Non-ergodic $\mathbb{Z}$-periodic billiards and infinite translation surfaces

Abstract

We give a criterion which proves non-ergodicity for certain infinite periodic billiards and directional flows on $\mathbb{Z}$-periodic translation surfaces. Our criterion applies in particular to a billiard in an infinite band with periodically spaced vertical barriers and to the Ehrenfest wind-tree model, which is a planar billiard with a $\mathbb{Z}^{2}$-periodic array of rectangular obstacles. We prove that, in these two examples, both for a full measure set of parameters of the billiard tables and for tables with rational parameters, for almost every direction the corresponding directional billiard flow is not ergodic and has uncountably many ergodic components. As another application, we show that for any recurrent $\mathbb{Z}$-cover of a square tiled surface of genus two the directional flow is not ergodic and has no invariant sets of finite measure for a full measure set of directions. In the language of essential values, we prove that the skew-products which arise as Poincaré maps of the above systems are associated to non-regular $\mathbb{Z}$-valued cocycles for interval exchange transformations.

Appendices

Appendix A: Stable space and coboundaries

In this Appendix we include for completeness the proof of Lemma 4.3 and Theorem 4.2 (see Sect. 4.3) along the lines of [16, 17, 57] (see also [13]).

The main idea of the proof of Theorem 4.2 is to show that for every p∈M _reg,ω the ergodic integrals $\int_{0}^{t} f(\varphi_{s} p) \,d s$ are bounded uniformly in t≥0 and hence deduce that $F^{v}_{f}$ is a coboundary. We will do so (as in [17]) by decomposing the ergodic integral along a special sequence of times, given by returns to a section $\mathcal{K}$ for the Teichmüller flow.

The construction of the section $\mathcal{K}$, which will be useful in both the proof of Theorem 4.2 and Lemma 4.3, is given in Sect. A.1. Some of the properties of $\mathcal{K}$ will not be used in the proof of Lemma 4.3, but only in the proof of Theorem 4.2.

1.1 A.1 Preliminary definitions and notation

Let μ be any $\mathit{SL}(2,{\mathbb{R}})$-invariant probability Borel measure on the moduli space $\mathcal{M}^{(1)}(M)$ ergodic for the Teichmüller flow $(G_{t})_{t\in{\mathbb{R}}}$. Since the measure μ is $\mathit{SL}(2,{\mathbb{R}})$-invariant and ergodic, we can assume that it is supported on a stratum $\mathcal{H}^{(1)} =\mathcal{H}^{(1)} (k_{1}, \ldots, k_{\kappa})$ for some k ₁,…,k _κ . Let us remark that since μ is a probability measure and it is ergodic for the Teichmüller flow, there exists a $(G_{t})_{t\in{\mathbb {R}}}$-invariant set $\mathcal{H}_{0}\subset{\mathcal{H}}^{(1)}$ of μ-measure one such that each $\omega\in\mathcal{H}_{0}$ is Oseledets regular for the Kontsevich-Zorich cocycle $(G^{KZ}_{t})_{t\in{\mathbb{R}}}$ (by the Oseledets’ theorem), every $\omega\in\mathcal{H}_{0}$ has neither vertical nor horizontal saddle connections and both the vertical and horizontal flow on (M,ω) are ergodic (these last two properties are classical and follow for example from [38]).

For any $\omega\in\mathcal{H}^{(1)}$ let M _reg,ω be the set of points which are regular both for the vertical and horizontal flow on (M,ω) (that, we recall, means that both flows are defined for all times).

Remark A.1

Remark that M _reg,ω has full measure on M and is invariant under $(G_{t})_{t \in{\mathbb{R}}}$, that is, $M_{reg, G_{t}\omega} = M_{reg,\omega}$ for all $t \in {\mathbb{R}}$.

For any $\omega\in\mathcal{H}^{(1)}$ and any point p∈M∖Σ let us denote by I _ω =I _ω (p) the arc of the horizontal flow on (M,ω) of total length 1 centered at p.

Remark A.2

Since the Teichmüller flow $(G_{t})_{t \in{\mathbb{R}}}$ preserves horizontal leaves and rescales the horizontal vector fields by $X_{h}^{\omega}=e^{t}X_{h}^{G_{t}\omega}$, we have that

$$t< s \quad\Rightarrow\quad I_{G_{s}\omega} (p) \subset I_{G_{t}\omega} (p) . $$

In the rest of the Appendix we will consider ω and p such that I _ω (p) satisfy the following property:

$$\begin{aligned} &\text{$I_{\omega} = I_{\omega}(p)$ has no self-intersections, does not intersect $\varSigma$} \\ &\phantom{I_{\omega} =}{}\text{and all but finitely many points from $I_{\omega}$ return to $I_{\omega}$} \\ &\phantom{I_{\omega} =}{}\text{under the vertical flow.} \end{aligned}$$

(A.1)

We will denote by T=T _ω :I _ω →I _ω the Poincaré map of the vertical flow $(\varphi_{t})_{t\in{\mathbb {R}}}$ on (M,ω), which is well defined by (A.1) and is an IET. Let us denote by $\tau_{\omega}:I_{\omega} \to{\mathbb{R}}^{+}$ the function which assigns to each point (apart from finitely many ones) its first return time. Let us also denote by I _j (ω), j=1,…,m, the subintervals exchanged by T _ω , by λ _j (ω) their lengths and by τ _j (ω) the first return time of the interval I _j (ω) to I _ω .

Since I _ω (p) does not contain any singularity and the set of singularities is discrete, let δ(ω)=δ(ω,p)>0 be maximal such that the strip

$$ \bigcup_{0\leq t < \delta({\omega})} \varphi_t I_{\omega}(p) $$

does not contain singularities, and thus is isometric to an Euclidean rectangle of height δ(ω) and width 1 in the flat coordinates given by ω.

For each j=1,…,m let $\gamma_{j}({\omega})\in H_{1}(M,{\mathbb {Z}})$ be the homology class obtained by considering the vertical trajectory of any point q∈I _j (ω) up to the first return time to I _ω and closing it up with a horizontal geodesic segment contained in I _ω .

Remark A.3

Suppose that a pair $(\omega_{0},p_{0})\in \mathcal{H}^{(1)}\times(M\setminus\varSigma)$ satisfies (A.1). Then there exists a sufficiently small neighborhood $\mathcal{U}\subset\mathcal{H}^{(1)}$ of ω ₀ such that for any $\omega\in\mathcal{U}$

• (i) the pair (ω,p ₀) also satisfies (A.1),

• (ii) the induced IET T _ω on I _ω (p ₀) has the same number m of exchanged intervals and the same combinatorial datum,

• (iii) the quantities λ _j (ω), τ _j (ω) for j=1,…,m and δ(ω,p ₀) change continuously with $\omega\in \mathcal{U}$,

• (iv) for every 1≤j≤m the homology class γ _j (ω) does not depend on $\omega\in \mathcal{U}$.

1.2 A.2 Proof of Lemma 4.3 and auxiliary lemmas

Lemma A.4

There exists p ₀∈M, a subset $\mathcal{K}\subset \mathcal{H}_{0}$ with positive transverse measure and positive constants A,C,c>0 such that for every $\omega\in\mathcal{K}$ the pair (ω,p ₀) satisfies (A.1),

$$ \frac{1}{c} \| \rho\|_\omega \leq \max _{1\leq j\leq m} \biggl| \int_{\gamma_j (\omega)} \rho \biggr| \leq c\| \rho \|_\omega\quad\mathit{for\ every}\ \rho\in H^1(M, { \mathbb{R}}), $$

(A.2)

$$ \lambda_j(\omega) \delta(\omega) \geq A \quad \mathit{and}\quad\frac{1}{C}\leq \tau_j(\omega)\leq C\quad \mathit{for\ any}\ 1\leq j\leq m. $$

(A.3)

Moreover, every $\omega\in\mathcal{K}$ is Birkhoff generic.

Proof

Choose $\omega_{0}\in\mathcal{H}_{0}$ in the support of the measure μ and let $p_{0}\in M_{reg,\omega_{0}}$. Then the pair (ω ₀,p ₀) satisfies (A.1). Moreover, one can show that {γ _j (ω ₀),1≤j≤m} generate the homology $H_{1}(M, {\mathbb{R}})$ (the proof is analogous to the proof of Lemma 2.17, Sect. 2.9 in [53]). In particular, their Poincaré dual classes $\{ \mathcal{P}\gamma_{j}(\omega_{0}), 1\leq j \leq m\}$ generate $H^{1}(M, {\mathbb{R}})$. Thus, it follows⁶ that there exists a constant c′>0 such that

$$ \frac{1}{c'} \| \rho\|_{\omega_0} \leq\max_{1\leq j\leq m} \biggl| \int_{\gamma_j(\omega_0) } \rho\biggr| = \max_{1\leq j\leq m} \bigl|\bigl\langle\mathcal{P}\gamma_j(\omega_0), \rho \bigr\rangle\bigr| \leq{c'} \| \rho\|_{\omega_0} $$

(A.4)

for all $\rho\in H^{1}(M, {\mathbb{R}})$. In view of Remark A.3, by choosing $\mathcal{U}$ to be a small compact neighbourhood of ω ₀ in $\mathcal{H}^{(1)}$, we have

$$ \gamma_j(\omega)=\gamma_j(\omega_0)\quad\text{for any}\ \omega\in\mathcal{U}\text{ and }1\leq j \leq m $$

(A.5)

and there exist constants A>0 and C>1 such that

$$ \lambda_j(\omega) \delta(\omega) \geq A \quad\text{and}\quad \frac{1}{C}\leq\tau_j(\omega)\leq C\quad\text{for all}\ \omega\in\mathcal{U},\ 1\leq j\leq m. $$

Furthermore, since $\mathcal{U}$ is compact, there exists a constant K>0 such that for any $\omega_{1}, \omega_{2} \in\mathcal{U}$, and any $\rho\in H^{1}(M, {\mathbb{R}})$ the Hodge norms satisfy $\| \rho\|_{\omega_{1}} \leq K \| \rho\|_{\omega_{2}}$ (it follows for example from [17], Sect. 2). Thus, by (A.4) and (A.5),

$$ \frac{1}{c} \| \rho\|_\omega \leq \max_{1\leq j\leq m} \biggl| \int_{\gamma_j (\omega)} \rho\biggr| \leq c\| \rho\|_\omega\quad\text{for all}\ \omega\in \mathcal{U}, \ \rho\in H^1(M, {\mathbb{R}}), $$

(A.6)

where c:=Kc′. Since ω ₀ belongs to the support of μ, $\mu(\mathcal{U})>0$. Let $\mathcal{S} \subset \mathcal{H}^{(1)}$ be a hypersurface containing ω ₀ and transverse to $(G_{t})_{t\in{\mathbb{R}}}$ and let $\mathcal{K}\subset \mathcal{S}\cap\mathcal{U}\cap\mathcal{H}_{0}$ be a subset with positive transverse measure and compact closure such that every $\omega\in\mathcal{K}$ is Birkhoff generic. Then $\mathcal{K}$ satisfies the conclusions of the Lemma. □

Proof of Lemma 4.3

Let $\mathcal{K}$ be the section from Lemma A.4. Since $(G_{t})_{t\in{\mathbb{R}}}$ is ergodic and $\mathcal{K}$ has positive transverse measure, there exists a full μ-measure set $\mathcal{M}'\subset \mathcal{H}^{(1)}$ such that for any $\omega\in\mathcal{M}'$ there exists a sequence $\{t_{k}\}_{k\in{\mathbb{N}}}$ of positive numbers such that t _k →+∞ and $G_{t_{k}}(\omega) \in\mathcal{K}$ for each $k \in{\mathbb{N}}$. Now taking $\gamma^{(k)}_{j} := \gamma_{j} (G_{t_{k}}\omega)$ and applying Lemma A.4 to every $G_{t_{k}}\omega\in\mathcal{K}$ we get (4.4). □

Notation

For each j=1,…,m, consider the set

$$R_j(\omega):=\bigl\{\varphi_up:p\in I_j( \omega),0\leq u\leq\delta(\omega)\bigr\} $$

(where I _j (ω) and δ(ω) are defined above Remark A.3). Remark that R _j (ω) is a rectangle in (M,ω) of base λ _j (ω) and height δ(ω) in the translation structure given by ω, since by the definition of δ(ω) it is contained in the rectangle of base I _ω and height δ(ω).

Lemma A.5

Suppose that $\omega\in\mathcal{K}$. Let ρ∈Ω ¹(M) be a form vanishing on the interval I _ω =I _ω (p ₀) and set $f:=i_{X_{v}}\rho$. Let p∈I _ω (p ₀) and let τ=τ _ω (p)>0 be its first return time to I _ω (p ₀) for the vertical flow $(\varphi _{t})_{t\in{\mathbb{R}}}$. If p∈I _j (ω) then

$$ \biggl|\int_{0}^{\tau}f( \varphi_tp)\,dt \biggr|= \biggl|\int_{\gamma _j(\omega)}\rho \biggr|\leq c\|\rho \|_\omega. $$

(A.7)

Moreover, the rectangle R _j (ω) has area ν _ω (R _j (ω))≥A, where A is the constant given by Lemma A.4, and if q∈R _j (ω) then

$$ \biggl|\int_{0}^{\tau}f(\varphi_tq)\,dt\biggr|\geq \biggl|\int_{\gamma _j(\omega)}\rho\biggr|- \|i_{X_h}\rho\|_{\infty}. $$

(A.8)

Proof

Let us assume that p∈I _j (ω). Let $\widetilde{\gamma}_{j}$ be the curve $\widetilde{\gamma}_{j}:[0,\tau]\to M$ given by $\widetilde{\gamma}_{j}(s)=\varphi_{s}p$ for 0≤s≤τ. By the definition of f,

$$ \int_0^{\tau} f(\varphi_s p) \,d s = \int_{\widetilde{\gamma}_j} \rho. $$

Recall that $\gamma_{j}(\omega)\in H_{1}(M,{\mathbb{Z}})$ denotes the homology class of the loop which is obtained by closing up $\widetilde{\gamma}_{j}$ with a a horizontal segment contained in I _ω . Thus, since ρ vanishes on I _ω , we obtain

$$\int_0^{\tau} f(\varphi_s p) \,d s = \int_{\widetilde{\gamma}_j} \rho=\int_{\gamma_j(\omega)}\rho. $$

Combining this with (A.2), we have (A.7).

Next remark that, by (A.3), the area of the rectangle R _j (ω) (defined before Lemma A.5) satisfies

$$\nu_\omega\bigl(R_j(\omega)\bigr)=\lambda_j( \omega)\delta(\omega)\geq A. $$

Let q∈R _j (ω). Then q=φ _u p for some p∈I _j (ω) and 0≤u≤δ(ω). Thus, since by definition of first return time τ=τ _ω (p) we have φ _τ p=T _ω p, where T _ω is the first return map of $(\varphi_{t})_{t\in {\mathbb{R}}}$ to I _ω , we can write

$$ \int_0^{\tau} f(\varphi_s p) \,d s - \int_0^{\tau} f(\varphi_s q) \,d s = \int_0^{u} f(\varphi_s p) \,d s - \int_{0}^{u} f\bigl(\varphi_s T_\omega(p) \bigr) \,d s. $$

(A.9)

Remark now that p,T _ω (p),φ _u p,φ _u T _ω (p) are corners of a rectangle R because they are contained in the rectangle of base I _ω and height δ(ω) in the translation structure given by ω. Denote by ∂ _v R and ∂ _h R the vertical and the horizontal part of the boundary of R respectively. Then $\int_{\partial_{v}R}\rho$ is equal to the RHS of (A.9) and $\int_{\partial_{h}R}\rho$ is bounded by $\| i_{X_{h}}\rho \|_{\infty}$. Thus, since ρ is closed and R is simply connected, we have ∫ _R dρ=0 and by Stoke’s theorem $0=\int_{\partial R}\rho=\int_{\partial_{v} R}\rho+\int_{\partial_{h} R}\rho$. It follows that

$$ \biggl| \int_0^{u} f(\varphi_s p) d s - \int_{0}^{u} f\bigl(\varphi_s T_\omega(p) \bigr) d s \biggr| = \biggl| \int_{\partial_v R} \rho\biggr| = \biggl| \int_{\partial_h R} \rho\biggr| \leq\| i_{X_{h}}\rho \|_{\infty}. $$

This, combined with (A.9) and (A.7), yields (A.8). □

Remark A.6

Recall that for any real t the vertical and horizontal vector fields $X_{v}^{\omega}$ and $X_{h}^{\omega}$ on (M,ω) rescale as follows under the Teichmüller geodesic flow $(G_{t})_{t\in\mathbb{R}}$:

$$X_v^\omega=e^{-t}X_v^{G_t\omega}, \qquad X_h^\omega=e^{t}X_h^{G_t\omega}. $$

Thus, the vertical and horizontal flows satisfy:

$$\varphi^{v,\omega}_s p=\varphi^{v,G_t\omega}_{e^{-t}s}p, \qquad \varphi^{h,\omega}_s p=\varphi^{h,G_t\omega}_{e^{t}s}p. $$

Notation

For 0≤t ₀<t ₁, consider the intervals $I_{G_{t_{0}}\omega}$, $I_{G_{t_{1}}\omega}$ defined at the beginning of the section, that, by Remark A.2, satisfy $I_{G_{t_{1}}\omega}\subset I_{G_{t_{0}}\omega}$ and for every regular point $p\in I_{G_{t_{0}}\omega}$ denote respectively by $\tau _{t_{0},t_{1}}^{+}(p)\geq0$ and $\tau_{t_{0},t_{1}}^{-}(p)\geq0$ the times of the first forward and respectively backward entrance of the vertical orbit of p to $I_{G_{t_{1}}\omega}$.

Lemma A.7

Suppose that for some $\omega\in\mathcal{H}^{(1)}$ there exists 0≤t ₀<t ₁ such that $G_{t_{0}}\omega,G_{t_{1}}\omega\in\mathcal {K}$. Then the entrance times $\tau_{t_{0},t_{1}}^{+}(p),\, \tau_{t_{0},t_{1}}^{-}(p)\geq0$ of p in $I_{G_{t_{1}}\omega}$ satisfy

$$ \tau_{t_0,t_1}^{+}(p)\leq e^{t_1}C, \qquad\tau_{t_0,t_1}^{-}(p)\leq e^{t_1}C. $$

(A.10)

Let ρ∈Ω ¹(M) be a form vanishing on the interval $I_{G_{t_{0}}\omega}$ and set $f:=i_{X_{v}}\rho$. Then for every $-\tau_{t_{0},t_{1}}^{-}(p) \leq s\leq\tau_{t_{0},t_{1}}^{+}(p)$ such that $\varphi_{s}p\in I_{G_{t_{0}}\omega}$ we have

$$\biggl|\int_0^{s}f(\varphi_{ t} p)dt\biggr|\leq cC^2e^{t_1-t_0}\|\rho \|_{G_{t_0}\omega}. $$

Proof

Let us assume that s≥0. The proof for s<0 is analogous. Denote by 0=s ₀<s ₁<⋯<s _K =s the consecutive return times (to $I_{G_{t_{0}}\omega}$) of the forward vertical orbit of p. For each pair s _i−1,s _i of consecutive return times of the vertical flow $(\varphi_{t})_{t\in\mathbb{R}}$ on (M,ω) to the interval $I_{G_{t_{0}}\omega}$, it follows from Remark A.6 that $e^{-t_{0}}s_{i-1}, e^{-t_{0}}s_{i}$ are consecutive return times of the vertical flow $(\varphi^{v,{G_{t_{0}}\omega}}_{t})_{t\in \mathbb{R}}$ on $(M, G_{t_{0}}\omega)$ to $I_{G_{t_{0}}\omega}$. Thus, since the first return time function of $(\varphi^{v}_{G_{t_{0}}\omega })_{t\in\mathbb{R}}$ to $I_{G_{t_{0}}\omega}$ assumes the finitely many values $\tau_{j}(G_{t_{0}}\omega)$ for i=1,…,K (see Sect. A.1), for all 0≤i<K we have

$$ e^{-t_0}s_{i}-e^{-t_0}s_{i-1}\geq\min_{1\leq j\leq m}\tau _j(G_{t_0}\omega). $$

(A.11)

Moreover, recalling the definition of $f=i_{X^{v}_{\omega}}\rho$ and using Remark A.6, it also follows that

$$\int_{s_{i-1}}^{s_{i}}f(\varphi_{t} p)dt = \int _{s_{i-1}}^{s_{i}}i_{X_v^{\omega}}\rho(\varphi_{t}p)dt = \int_{e^{-t_0}s_{i-1}}^{e^{-t_0}s_i}i_{X_v^{G_{t_0}\omega}}\rho \bigl(\varphi^{v,G_{t_0}\omega}_{t}p\bigr)dt. $$

Thus, by Lemma A.5 applied to $G_{t_{0}}\omega\in\mathcal {K}$, we have

$$\biggl|\int_{s_{i-1}}^{s_i}f(\varphi_{ t} p)dt\biggr|\leq c\|\rho\| _{G_{t_0}\omega}, $$

for each 1≤i≤K. Therefore,

$$\biggl|\int_{s_{0}}^{s_K}f(\varphi_{ t} p)dt\biggr|\leq\sum _{i=1}^K\biggl|\int_{s_{i-1}}^{s_i}f(\varphi_{ t} p)dt\biggr|\leq K c\| \rho\|_{G_{t_0}\omega}. $$

We need to show that $K\leq C^{2}e^{t_{1}-t_{0}}$. From (A.11) we get

$$s_K\geq Ke^{t_0}\min_{1\leq j\leq m}\tau_j(G_{t_0}\omega). $$

Moreover, the orbit segment

$$\bigl\{\varphi^{v,\omega}_tp:s_0<t<s_K \bigr\}=\bigl\{\varphi^{v,G_{t_1}\omega}_tp: 0<t<e^{-t_1}s_K \bigr\} $$

does not intersect the interval $I_{G_{t_{1}}\omega}$. It follows that

$$e^{-t_1}s_{K} \leq\max_{1\leq j\leq m}\tau_j(G_{t_1}\omega). $$

Therefore,

$$K\leq\frac{e^{t_1}\max_{1\leq j\leq m}\tau_j(G_{t_1}\omega)}{e^{t_0}\min_{1\leq j\leq m}\tau_j(G_{t_0}\omega)}. $$

In view of (A.3), it follows that $K\leq e^{t_{1}-t_{0}}C^{2}$ and $s\leq e^{t_{1}}C$. □

Let us recall that to each smooth $f: M \to\mathbb{R}$ one can associate a cocycle $F^{v}_{f}$ over the flow $(\varphi_{t})_{t\in\mathbb{R}}$ that for x∈M _reg and $t \in\mathbb{R}$ is given by $F^{v}_{f}(t,x):=\int_{0}^{t} f(\varphi_{s} x)\, ds$ (see (3.2)).

Lemma A.8

If a smooth form ρ∈Ω ¹(M) is exact then the cocycle $F^{{v}}_{f}$ associated to $f=i_{X_{v}}\rho$ is a coboundary. Moreover, for every smooth form ρ∈Ω ¹(M) and any simply connected subset D⊂M there exists ρ′∈Ω ¹(M) vanishing on D and such that [ρ′]=[ρ].

Proof

If ρ∈Ω ¹(M) is exact then ρ=dh for some smooth function $h:M\to{\mathbb{R}}$. Thus

$$f=i_{X_{v}}\rho=i_{X_{v}}dh=\mathcal{L}_{X_{v}}h. $$

Therefore,

$$F^v_f(t,x) = \int_0^t f(\varphi_s x)\,ds=\int_0^t \mathcal {L}_{X_{v}}h(\varphi_s x)\,ds=h(\varphi_t x) - h(x), $$

so $F^{{v}}_{f}$ is a coboundary.

Let ρ∈Ω ¹(M) be an arbitrary form. Since D⊂M is simply connected, there exists a smooth function $h:M\to{\mathbb{R}}$ such that dh=ρ on D. Then ρ′:=ρ−dh is cohomologous to ρ and vanishes on D. □

1.3 A.3 Decomposition of ergodic integrals and proof of Theorem 4.2

Proof of Theorem 4.2

Let p ₀∈M and $\mathcal{K}$ be the point and the section given by Lemma A.4. Since $(G_{t})_{t\in{\mathbb{R}}}$ is ergodic and $\mathcal{K}$ has positive transverse measure, there exists a full μ-measure set $\mathcal{M}'\subset\mathcal{H}^{(1)}$ such that for any $\omega\in \mathcal{M}'$ there exists a sequence {t _k } _k≥0 of positive numbers such that t _k →+∞ and $G_{t_{k}}(\omega) \in\mathcal {K}$ for each k≥0. Let us show that $\mathcal{M}'$ satisfies the conclusion of the theorem.

Let us remark first that, since both the property of being Oseledets regular and having no vertical saddle connections are $(G_{t})_{t\in{\mathbb{R}}}$-invariant, any $\omega\in\mathcal{M}'$ is Oseledets regular and has no vertical saddle connections by the definition of $\mathcal{K}$.

Fix $\omega\in\mathcal{M}'$ and let t ₀ be the minimum t≥0 such that $G_{t}(\omega) \in\mathcal{K}$ and let $\{t_{k}\}_{k\in {\mathbb{N}}}$ be the sequence of successive returns to $\mathcal {K}$. Let ρ be a closed smooth form such that $[\rho]\in E_{\omega}^{-}(M,{\mathbb{R}})$. Let $(\varphi_{t})_{t\in{\mathbb{R}}}$ be the vertical flow on (M,ω) and consider the function $f= i_{X_{v}} \rho$. We want to show that the associated cocycle $F_{f}^{v}$ (whose definition is recalled before Lemma A.8) is a coboundary for $(\varphi_{t})_{t\in{\mathbb{R}}}$. In view of Lemma A.8, we can also assume that ρ vanishes on the interval $I_{G_{t_{0}}\omega}$.

Let us consider the sequence of intervals $\{I_{G_{t_{k}}\omega}\}_{k\geq0}$ centered at p ₀. By Remark A.2, $\{I_{G_{t_{k}}\omega}\}_{k\geq0}$ is a decreasing sequence of nested intervals. Fix a regular point p∈M _reg,ω . For any t>0, the trajectory Φ _t :={φ _s p:0≤s≤t} can be inductively decomposed into principal return trajectories as follows (analogously to Lemma 9.4 in [17]). Let $K \in{\mathbb{N}}$ be the maximum k≥0 such that Φ _t intersect $I_{G_{t_{k}}\omega}$. For every k=0,…,K let 0≤l _k ≤r _k ≤t be the times of the first and the last intersection of Φ _t with $I_{G_{t_{k}}\omega}$. Then, since, by Remark A.2, the intervals $\{ I_{G_{t_{k}}\omega}\}_{k}$ are nested,

$$0\leq l_0\leq l_1\leq\cdots\leq l_K\leq r_K\leq\cdots\leq r_1\leq r_0\leq t. $$

Moreover, $l_{i}-l_{i-1}=\tau_{t_{i-1},t_{i}}^{+}(\varphi_{l_{i-1}}p)$, $r_{i-1}-r_{i}=\tau_{t_{i-1},t_{i}}^{-}(\varphi_{r_{i-1}}p)$ for i=1,…,K and $r_{K}-l_{K}\leq\tau_{t_{K},t_{K+1}}^{+}(\varphi_{l_{K}}p)$, where the functions $\tau_{t_{i-1},t_{i}}^{\pm}$ are defined before Lemma A.7. By Lemma A.7, for every 1≤i≤K we have

$$ \biggl|\int_{l_{i-1}}^{l_{i}}f(\varphi_{ s}p)ds\biggr|= \biggl|\int _0^{l_{i}-l_{i-1}}f(\varphi_{ s} \varphi_{l_{i-1}}p)ds\biggr|\leq c\,C^2e^{t_i-t_{i-1}}\|\rho\|_{G_{t_{i-1}}\omega}, $$

(A.12)

$$ \biggl|\int_{r_{i}}^{r_{i-1}}f(\varphi_s p)ds\biggr|= \biggl|\int _{r_{i}-r_{i-1}}^0f(\varphi_{ s} \varphi_{r_{i-1}}p)ds\biggr|\leq c\,C^2e^{t_i-t_{i-1}}\|\rho\|_{G_{t_{i-1}}\omega} $$

(A.13)

and

$$ \biggl|\int_{l_{K}}^{r_{K}}f(\varphi_{ s}p)ds\biggr|= \biggl|\int _0^{r_{K}-l_{K}}f(\varphi_{ s} \varphi_{l_{K}}p)ds\biggr|\leq cC^2e^{t_{K+1}-t_K}\|\rho\|_{G_{t_K}\omega}. $$

(A.14)

Moreover, since $l_{0}\leq\tau_{t_{0},t_{1}}^{-}(\varphi_{l_{0}}p)$ and $t-r_{0}\leq\tau_{t_{0},t_{1}}^{+}(\varphi_{r_{0}}p)$, by (A.10), we have $l_{0},\,t-r_{0}\leq e^{t_{1}}C$. Thus

$$ \biggl|\int_{0}^{l_0}f(\varphi_{ s}p)ds\biggr|\leq e^{t_1}C\|f\|_{\infty}, \qquad\biggl|\int_{r_0}^{t}f(\varphi_{ s}p)ds\biggr|\leq e^{t_1}C\|f\|_{\infty}. $$

(A.15)

Summing (A.12)–(A.15) we get

$$ \biggl|\int_{0}^{t}f(\varphi_{ s}p)ds\biggr|\leq 2\sum_{k=0}^\infty cC^2e^{t_{K+1}-t_K}\|\rho\|_{G_{t_K}\omega}+ 2e^{t_1}C\|f\|_{\infty}. $$

(A.16)

Since, by assumption, $[\rho] \in E_{\omega}^{-}(M,{\mathbb{R}})$ (recall (4.3)), it follows that there exists constants C ₁,θ>0 such that $\|\rho\|_{G_{t_{K}}\omega} \leq C_{1} e^{-\theta t_{k}}$ for all k≥0. Using this inequality together with (A.16), we get that there exists C ₂>0 such that for any t≥0, one has

$$ \biggl| \int_0^t f(\varphi_s p) d s \biggr| \leq C_2 \sum_{k=0}^\infty e^{( t_{k+1}-t_k)} e^{-\theta t_k} +C_2 = C_2 \sum_{k=0}^\infty e^{( \frac{ t_{k+1}-t_k}{t_k} - \theta )t_k } + C_2 . $$

(A.17)

Since $\mathcal{K}$ has positive transverse measure and ω is Birkhoff generic (since Birkhoff generic points are $(G_{t})_{t\in{\mathbb{R}}}$-invariant and $G_{t_{0}}\omega\in\mathcal{K}$ which by construction consists only of Birkhoff generic points), by Birkhoff ergodic theorem we have $\lim_{k \to\infty} t_{k} /k = 1/\mu_{tr}(\mathcal{K})$, where $\mu_{tr}(\mathcal{K})>0$ is the transverse measure of $\mathcal{K}$. Thus, if k is sufficiently large, (t _k+1−t _k )/t _k −θ≤−θ/2, which shows that the above series is convergent and the ergodic integrals in (A.17) are uniformly bounded for all t≥0 and p∈M _reg,ω . By Lemma 3.7 this implies that $F^{{v}}_{f}$ is a coboundary. This concludes the proof of the first part of Theorem 4.2.

Let us now prove the second part of Theorem 4.2. Let us assume in addition from now on that μ is KZ-hyperbolic. Let $\omega\in\mathcal{M}'$, p∈M _reg,ω and, as before, let us denote by $(\varphi_{t})_{t\in {\mathbb{R}}}$ the vertical flow on (M,ω). Let ρ∈Ω ¹(M) be a smooth closed one form such that $[\rho]\notin E_{\omega}^{-}(M,{\mathbb{R}})$. Again, by Lemma A.8, we can assume that ρ vanishes on the interval $I_{G_{t_{0}}\omega}$.

For every $k\in{\mathbb{N}}$, consider the return times $\tau _{j}(G_{t_{k}}\omega)$ (see the definition in Sect. A.1) and let

$$ T_k:= e^{t_k} \tau_j(G_{t_k}\omega), $$

(A.18)

so that, by Remark A.6, T _k is the return time of a point $p \in I_{j}(G_{t_{k}}\omega)$ to $I_{G_{t_{k}}\omega}$ under the vertical flow $(\varphi_{t})_{t\in\mathbb{R}}$ for (M,ω). Since $G_{t_{k}}\omega\in\mathcal{K}$ for $k\in{\mathbb{N}}$, by Lemma A.5, for every $k\in{\mathbb{N}}$ and j=1,…,m there exists a rectangle $R_{j}(G_{t_{k}}\omega)$ in (M,ω) such that $\nu_{\omega}(R_{j}(G_{t_{k}}\omega))=\nu_{G_{t_{k}}\omega }(R_{j}(G_{t_{k}}\omega))\geq A$ and

$$ \biggl|\int_{0}^{T_k}f(\varphi_tq)\,dt\biggr|\geq \biggl|\int_{\gamma _j(G_{t_k}\omega)}\rho\biggr|- e^{-t_k}\|i_{X_h}\rho\|_{\infty}\quad\text{for}\ q\in R_j(G_{t_k}\omega). $$

(A.19)

Let us now prove that the cocycle $F^{v}_{f}$ is not a coboundary. Assume by contradiction that $F^{v}_{f}$ is a coboundary with a measurable transfer function $u: M \to{\mathbb{R}}$. Then there exists a constant B>0, depending on the constant A given by Lemma A.4, such that the set

$$\varLambda_B := \bigl\{ q \in M : \bigl|u(q)\bigr|\geq B\bigr\}\quad\text{satisfies}\quad \nu_{\omega}(\varLambda_B)\leq A/2. $$

Thus, for any fixed $k\in{\mathbb{N}}$ and 1≤j≤m, for all q in a set of ν _ω -measure greater than 1−A (more precisely, for all $q \notin\varLambda_{B} \cup \varphi_{-T_{k}}\varLambda_{B}$), we have

$$ \biggl\vert\int_0^{T_k} f \bigl(\varphi^{v}_s q\bigr)\,ds \biggr\vert =\bigl\vert F^{v}_f ( T_k,q) \bigr\vert= \bigl\vert u(\varphi_{T_k} q) - u(q) \bigr\vert\leq2 B. $$

(A.20)

Since $\nu_{\omega}(R_{j}(G_{t_{k}}\omega)) \geq A$, there exists $q_{j} \in R_{j}(G_{t_{k}}\omega)$ satisfying (A.20). In view of (A.19) and (A.2) (applied to $G_{t_{k}}\omega\in\mathcal{K}$) and by definition (A.18) of T _k , it follows that

$$\begin{aligned} \frac{1}{{c}}\|\rho\|_{G_{t_k} \omega} \leq&\max_{1\leq j\leq m}\biggl| \int_{\gamma_j(G_{t_k} \omega)} \rho\biggr| \leq\max_{1\leq j\leq m} \biggl| \int_0^{T_k} f(\varphi_s q_j) \,d s\biggr| +\|i_{X_h}\rho\|_{\infty} \\ \leq&2B+ e^{-t_k}\|i_{X_h} \rho\|_{\infty} \leq{2B}+ \| i_{X_h} \rho\|_{\infty}. \end{aligned}$$

Thus, $\liminf_{t \to+\infty}\| \rho\|_{G_{t} \omega}<\infty$. Since μ is KZ-hyperbolic, recalling the definition of the stable space (4.3), this implies that $[\rho]\in E_{\omega}^{-}(M,{\mathbb{R}})$, contrary to the assumptions. Thus, we conclude that $F^{v}_{f}$ cannot be a coboundary. □

Appendix B: Ergodic decomposition and Mackey action

Given an ergodic automorphism T of a standard probability space $(X,\mathcal{B},\mu)$, a locally compact abelian second countable group G and a cocycle ψ:X→G for T, consider the skew-product extension T _ψ :X×G→X×G. If the skew product is not ergodic then the structure of its ergodic components (defined below) can be studied by looking at properties of the so called Mackey action.

Let (τ _g ) _g∈G denote the G-action on $(X\times G,\mathcal{B}\times\mathcal{B}_{G},\mu\times m_{G})$ given by τ _g (x,h)=(x,h+g). Then (τ _g ) _g∈G commutes with the skew product T _ψ . Fix a probability Borel measure m on G equivalent to the Haar measure m _G . Then the probability measure μ×m is quasi-invariant under T _ψ and (τ _g ) _g∈G , i.e. (T _ψ )_∗(μ×m) and (τ _g )_∗(μ×m) for any g∈G are equivalent to μ×m (or, in other words, T _ψ and (τ _g ) _g∈G are non-singular actions on $(X\times G,\mathcal{B}\times\mathcal{B}_{G},\mu\times m)$). Denote by $\mathcal{I}_{\psi}\subset\mathcal{B}\times\mathcal{B}_{G}$ the σ-algebra of T _ψ -invariant subsets. Since $(X\times G,\mathcal{B}\times\mathcal{B}_{G},\mu\times m)$ is a standard probability Borel space, the quotient space $((X\times G)/\mathcal{I}_{\psi},\mathcal{I}_{\psi},\mu\times m|_{\mathcal{I}_{\psi}})$ is well-defined (and is also standard). This space is called the space of ergodic components and it will be denoted by $(Y, \mathcal{C}, \nu)$. Since (τ _g ) _g∈G preserves $\mathcal{I}_{\psi}$ it also acts on $(Y, \mathcal{C}, \nu)$. This non-singular G-action is called the Mackey action (and is denoted by $(\tau^{\psi}_{g})_{g\in G}$) associated to the skew product T _ψ , and it is always ergodic. Moreover, there exists a measurable map $Y\ni y\mapsto\overline{\mu}_{y}$ taking values in the space of probability measures on $(X\times G,\mathcal{B}\times\mathcal{B}_{G})$ such that

• $\mu\times m=\int_{Y}\overline{\mu}_{y}\,d\nu(y)$;

• $\overline{\mu}_{y}$ is quasi-invariant and ergodic under T _ψ for ν-a.e. y∈Y;

• $\overline{\mu}_{y}$ is equivalent to a σ-finite measure μ _y invariant under T _ψ .

Then T _ψ on $(X\times G,\mathcal{B}\times\mathcal{B}_{G},{\mu}_{y})$ for y∈Y are called ergodic components of T _ψ .

Theorem B.1

[44, 55]

Suppose that T:(X,μ)→(X,μ) is ergodic and let ψ:X→G be a cocycle. Then

• (i) ψ is recurrent if and only if the measure μ _y is continuous for ν-a.e. y∈Y;

• (ii) ψ is non-recurrent if and only if μ _y is purely atomic for ν-a.e. y∈Y;

• (iii) ψ is regular if and only if the Mackey action $(\tau^{\psi}_{g})_{g\in G}$ is strictly transitive, i.e. the measure ν is supported on a single orbit of $(\tau^{\psi}_{g})_{g\in G}$.

If ψ is not recurrent then almost every ergodic component T _ψ :(X×G,μ _y )→(X×G,μ _y ) is trivial, i.e. it is strictly transitive.

If ψ is regular then the structure of ergodic components is trivial, i.e. if we fix one ergodic component then every other ergodic component is the image of the fixed component by a transformation τ _g . In particular, all ergodic components are isomorphic.

As an immediate consequence of Theorem B.1 we obtain that if a cocycle is recurrent and non-regular then the structure of ergodic components of the skew product and the dynamics inside ergodic components are highly non-trivial.

Proof of Proposition 3.4

Since the measure ν is ergodic for the Mackey ${\mathbb {Z}}$-action, it is either continuous or purely discrete. If ν is discrete then, by ergodicity, ν is supported by a single orbit, in contradiction with (iii). Consequently, ν is continuous and the skew product T _ψ has uncountably many ergodic components. Indeed, if T _ψ has at most countably many ergodic components then the measure ν is supported on an at most countable set, so ν is purely discrete.

The continuity of almost every measure μ _y follows directly from (i). This also shows that ν-a.e. ergodic component is not supported by a countable set, since if an ergodic component representing by y∈Y has at most countably many elements then the measure μ _y is also discrete. □