,

$$ \begin{align*} G:= \bigcap_n A\setminus \partial S_n\,\cap\, \bigcap_n A\setminus \partial T_n \end{align*} $$

is residual in ${\operatorname {Aut}}\,X$ . We claim that for all $f\in G$ , the stable manifold $W^s_f(\eta (f))$ is dense in $X\setminus {\operatorname {rne}}(f)$ and the unstable manifold $W^u_f(\eta (f))$ is dense in $X\setminus {\operatorname {rne}}(f^{-1})$ . By Lemma 2.4, this proves the theorem.

Fix $f\in G$ and $U_n$ intersecting $X\setminus {\operatorname {rne}}(f)$ . We will show that $f\in S_n$ . By the definition of G, this follows if we show that $f\in \overline S_n$ . Let $K\subset X$ be compact and let $\epsilon>0$ . We will show that there is $g\in S_n$ such that $d_K(f,g)<\epsilon $ . An analogous argument shows that $f\in T_n$ .

Find $j\geq 1$ such that $f\in B_j$ . Let H be a holomorphically convex compact set containing $K\cup H_j$ . By definition of ${\operatorname {rne}}(f)$ , there is $x\in U_n$ and an automorphism $\tilde f\in B_j$ with $d_H(f,\tilde f)< \epsilon /2$ , such that the orbit $({\tilde f}^n(x))$ is not contained in H. Let $n_0\geq 0$ be the smallest non-negative integer such that ${\tilde f}^{n_0}(x)\in X\setminus H$ . The stable manifold $W^s_{\tilde f} (\eta (\tilde f))$ is not contained in H, so we may choose $y\in W^s_{\tilde f} (\eta (\tilde f))\setminus H$ such that $\tilde f^n(y)\in H$ for all $n\geq 1$ .

Let W be a Runge neighbourhood of H containing neither $\tilde f^{n_0}(x)$ nor y. (We take a Runge open set to be Stein by definition.) Let V be a neighbourhood of $\tilde f^{n_0}(x)$ and let $\varphi : [0,1]\times V\to X$ be a $C^1$ isotopy such that for all $t\in [0,1]$ :

1. (1) $\varphi _t : V\to X$ is holomorphic and injective;

2. (2) $\varphi _t(V)$ is disjoint from W;

3. (3) $W\cup \varphi _t(V)$ is Runge;

4. (4) $\varphi _0$ is the inclusion of V into X;

5. (5) $\varphi _1(\tilde f^{n_0}(x))=y$ .

Find $m_0\geq 1$ such that $\tilde f^{m_0}(y)$ belongs to the local stable manifold $\Gamma ^s_{\tilde f}(\eta (\tilde f),r)$ , where r is given by Lemma 2.2. There are relatively compact neighbourhoods $U\subset U_n$ of x and Z of $\tilde f^{m_0}(y)$ such that ${\tilde f}^{n_0}(U)\Subset V$ and $Z\Subset \tilde f^{m_0}\circ \varphi _1\circ {\tilde f}^{n_0}(U)$ .

By the Andersén–Lempert theorem [Reference Forstnerič10, Theorem 4.10.5], there is a sequence $(\Phi _j)$ in ${\operatorname {Aut}}\,X$ such that $\Phi _j\to {\operatorname {id}}$ on W and $\Phi _j\to \varphi _1$ on V, uniformly on compact subsets. For j large enough, the automorphism $g=\tilde f\circ \Phi _j$ satisfies the following conditions:

1. (1) $g\in B_j$ ;

2. (2) $d_{H}(\tilde f,g)<\epsilon /2$ ;

3. (3) $Z\Subset g^{m_0+n_0}(U)$ ;

4. (4) $ W^s_{g}(\eta (g))\cap Z\neq \varnothing $ .

Hence, $U_n$ intersects the stable manifold $W^s_{g}(\eta (g))$ .

Proof. By Theorem 4.1, $J_f^+ =\overline {W_f^s(q)}=\overline {W_{f^k}^s(q)}$ and $J_f^- =\overline {W_f^u(q)}=\overline {W_{f^k}^u(q)}$ . Let $D_s$ be a polydisc transverse to $W_{f^k}^s(q)$ contained in U and let $D_u$ be a polydisc transverse to $W_{f^k}^u(q)$ contained in W. By the lambda lemma applied to the map $f^k$ , there is a point in $D_s\subset U$ whose $f^k$ -orbit enters V and subsequently lands in $D_u\subset W$ .

Proof. (a) By Theorem 4.1, f has a saddle fixed point q such that $J_f^+$ is the closure of the stable manifold $W_f^s(q)$ , which is connected.

(b) Let B be the basin of attraction of an attracting cycle of f. (By Remark 4.3, there are infinitely many such basins.) Let k denote the length of the cycle. Let $B_0$ be a connected component of B. Then, $B_0$ is the basin of attraction with respect to $f^k$ of a point p in the cycle. We have $p\in J_f^-$ . Let U be a neighbourhood of a point in $J_f^+$ . By Corollary 4.6, $f^{jk}(U)$ intersects $B_0$ for some $j\geq 0$ , so U intersects $B_0$ . This shows that $J_f^+\subset \partial B_0$ . The opposite inclusion is evident.

(c) Being the boundary of an open set by item (b), $J_f^+$ has empty interior. It follows that $J_f$ has empty interior too.

(d) By Remark 4.3, f has infinitely many attracting fixed points. Each basin of attraction of f is a connected component of $F_f^+ = X\setminus J_f^+$ .

(e) Let W be a connected component of $F_f^+={\operatorname {rne}}(f)$ . The characterization of pseudoconvex domains in ${\mathbb {C}}^n$ in terms of Hartogs figures extends to Stein manifolds. Let H be a Hartogs figure in a polydisc P of dimension $\dim X$ and let $\phi :P\to X$ be a holomorphic map with $\phi (H)\subset W$ . The maximum principle shows that $\phi (P)\subset {\operatorname {rne}}(f)$ , so by connectedness, $\phi (P)\subset W$ .

Proof. Corollary 4.6 yields $J_f\subset \Omega _f$ . Conversely, let x be a non-wandering point of an automorphism f of X outside $J_f$ . For generic f, by Corollary 4.4, x lies in ${\operatorname {rne}}(f)$ or ${\operatorname {rne}}(f^{-1})$ . If $x\in {\operatorname {rne}}(f)$ , then $x\in {\operatorname {att}}(f)$ , and if $x\in {\operatorname {rne}}(f^{-1})$ , then $x\in {\operatorname {att}}(f^{-1})={\operatorname {rep}}(f)$ by [Reference Arosio and Lárusson3, Step 1 of the proof of Theorem 2].

Proof. By Theorem 4.8, $\Omega _f = J_f \cup {\operatorname {att}}(f) \cup {\operatorname {rep}}(f)$ . By Corollary 4.7, $J_f$ has empty interior. Finally, ${\operatorname {att}}(f)\cup {\operatorname {rep}}(f)$ is discrete.

Proof. It is clear that a point in the proper basin of an attracting or repelling cycle is not chain-recurrent. One cannot escape from an attracting cycle along an $\epsilon $ -pseudo-orbit for f if $\epsilon $ is small enough. Likewise, one cannot reach a repelling cycle along an $\epsilon $ -pseudo-orbit for f if $\epsilon $ is small enough. Thus, each attracting cycle and each repelling cycle is a chain-recurrence class of its own.

By Remark 1.2(3),

$$ \begin{align*} X=J_f^+\sqcup {\operatorname{nrc}}(f)\sqcup {\operatorname{bas}}(f)=J_f^-\sqcup {\operatorname{nrc}}(f^{-1})\sqcup {\operatorname{bas}}(f^{-1}),\end{align*} $$

so

$$ \begin{align*} X\setminus ({\operatorname{bas}}(f)\cup {\operatorname{bas}}(f^{-1}))= \big(J_f^+ \cup {\operatorname{nrc}}(f)\big) \cap \big(J_f^- \cup {\operatorname{nrc}}(f^{-1})\big). \end{align*} $$

It follows immediately from Corollary 4.6 that $J_f$ lies in a single chain-recurrence class.

Now take $p\in {\operatorname {nrc}}(f) \cap J_f^-$ . The forward orbit of p is relatively compact because ${\operatorname {nrc}}(f)\subset F_f^+={\operatorname {rne}}(f)$ , so p has an $\omega $ -limit point $s\in J_f^+$ . By Corollary 4.6, arbitrarily close to s is a point whose forward orbit comes arbitrarily close to a saddle fixed point and subsequently comes arbitrarily close to p. This shows that p is chain-recurrent and lies in the same chain-recurrence class as $J_f$ . The case of $p\in {\operatorname {nrc}}(f^{-1}) \cap J_f^+$ is analogous.

Finally, suppose that $p\in {\operatorname {nrc}}(f)\cap {\operatorname {nrc}}(f^{-1})$ . Then, p has an $\alpha $ -limit point $r\in J_f^-$ and an $\omega $ -limit point $s\in J_f^+$ . By Corollary 4.6, arbitrarily close to s is a point whose forward orbit comes arbitrarily close to a saddle fixed point and subsequently comes arbitrarily close to r. Thus, p is chain-recurrent and lies in the same chain-recurrence class as $J_f$ .

Proof. To see that items (v) and (vi) are equivalent, note that given ${\operatorname {nrc}}(f^{-1}) \subset {\operatorname {bas}}(f)$ , ${\operatorname {nrc}}(f)\subset {\operatorname {bas}}(f^{-1})$ is equivalent to $J_f^- \subset J_f^+\cup {\operatorname {bas}}(f)$ , that is, $J_f^-\setminus J_f^+\subset {\operatorname {bas}}(f)$ .

Proof. We may assume that every automorphism close enough to f on K lies in W. Choose a holomorphically convex compact set L such that $p\in L^\circ $ and $f(K)\subset L^\circ $ , so that $g(K)\subset L$ for every automorphism g close enough to f on K.

By Remark 6.1, there are $g\in {\operatorname {Aut}}\, X$ and $q\in X$ arbitrarily close to f and p, respectively, such that the forward g-orbit of q is not contained in L, say $g^n(q)\notin L$ , with $n\geq 1$ as small as possible, and the backward g-orbit of q is not contained in L, say $g^{-m}(q)\notin L$ , with $m\geq 1$ as small as possible.

By [Reference Varolin18, Theorem 2], there is $\phi \in {\operatorname {Aut}}\, X$ as close to the identity as we wish on L, such that $\phi $ fixes $g^{-m}(q),\ldots , g^{n-1}(q)$ and $\phi (g^n(q))=g^{-m}(q)$ . Furthermore, we may take $\phi $ to have any invertible derivative at $g^n(q)$ . Then, $\phi \circ g$ has q as a periodic point and $\phi \circ g$ is as close to g as we wish on $g^{-1}(L)\supset K$ . Since X has the density property, it has automorphisms arbitrarily close to the identity that interchange p and any sufficiently nearby point. We conjugate $\phi \circ g$ by such an automorphism to obtain the desired automorphism h.

Proof. Let $\{U_n:n\geq 1\}$ be a countable basis for the topology of X. Let $S_n$ be the open set of all $f\in {\operatorname {Aut}}\,X$ such that f has an attracting cycle intersecting $U_n$ . Then, $G=\bigcap {\operatorname {Aut}} X\setminus \partial S_n$ is a residual subset of ${\operatorname {Aut}}\,X$ . We will show that if $f\in G$ and $p\in {\operatorname {tam}}(f)\setminus ({\operatorname {rne}}(f)\cup {\operatorname {rne}}(f^{-1}))$ , then $p\in \overline {{\operatorname {att}}(f)}$ .

Let $f\in G$ . If we show that for all n such that $p\in U_n$ we have $f\in S_n$ , it immediately follows that $p\in \overline {{\operatorname {att}}(f)}$ . By definition of G, it suffices to show that $f\in \overline S_n$ , which is immediate from Lemma 6.2. The proofs for ${\operatorname {rep}}(f)$ and ${\operatorname {sad}}(f)$ are analogous.

Proof. (a) That $J_f^*$ is not compact for generic f is immediate from Remark 4.3.

(b) By Lemma 6.3, $J_f\cap {\operatorname {tam}}(f) \subset \overline {{\operatorname {sad}}(f)}$ . Conversely, we clearly have $\overline {{\operatorname {sad}}(f)}\subset J_f$ . We will sketch a proof that $\overline {{\operatorname {sad}}(f)}\subset {\operatorname {tam}}(f)$ . Let $x\in \overline {{\operatorname {sad}}(f)}$ and take $q\in {\operatorname {sad}}(f)$ close to x. Choose $y\in W_f^s(q)$ and $z\in W_f^u(q)$ outside a big compact set. By the lambda lemma, there are points $y'$ close to y, $x'$ close to q and $z'$ close to z such that the forward f-orbit of $y'$ contains $x'$ and $z'$ (in this order). With such points $x'$ , we can form a sequence $(x_n)$ with $x_n\to x$ such that there are integers $j_n, k_n\geq 0$ with $f^{j_n}(x_n)\to \infty _X$ and $f^{-k_n}(x_n)\to \infty _X$ . This shows that $x\in {\operatorname {tam}}(f)$ (see Remark 6.1).

(c) Lemma 6.3 together with condition (b) imply that $J_f^* \subset \overline {{\operatorname {att}}(f)} \cap \overline {{\operatorname {rep}}(f)}$ .

(d) Take a countable basis $\{U_n : n\geq 1\}$ for the topology of X. Let $S_{m,n}$ be the open subset of ${\operatorname {Aut}}\,X$ of automorphisms with a saddle cycle through $U_m$ and $U_n$ . Then, $G=\bigcap {\operatorname {Aut}}\,X\setminus \partial S_{m,n}$ is a residual subset of ${\operatorname {Aut}}\,X$ and we claim that every $f\in G$ is chaotic on $J_f^*$ . So let $f\in G$ , and take $U_m$ and $U_n$ both intersecting $J_f^*$ . We need $f\in S_{m,n}$ , that is, we need a saddle cycle for f through $U_m$ and $U_n$ (notice that the saddle cycle is necessarily contained in $J_f^*$ ). By the definition of G, it suffices to show that $f\in \overline S_{m,n}$ , which follows as in [Reference Arosio and Lárusson2, §5].

(e) By Theorem 3.1, every periodic point of f is hyperbolic. Let $\Lambda $ be a forward invariant subset of X on which f is chaotic. Then, $\Lambda $ lies in the closure of the set of periodic points of f. If there is an attracting cycle in $\Lambda $ , then clearly f cannot be chaotic on $\Lambda $ . Since f is invertible, the same is true if there is a repelling cycle in $\Lambda $ . Thus, $\Lambda \subset \overline {{\operatorname {sad}}(f)}$ .

(f) If $J_f^*=\overline {{\operatorname {sad}}(f)}$ had an isolated point, it would be a saddle point. Being non-compact, $J_f^*$ could not consist of the cycle of that saddle point alone. However, then f would not be chaotic on $J_f^*$ .

(g) Since $J_f$ and $J_f^*$ are completely invariant, so is $M_f$ . Attracting and repelling periodic points lie outside $J_f$ and saddle periodic points lie in $J_f^*$ , so none lie in $M_f$ by Theorem 3.1. Suppose that the total orbit of $x\in J_f$ is not relatively compact. Say the forward orbit of x is not relatively compact. Let U be a neighbourhood of x and $K\subset X$ be compact. We will show that there is $y\in U$ such that neither the forward nor the backward orbit of y is contained in K. This implies that x is tame. Find $m\geq 1$ such that $f^m(x)\notin K$ and choose a neighbourhood $V\subset U$ of x such that $f^m(V)\cap K=\varnothing $ . Since $x\notin F_f^-$ , by Montel’s theorem, there is $y\in V$ whose backward orbit is not contained in K; neither is the forward orbit of y.

Proof. Let x and y be saddle periodic points, take $p\in W_f^s(x)\cap W_f^u(y)$ and let $K\subset X$ be compact. We show that there are orbits of f that come arbitrarily close to p and leave K in forward and backward time. (The definition of tameness would allow us to perturb f, but it is not necessary.) By passing to an iterate of f, we can assume that x and y are fixed.

Take $z\in W_f^s(y)\setminus K$ and a small polydisc $D_1$ transverse to $W_f^s(y)$ centred at z. As a polydisc $D_2$ transverse to $W_f^u(y)$ centred at p, choose simply a small portion of $ W_f^s(x)$ around p. By the lambda lemma, there is an orbit starting in $D_1$ and ending in a point $p' \in D_2 \subset W_f^s(x)$ . Choose a small polydisc $D_3$ transverse to $W_f^s(x)$ centred at $p'$ . Choose also $w\in W_f^u(x)\setminus K$ and a small polydisc $D_4$ transverse to $W_f^u(x)$ centred at w. By the lambda lemma, there is an orbit starting at a point $p"\in D_3$ and ending in $D_4$ . If $D_3$ is small enough, $p"$ will be close enough to $p'$ that the backward orbit of $p"$ leaves K.

Proof. We prove the theorem for a saddle fixed point. The case of a periodic point is similar. Let d be a distance inducing the topology on X. Find $\epsilon>0$ and a compact subset $K\subset X$ such that

$$ \begin{align*} \{g\in \mathrm{Aut}(X): d_K(f,g)<\epsilon\}\subset W. \end{align*} $$

We may assume that the local stable and unstable manifolds $\Gamma ^s_f(p,r)$ and $\Gamma ^u_f(p,r)$ are contained in a coordinate ball centred at p. Let $r>0$ be given by Lemma 2.2. By enlarging K if necessary, we may assume that K is holomorphically convex and contains the polydisc $\overline {\Delta ^n(0,r)}$ in its interior.

Let $x_0 \in W_f^s(p)\setminus K$ such that $f^n(x_0) \in K$ for all $n\geq 1$ . Let $y_0\in W_f^u(p)\setminus K$ such that $f^{-n}(y_0)\in K$ for all $n\geq 1$ . Find $n_s\geq 1$ such that $f^{n_s}(x_0)\in \Gamma ^s_f(p,r)$ and $n_u\geq 1$ such that $f^{-n_u}(y_0)\in \Gamma ^u_f(p,r)$ .

Let U be a Runge neighbourhood of K containing neither $x_0$ nor $y_0$ . Let V be a neighbourhood of $y_0$ and let $\varphi : [0,1]\times V\to X$ be a $C^1$ isotopy such that for all $t\in [0,1]$ :

1. (1) $\varphi _t : V\to X$ is holomorphic and injective;

2. (2) $\varphi _t(V)$ is disjoint from U;

3. (3) $U\cup \varphi _t(V)$ is Runge;

4. (4) $\varphi _0$ is the inclusion of V into X;

5. (5) $\varphi _1(y_0)=x_0$ and $d_{y_0}\varphi _1(T_{y_0}W_f^u(p))$ is transverse to $T_{x_0}W_f^s(p).$

By the Andersén–Lempert theorem [Reference Forstnerič10, Theorem 4.10.5], there is a sequence $(\Phi _j)$ in ${\operatorname {Aut}}\,X$ such that $\Phi _j\to {\operatorname {id}}$ on U and $\Phi _j\to \varphi _1$ on V, uniformly on compact subsets. Let $ f_j:= f\circ \Phi _j$ .

Let $\psi :D\to \Gamma ^u_f(p,r)$ be a holomorphic graph parametrization of the local unstable manifold $\Gamma ^u_f(p,r)$ near $f^{-n_u}(y_0)$ , defined on a small polydisc D. If D is small enough, then $(f^{n_u}\circ \psi )(D)$ is a holomorphically embedded piece of the unstable manifold $W^u_f(p)$ containing $y_0$ and contained in the neighbourhood V. Then, $(\varphi _1\circ f^{n_u}\circ \psi )(D)$ is an embedded complex submanifold which intersects the stable manifold $W^u_f(p)$ transversally at $x_0$ . Finally, $(f^{n_s}\circ \varphi _1\circ f^{n_u}\circ \psi )(D)$ is an embedded complex submanifold which intersects the local stable manifold $\Gamma ^s_f(p,r)$ transversally at $f^{n_s}(x_0)$ .

For large j, let $p_j:=\eta (f_j)$ be the saddle fixed point near p given by Lemma 2.1. Then, by Lemma 2.2, there is a holomorphic graph parametrization $\psi _j:D\to \Gamma ^u_{ f_j}(p_j,r)$ which converges uniformly to $\psi $ as $j\to \infty $ . It follows that the holomorphic map $ f_j^{n_s+n_u}\circ \psi _j$ converges uniformly to $f^{n_s}\circ \varphi _1\circ f^{n_u}\circ \psi $ , so if j is large enough, the embedded complex submanifold $ (f_j^{n_s+n_u}\circ \psi )(D)$ intersects the local stable manifold $\Gamma ^s_{f_j}(p_j,r)$ transversally in a homoclinic point for $f_j$ . Conjugation by a small perturbation of the identity makes the point $p_j$ coincide with p.

Proof. Let $\{U_n:n\geq 1\}$ be a countable basis for the topology on X. Let $S_n\subset {\operatorname {Aut}}\,X$ be the set of automorphisms that have a saddle periodic point in $U_n$ whose stable and unstable manifolds intersect transversely. By Lemma 2.3, $S_n$ is open, so the subset $G:=\bigcap {\operatorname {Aut}}\,X \setminus \partial S_n$ of ${\operatorname {Aut}}\,X$ is residual.

Let $f\in G$ and take $n\geq 0$ with $U_n\cap J_f^*\neq \varnothing $ . By the definition of $J_f^*$ , f has a saddle periodic point in $U_n$ , so by Theorem 7.3, $f\in \overline S_n$ . Also, by the definition of G, $f\not \in \partial S_n$ , so $f\in S_n$ . Let $p\in U_n$ be a saddle periodic point of f with a transverse homoclinic point q. Then, if $m\geq 0$ is large enough, $f^m(q)$ is also a transverse homoclinic point and is contained in $U_n$ .

Proof. It is easy to see that properties (i) and (ii) are equivalent and that properties (iii)–(vi) are equivalent. Also (ii) implies (iii). It remains to show that (v) implies (ii). Since $J_f$ is not compact, there is $y\in J_f\setminus K$ . Let $x\in J_f$ . Since f is chaotic on $J_f$ , there is a cycle visiting both an arbitrary neighbourhood of x and an arbitrary neighbourhood of y.