Global Diffusion on a Tight Three-Sphere

Abstract

We consider an integrable Hamiltonian system weakly coupled with a pendulum-type system. For each energy level within some range, the uncoupled system is assumed to possess a normally hyperbolic invariant manifold diffeomorphic to a three-sphere, which bounds a strictly convex domain, and whose stable and unstable invariant manifolds coincide. The Hamiltonian flow on the three-sphere is equivalent to the Reeb flow for the induced contact form. The strict convexity condition implies that the contact structure on the three-sphere is tight. When a small, generic coupling is added to the system, the normally hyperbolic invariant manifold is preserved as a three-sphere, and the stable and unstable manifolds split, yielding transverse intersections. We show that there exist trajectories that follow any prescribed collection of invariant tori and Aubry–Mather sets within some global section of the flow restricted to the three-sphere. In this sense, we say that the perturbed system exhibits global diffusion on the tight three-sphere.

Appendices

Appendix A: Background on Symplectic Dynamics

1.1 A.1 Contact Geometry

Given a compact, connected, oriented, $3$-dimensional manifold $M$, a contact form $\lambda$ on $M$ is a $1$-form on $M$ such that $\lambda \wedge d\lambda$ is a volume form on $M$. The contact structure associated to $\lambda$ is the plane bundle in $TM$ given by $\xi =\text {ker}(\lambda )=\{(x,h)\in TM\,|\,\lambda (x)(h)=0\}$. The restriction $d\lambda _{\mid \xi \oplus \xi }$ defines a symplectic structure on each fiber of $\xi \rightarrow M$. The characteristic distribution of $M$ is the $1$-dimensional distribution

$$\begin{aligned} L=\{(x,h)\in TM\,|\,x\in M,\, \omega (h,k)=0\text { for all } k\in T_xM\}. \end{aligned}$$

The corresponding $1$-dimensional foliation is called the characteristic foliation.

The Reeb vector field $X$ associate to $\lambda$ is the vector field on $M$ uniquely defined by $i_{X}(d\lambda )=0$ and $i_{X}(\lambda )=1$. The Reeb vector field $X$ spans the characteristic distribution $L$ which has the canonical section $X$. The flow lines of $X$ are contained in the leaves of the characteristic foliation. We have that $TM$ naturally splits as

$$\begin{aligned} TM=L\oplus \xi ={\mathbb {R}}X\oplus \xi . \end{aligned}$$

A contact structure $\lambda$ is said to be tight provided that there are no overtwisted disks in $M$, that is, there is no embedded disk $D\subseteq M$ such that $T\partial D\subseteq \xi$ and $T_pD\ne \xi _p$ for all $p\in \partial D$. As an example, the $1$-form on ${\mathbb {R}}^4$ of coordinates $(q_1,p_1,q_2,p_2)$,

$$\begin{aligned} \lambda _0=\frac{1}{2}(q_1dp_1-p_1dq_1+q_2dp_2-p_2dq_2), \end{aligned}$$

gives a tight contact form on the $3$-dimensional sphere $S^3\subseteq {\mathbb {R}}^4$ when restricted to $S^3$. By a theorem of Eliashberg [20], every tight contact form on $S^3$ is diffeomorphic to $g\lambda _0$, for some $C^1$-differentiable $g:S^3\rightarrow {\mathbb {R}}{\setminus }\{0\}$. The sphere $S^3$ equipped with this distinguished contact structure is called the tight three-sphere.

If we denote by $\phi ^t$ the flow of $X$, we have that $(\phi ^t)^*\lambda =\lambda$, and $(D\phi ^t)_m:\xi _m\rightarrow \xi _{\phi ^t(m)}$ is symplectic with respect to $d\lambda$.

1.2 A.2 The Conley–Zehnder Index

To each contractible $T$-periodic solution $x(t)$ of the Reeb vector field, there is assigned the so called Conley–Zehnder index. The Conley–Zehnder index generalizes the usual Morse index for closed geodesics on a Riemannian manifold. Roughly speaking, the index measures how much neighboring trajectories of the same energy wind around the orbit.

We now recall the definition of the index. Assume that $x$ is a $T$-periodic solution, which is contractible. The derivative map $D\phi ^t:T_{x(0)}M\rightarrow T_{x(t)}M$ maps the contact plane $\xi _{x(0)}$ to $\xi _{x(t)}$ and is symplectic with respect to $d\lambda$. We assume that $x(t)$ is non-degenerate, meaning that $D\phi ^T$ does not contain $1$ in the spectrum. Choose a smooth disk $u:D\rightarrow M$ s.t. $u(e^{2i\pi t/T})=x(t)$, where $D=\{z\in \mathbb {C}\,|\,|z|\le 1\}$. Then choose a symplectic trivialization $\beta : u^*\xi \rightarrow D\times {\mathbb {R}}^2$. We associate to $x(t)$ an arc of symplectic matrices $\Phi :[0,T]\rightarrow Sp(1)$, where $Sp(1)=\{A\in GL(2)\,|\, A^TJA=J\}$, by

$$\begin{aligned} \Phi (t)= \beta (e^{2i\pi t/T})\circ (D\phi ^t)_{\mid \xi _{x(t)}}\circ \beta (1)^{-1}. \end{aligned}$$

The arc starts at the identity $\Phi (0) = \text {Id}$ and ends at $\Phi (T)$, with $\det (\Phi (T)-I)\ne 0$, due to the non-degeneracy condition. Take $z\in \mathbb {C}$ and compute the winding number of $\Phi (t)z$,

$$\begin{aligned} \Delta (z)=\theta (T)-\theta (0)\in {\mathbb {R}}, \end{aligned}$$

where $\theta (t)$ is a continuous argument of $\Phi (t)z$, i.e., $\Phi (t)z=r(t)e^{2\pi i \theta (t)}$. Then define the winding interval of the arc $\Phi (t)$ by

$$\begin{aligned} I(\Phi )=\{\Delta (z)\,|\,z\in \mathbb {C}{\setminus } \{0\}\}. \end{aligned}$$

Equivalently, we can put $\Phi (t)e^{2\pi i s}=r(t,s)e^{2\pi i\theta (t,s)}$ for all $s\in [0,1]$, where $\theta (0,s)=s$, and define $\Delta (s)=\theta (T,s)-\theta (0,s)=\theta (T,s)-s$, and $I(\Phi )=\{\Delta (s)\,|\,s\in [0,1]\}$. The length of the winding interval is less than $1/2$. Then the winding interval either lies between two consecutive integers or contains precisely one integer.

We define

$$\begin{aligned} \mu (\Phi )=\left\{ \begin{array}{l@{\quad }l@{\quad }l} 2k, &{} \hbox {if }&{}k\in I(\Phi ); \\ 2k+1, &{} \hbox {if }&{} I(\phi )\subset (k,k+1). \end{array} \right. \end{aligned}$$

Then we define the Conley–Zehnder index $\mu (x,T, [u])$ of $(x,T)$ by $\mu (x,T, [u]) =\mu (\Phi )$. It depends on $x$ and $T$ and on the homotopy class of the choice of the disk map $u:D\rightarrow M$ satisfying $u(e^{2\pi it/T})=x(t)$. In the case when $\pi _2(M)=0$ (e.g., if $M=S^3$), the index is independent of the choice of the disk map $u$.

1.3 A.3 Existence of Global Surfaces of Section

Consider ${\mathbb {R}}^4=\{x=(q_1,p_1,q_2,p_2) \,|\, q_1,p_1,q_2,p_2\in {\mathbb {R}}\}$ endowed with the standard symplectic form $\omega =\sum _{i=1}^{2}dq_j\wedge dp_j$, and $H:{\mathbb {R}}^4\rightarrow {\mathbb {R}}$ a $C^r$-differentiable Hamiltonian function. If $c\in {\mathbb {R}}$ is a regular value for $H$, then $S_c=\{x\,|\, H(x)=c\}$ is a $3$-dimensional manifold invariant under the Hamiltonian flow of $H$. Assume that $S_c$ is compact and connected.

The manifold $S_c$ is said to bound a strictly convex domain provided that there exists $\delta >0$ such that $D^2H(x)- \delta \cdot \text {id}$ is positive definite for all $x\in {\mathbb {R}}^4$. This is equivalent with the conditions that $W=\{x\,|\, H(x)\le c\}$ is bounded, and $D^2H(x)(h,h)>0$ for each $x\in W$ and each non-zero vector $h$.

An energy manifold $S_c$ of the Hamiltonian $H$ is said to be of contact type if there exists a one-form $\lambda$ on $S_c$ such that $d\lambda =-j^*\omega$ and $i_{X_H}(\lambda )\ne 0$ hold on $S_c$, where $j:S_c\rightarrow {\mathbb {R}}^4$ is the inclusion map.

Assume now that $S_c$ is diffeomorphic to $S^3$, that it is of contact type, and that the contact structure is tight. The manifold $S_c$ is said to be dynamically convex if for every periodic solution $(x,T)$ of the Reeb vector field, we have $\tilde{\mu }(x,T)\ge 3$.

If $S_c$ is equipped with the contact form ${\lambda _0}_{\mid S_c}$, encloses $0_{{\mathbb {R}}^4}$ and is strictly convex, then it is dynamically convex. The converse is not true.

The following result provides sufficient conditions for the existence of a disk-like surface of section. Given a closed $3$-dimensional manifold and a flow $\phi$ with no rest points, we say that a topologically embedded $2$-dimensional disk ${\mathcal {D}}$ is a disk-like global surface of section provided that: (i) the boundary $\text {bd}({\mathcal {D}})$ is a periodic orbit (called spanning orbit), (ii) the interior of the disk $\text {int}({\mathcal {D}})$ is a smooth manifold transverse to the flow, and (iii) every orbit, other than the spanning orbit, intersects $\text {int}({\mathcal {D}})$ in forward and backward time.

Theorem A.1

[29] Assume that $S_c$ is diffeomorphic to $S^3$, is equipped with a tight contact structure, and is dynamically convex. Then there exits a global disk-like surface of section ${\mathcal {D}}$ and an associated global return map $f:\text {int}({\mathcal {D}})\rightarrow \text {int}({\mathcal {D}})$ that is smoothly conjugated to an area preserving mapping of the open unit disk in ${\mathbb {R}}^2$. The spanning orbit $\chi$ of prime period $T$ has Conley–Zehnder index $\tilde{\mu }(\chi ,T)=3$.

We note that a generalization of this result to non-dynamically convex tight contact forms on the three-sphere appears in [31].

Appendix B: Background on the Scattering Map

Consider a flow $\Phi : M\times {\mathbb {R}}\rightarrow M$ defined on a manifold $M$ that possesses a normally hyperbolic invariant manifold $\Lambda \subseteq M$.

As the stable and unstable manifolds of $\Lambda$ are foliated by stable and unstable manifolds of points, respectively, for each $x\in W^u(\Lambda )$ there exists a unique $x^u\in \Lambda$ such that $x\in W^u(x^u)$, and for each $x\in W^s(\Lambda )$ there exists a unique $x^s\in \Lambda$ such that $x\in W^s(x^s)$. We define the wave maps $\Omega ^s:W^s(\Lambda )\rightarrow \Lambda$ by $\Omega ^s(x)=x^u$, and $\Omega ^u:W^u(\Lambda )\rightarrow \Lambda$ by $\Omega ^u(x)=x^s$. The maps $\Omega ^s$ and $\Omega ^u$ are $C^{\ell }$-smooth.

We now describe the scattering map, following [17]. Assume that $W^u(\Lambda )$ has a transverse intersection with $W^s(\Lambda )$ along a $l$-dimensional homoclinic manifold $\Gamma$. The manifold $\Gamma$ consists of a $(l-1)$-dimensional family of trajectories asymptotic to $\Lambda$ in both forward and backwards time. The transverse intersection of the hyperbolic invariant manifolds along $\Gamma$ means that $\Gamma \subseteq W^u(\Lambda ) \cap W^s(\Lambda )$ and, for each $x\in \Gamma$, we have

$$\begin{aligned} T_xM= & {} T_xW^u(\Lambda )+T_xW^s(\Lambda ),\nonumber \\ T_x\Gamma= & {} T_xW^u(\Lambda )\cap T_xW^s(\Lambda ). \end{aligned}$$

(B.1)

Let us assume the additional condition that for each $x\in \Gamma$ we have

$$\begin{aligned} T_xW^s(\Lambda )= & {} T_xW^s(x^s)\oplus T_x(\Gamma ),\nonumber \\ T_xW^u(\Lambda )= & {} T_xW^u(x^u)\oplus T_x(\Gamma ), \end{aligned}$$

(B.2)

where $x^u,x^s$ are the uniquely defined points in $\Lambda$ corresponding to $x$.

The restrictions $\Omega ^s_\Gamma ,\Omega ^u_\Gamma$ of $\Omega ^s,\Omega ^u$, respectively, to $\Gamma$ are local $C^{\ell -1}$ – diffeomorphisms. By restricting $\Gamma$ even further, if necessary, we can ensure that $\Omega ^s_\Gamma ,\Omega ^u_\Gamma$ are $C^{\ell -1}$-diffeomorphisms. A homoclinic manifold $\Gamma$ for which the corresponding restrictions of the wave maps are $C^{\ell -1}$-diffeomorphisms will be referred as a homoclinic channel.

Definition B.1

Given a homoclinic channel $\Gamma$, the scattering map associated to $\Gamma$ is the $C^{\ell -1}$-diffeomorphism $S^\Gamma =\Omega _\Gamma ^s\circ (\Omega _\Gamma ^u)^{-1}$ defined on the open subset $U^u:=\Omega _\Gamma ^u(\Gamma )$ in $\Lambda$ to the open subset $U^s:=\Omega _\Gamma ^s(\Gamma )$ in $\Lambda$.

Proposition B.2

Assume that $T_1$ and $T_2$ are two invariant submanifolds of complementary dimensions in $\Lambda$. Then $W^u(T_1)$ has a transverse intersection with $W^s(T_2)$ in $M$ if and only if $S(T_1)$ has a transverse intersection with $T_2$ in $\Lambda$.

Appendix C: Linearization of Normally Hyperbolic Flows

Let $M$ be a $C^r$-smooth, $m$-dimensional manifold (without boundary), with $r\ge 1$, and $\phi :M\times {\mathbb {R}}\rightarrow M$ a $C^r$-smooth flow on $M$. A submanifold (possibly with boundary) $\Lambda$ of $M$ is said to be a normally hyperbolic invariant manifold for $\phi ^t$ if $\Lambda$ is invariant under $\phi ^t$, there exists a splitting of the tangent bundle of $TM$ into sub-bundles

$$\begin{aligned} TM=E^u\oplus E^s\oplus T\Lambda , \end{aligned}$$

that are invariant under $D\phi ^t$ for all $t\in {\mathbb {R}}$, and there exist a constant $C>0$ and rates $0<\beta <\alpha$, such that for all $x\in \Lambda$ we have

$$\begin{aligned}&\Vert D\phi ^t(x)(v)\Vert \le Ce^{-\alpha t}\Vert v\Vert \text { for all } t\ge 0, \text { if and only if } v\in E^s_x,\\&\Vert D\phi ^t(x)(v)\Vert \le Ce^{\alpha t}\Vert v\Vert \text { for all } t\le 0, \text { if and only if }v\in E^u_x,\\&\Vert D\phi ^t(x)(v)\Vert \le Ce^{\beta |t| \Vert v\Vert } \text { for all } t\in {\mathbb {R}}, \text { if and only if }v\in T_x\Lambda . \end{aligned}$$

In [28] it is proved that in some neighborhood of $\Lambda$ the flow $\phi ^t$ is conjugate with its linearization. That is, there exists a neighborhood $\mathcal {U}$ of $\Lambda$ and a homeomorphism $h$ from $\mathcal {U}$ to some neighborhood $\mathcal {V}$ of the zero section of the normal bundle to $\Lambda$ such that

$$\begin{aligned} D\phi ^t\circ h=h\circ \phi ^t. \end{aligned}$$

The homeomorphism $h$ defines a system of coordinates $(x_c,x_s,x_u)\in \Lambda \oplus E^s\oplus E^u$. The flow written in these coordinates takes the form

$$\begin{aligned} h(\phi ^t(x))=(\phi ^t(x_c), D\phi ^t_{x_c}(x_s), D\phi ^t_{x_c}(x_u)), \end{aligned}$$

for $x\in \mathcal {U}$ and $h(x)=(x_c,x_s,x_u)\in \mathcal {V}$.

Appendix D: Correctly Aligned Windows

We follow [15, 23–25, 52].

Definition D.1

An $(m_1,m_2)$-window in an $m$-dimensional manifold $M$, where $m_1+m_2=m$, is a compact subset $R$ of $M$ together with a $C^0$-parametrization given by a homeomorphism $\rho$ from some open neighborhood $U_{R}$ of $[0,1]^{m_1}\times [0,1]^{m_2}\subseteq {\mathbb {R}}^{m_1}\times {\mathbb {R}}^{m_2}$ to an open subset of $M$, with $R=\rho ([0,1]^{m_1}\times [0,1]^{m_2})$, and with a choice of an ‘exit set’

$$\begin{aligned} R^\mathrm{ex} =\rho \left( \partial [0,1]^{m_1}\times [0,1]^{m_2} \right) \end{aligned}$$

and of an ‘entry set’

$$\begin{aligned} R^\mathrm{en} =\rho \left( [0,1]^{m_1}\times \partial [0,1]^{m_2}\right) . \end{aligned}$$

Let $f$ be a continuous map on $M$ with $f(\text {im}(\rho _1))\subseteq \text {im}(\rho _2)$. Denote $f_\rho =\rho _2^{-1}\circ f\circ \rho _1$.

Definition D.2

Let $R_1$ and $R_2$ be $(m_1,m_2)$-windows, and let $\rho _1$ and $\rho _2$ be the corresponding local parametrizations. We say that $R_1$ is correctly aligned with $R_2$ under $f$ if the following conditions are satisfied:

• (i) There exists a continuous homotopy $h:[0,1]\times U_{R_1} \rightarrow {\mathbb R}^{m_1} \times {\mathbb R}^{m_2}$, such that the following conditions hold true

  $$\begin{aligned} \begin{array}{cc} &{} h_0=f_\rho , \\ &{} h([0,1],\partial [0,1]^{m_1}\times [0,1]^{m_2}) \cap [0,1]^{m_1}\times [0,1]^{m_2} = \emptyset , \\ &{} h([0,1], [0,1]^{m_1}\times [0,1]^{m_2}) \cap [0,1]^{m_1}\times \partial [0,1]^{m_2} = \emptyset , \end{array} \end{aligned}$$

  and

• (ii) There exists $y_0\in [0,1]^{m_2}$ such that the map $A_{y_0}:{\mathbb {R}}^{m_1}\rightarrow {\mathbb {R}}^{m_1}$ defined by $A_{y_0}(x)=\pi _{m_1}\left( h_{1}(x, y_0)\right)$ satisfies

  $$\begin{aligned} A_{y_0}\left( \partial [0,1]^{m_1}\right) \subseteq \mathbb {R}^{m_1}{\setminus } [0,1]^{m_1},\\ \deg ({A_{y_0}},0)\ne 0, \end{aligned}$$

  where $\pi _{m_1}: {\mathbb {R}}^{m_1}\times {\mathbb {R}}^{m_2}\rightarrow {\mathbb {R}}^{m_1}$ is the projection onto the first component, and $\deg$ is the Brouwer degree of the map $A_{y_0}$ at $0$.

Theorem D.3

Let $\{R_i\}_{i\in {\mathbb {Z}}}$, be a collection of $(m_1,m_2)$-windows in $M$, and let $f_i$ be a collection of continuous maps on $M$. If for each $i\in {\mathbb {Z}}$, $R_i$ is correctly aligned with $R_{i+1}$ under $f_i$, then there exists a point $p\in R_0$ such that

$$\begin{aligned} (f_{i}\circ \cdots \circ f_{0})(p)\in R_{i+1}, \text { for all } i\in {\mathbb {Z}}. \end{aligned}$$

Moreover, under the above conditions, and assuming that for some $k>0$ we have $R_{i}=R_{(i\,\mathrm{mod}\, k)}$ and $f_{i}=f_{(i\,\mathrm{mod}\, k)}$ for all $i\in {\mathbb {Z}}$, then there exists a point $p$ as above that is periodic in the sense

$$\begin{aligned} (f_{k-1}\circ \cdots \circ f_{0})(p)=p. \end{aligned}$$

Assume that $f:M\rightarrow M$ is a diffeomorphism on a manifold $M$, $\Lambda \subseteq M$ is an $l$-dimensional normally hyperbolic invariant manifold, and $S:U\rightarrow V$ is a scattering map associated to some homoclinic channel $\Gamma$.

Lemma D.4

Let $\{R_i,R'_i\}_{i\in {\mathbb {Z}}}$ be a bi-infinite sequence of $l$-dimensional windows contained in $\Lambda$. Assume that the following properties hold for all $i\in {\mathbb {Z}}$:

• (i) $R_{i}\subseteq U$ and $R'_{i}\subseteq V$.

• (ii) $R_{i}$ is correctly aligned with $R'_{i+1}$ under the scattering map $S$.

• (iii) for each pair $R'_{i+1},R_{i+1}$ and for each $L>0$ there exists $L'>L$ such that $R'_{i+1}$ is correctly aligned with $R_{i+1}$ under the iterate $f_{\mid \Lambda }^{L'}$ of the restriction $f_{\mid \Lambda }$ of $f$ to $\Lambda$.

Fix any bi-infinite sequence of positive real numbers $\{\varepsilon _i\}_{i\in {\mathbb {Z}}}$. Then there exist an orbit $(f^{n}(z))_{n\in {\mathbb {Z}}}$ of some point $z\in M$, an increasing sequence of integers $(n_i)_{i\in {\mathbb {Z}}}$, and some sequences of positive integers $\{N_i\}_{i\in {\mathbb {Z}}}, \{K_i\}_{i\in {\mathbb {Z}}}, \{M_i\}_{i\in {\mathbb {Z}}}$, such that, for all $i\in {\mathbb {Z}}$:

$$\begin{aligned}&d(f^{n_i}(z),\Gamma )<\varepsilon _i,\\&d(f^{n_i+N_{i+1}}(z), f_{\mid \Lambda }^{N_{i+1}}(R'_{i+1}))<\varepsilon _{i+1},\\&d(f^{n_{i}-M_{i}}(z), f_{\mid \Lambda }^{-M_{i}}(R_{i}))<\varepsilon _{i},\\&n_{i+1}=n_i+N_{i+1}+K_{i+1}+M_{i+1}. \end{aligned}$$

Appendix E: Topological Method for the Diffusion Problem

In this section we recall the main result from [24]. Assume the following:

• (C1) $M$ is a $n$-dimensional $C^r$-differentiable Riemannian manifold, and $f:M\rightarrow M$ is a $C^r$-smooth map, for some $r\ge 2$.

• (C2) There exists a submanifold $\Lambda$ in $M$, diffeomorphic to an annulus $\Lambda \simeq \mathbb {T}^1\times [0,1]$. We assume that $f$ is normally hyperbolic to $\Lambda$ in $M$. Denote the dimensions of the stable and unstable manifolds of a point $x \in \Lambda$ by $\text {dim}(W^s(x)) = n_s$ and $\text {dim}(W^u(x)) = n_u$. Then, $n = 2 + n_s + n_u$.

• (C3) On $\Lambda$ there is a system of angle-action coordinates $(\phi ,I)$, with $\phi \in {\mathbb T}^1$ and $I \in [0,1]$. The restriction $f|_{\Lambda }$ of $f$ to $\Lambda$ is a boundary component preserving, area preserving, monotone twist map, with respect to the angle-action coordinates $(\phi ,I)$.

• (C4) The stable and unstable manifolds of $\Lambda$, $W^s(\Lambda )$ and $W^u(\Lambda )$, have a differentiably transverse intersection along a $2$-dimensional homoclinic channel $\Gamma$. We assume that the scattering map $S:U^-\rightarrow U^+$ associated to $\Gamma$ is well defined.

• (C5) There exists a bi-infinite sequence of Lipschitz primary invariant tori $\{T_i\}_{i\in {\mathbb {Z}}}$ in $\Lambda$, and a bi-infinite, increasing sequence of integers $\{i_k\}_{k\in {\mathbb {Z}}}$ with the following properties:

  • (i) Each torus $T_i$ intersects the domain $U^-$ and the range $U^+$ of the scattering map $S$ associated to $\Gamma$.

  • (ii) For each $i\in \{{i_{k}+1}, \ldots , {i_{k+1}-1}\}$, the image of $T_i\cap U^-$ under the scattering map $S$ is topologically transverse to $T_{i+1}$.

  • (iii) For each torus $T_i$ with $i\in \{{i_{k}+2}, \ldots , {i_{k+1}-1}\}$, the restriction of $f$ to $T_i$ is topologically transitive.

  • (iv) Each torus $T_i$ with $i\in \{{i_{k}+2}, \ldots , {i_{k+1}-1}\}$, can be $C^0$-approximated from both sides by other primary invariant tori from $\Lambda$.

  We will refer to a finite sequence $\{T_i\}_{i=i_k+1,\ldots , i_{k+1}}$ as above as a transition chain of tori.

• (C6) The region in $\Lambda$ between $T_{i_k}$ and $T_{i_{k}+1}$ is a BZI.

• (C7) Inside each region between $T_{i_k}$ and $T_{i_{k}+1}$ there is prescribed a finite collection of Aubry–Mather sets $\{\Sigma _{\rho ^k_1}, \Sigma _{\rho ^k_2},\ldots , \Sigma _{\rho ^k_{s_k}}\}$, where $s_k\ge 1$, and $\rho ^k_s$ denotes the rotation number of $\Sigma _{\rho ^k_s}$. These Aubry–Mather sets are assumed to be vertically ordered, relative to the $I$-coordinate on the annulus.

Instead of (C6) we can consider the following condition:

(C6$'$):

The region $\Lambda _k$ in $\Lambda$ between $T_{i_k}$ and $T_{i_{k}+1}$ contains finitely many invariant primary tori $\{\Upsilon _{h^k_1}, \ldots , \Upsilon _{h^k_{l_k}}\}$, where $l_k\ge 1$, satisfying the following properties:

(i):

Each $\Upsilon _{h^k_j}$ falls in one of the following two cases:

(a):

$\Upsilon _{h^k_j}$ is an isolated invariant primary torus.

(b):

There exists a hyperbolic periodic orbit in $\Lambda$ such that its stable and unstable manifolds coincide.

(ii):

The invariant primary tori $\{\Upsilon _{h^k_1}, \ldots , \Upsilon _{h^k_{l_k}}\}$ are vertically ordered, relative to the $I$-coordinate on the annulus.

(iii):

For each $\Upsilon _{h^k_j}$, $j=1,\ldots , l_k$, the inverse image $S^{-1}(\Upsilon _{h^k_j}\cap U^+)$ forms with $\Upsilon _{h^k_j}$ a topological disk $D_{h^k_j} \subseteq U^-$ below $\Upsilon _{h^k_j}$, such that $S(D_{h^k_j})\subseteq U^+$ is a topological disk above $\Upsilon _{h^k_j}$, which is bounded by $\Upsilon _{h^k_j}$ and $S(\Upsilon _{h^k_j}\cap U^-)$.

Theorem E.1

Let $f:M\rightarrow M$ be a $C^r$-differentiable map, and let $(T_i)_{i\in {\mathbb {Z}}}$ be a sequence of invariant primary tori in $\Lambda$, satisfying the properties (C1)–(C6), or (C1)–(C5) and (C6 $'$), from above. Then for each sequence $(\epsilon _i)_{i\in {\mathbb {Z}}}$ of positive real numbers, there exist a point $z\in M$ and a bi-infinite increasing sequence of integers $(N_i)_{i\in {\mathbb {Z}}}$ such that

$$\begin{aligned} d(f^{N_i}(z), T_{i})<\epsilon _i, \text { for all } i\in {\mathbb {Z}}. \end{aligned}$$

(E.1)

In addition, if condition (C7) is assumed, and some positive integers $\{n^k_s\}_{s=1,\ldots , s_k}$, $k\in {\mathbb {Z}}$ are given, then there exist $z\in M$ and $(N_i)_{i\in {\mathbb {Z}}}$ as in (E.1), and positive integers $\{m^k_s\}_{s=1,\ldots , s_k}$, $k\in {\mathbb {Z}}$, such that, for each $k$ and each $s\in \{1,\ldots , s_k\}$, we have

$$\begin{aligned} \pi _\phi (f^j(w^k_s))<\pi _\phi (f^j(z))<\pi _\phi (f^j(\bar{w}^k_s)), \end{aligned}$$

(E.2)

for some $w^k_s,\bar{w}^k_s\in \Sigma _{\omega ^k_s}$ and for all $j$ with

$$\begin{aligned} N_{i_k}+\sum _{t=0}^{s-1} n^k_t+\sum _{t=0}^{s-1}m^k_t\le j\le N_{i_k}+\sum _{t=0}^{s} n^k_t+\sum _{t=0}^{s-1}m^k_t. \end{aligned}$$