2.1 Linear cocycles over continuous transformations

Let X be a topological space and $f: X\to X$ a homeomorphism. A function $\mathcal {A}=\mathcal {A}(x,n)$ , $x\in X$ and $n\in \mathbb {Z}$ with values in $\mathrm {GL}(d,\mathbb {R})$ is called a continuous linear cocycle if the following properties hold:

1. 1. for each $x\in X$ , we have $\mathcal {A}(x,0)=\mathrm {Id}$ , and given $n,k\in \mathbb {Z}$ ,

   $$\begin{align*}\mathcal{A}(x, n+k)=\mathcal{A}(f^k(x), n)\mathcal{A}(x, k); \end{align*}$$

2. 2. for every $n\in \mathbb {Z}$ , the function $\mathcal {A}(\cdot ,n)\colon X\to \mathrm {GL}(d,\mathbb {R})$ is continuous.

Given a continuous function $A\colon X\to \mathrm {GL}(d,\mathbb {R})$ , we define a cocycle by

$$\begin{align*}\mathcal{A}(x, n) \mathrel{:=} \begin{cases} A(f^{n-1}(x))\cdots A(f(x))A(x) &\text{if } n>0\\ \text{Id} &\text{if } n=0,\\ {A(f^n(x))}^{-1}\cdots {A(f^{-2}(x))}^{-1}{A(f^{-1}(x))}^{-1} &\text{if } n<0. \end{cases} \end{align*}$$

We call the map $A(x)$ the generator of the cocycle $\mathcal {A}(x,n)$ and we say that the cocycle is generated by the function $A(x)$ . Every cocycle $\mathcal {A}(x,n)$ is generated by the function $A(x) \mathrel {:=} \mathcal {A}(x,1)$ .

2.2 Lyapunov exponents and regularity of cocycles

Given a continuous cocycle $\mathcal {A}$ over the homeomorphism f, we say that a point $x \in X$ is regular if there exist a list of numbers

(2.1) $$ \begin{align} \chi_1(x)>\chi_2(x)>\cdots>\chi_{s(x)}(x) \end{align} $$

and a splitting

(2.2) $$ \begin{align} \mathbb{R}^d=\bigoplus_{i=1}^{s(x)}E_i(x) \end{align} $$

such that the following conditions are satisfied:

• • for every i with $1 \le i \le s(x)$ , the two-sided sequence

  $$\begin{align*}\frac1n \log\|\mathcal{A}(x,n)v\| \end{align*}$$
  converges to $\chi _i(x)$ as $n \to \pm \infty $ uniformly in $v \in S^{d-1} \cap E_i(x)$ ;

• • for every partition of the set $\{1,\dots ,s(x)\}$ into two nonempty (disjoint) subsets I, J, we have

  $$\begin{align*}\lim_{n \to \pm \infty} \frac{1}{|n|} \log \measuredangle \left( \mathcal{A}(x,n) \left(\bigoplus_{i \in I} E_i(x)\right), \mathcal{A}(x,n) \left(\bigoplus_{j \in J} E_j(x)\right) \right) = 0 \, , \end{align*}$$
  where the angle $\measuredangle (E,F)$ between two subspaces is defined as the least nonnegative angle between two vectors in the corresponding spaces.

If x is a regular point, it can be shown that the Lyapunov exponents (2.1) and the Oseledets splitting (2.2) are uniquely defined. Let $\mathcal {R}$ denote the set of regular points. It can be shown that $\mathcal {R}$ is a Borel set and that the number $s(x)$ , the Lyapunov exponents $\chi _i(x)$ , and Oseledets spaces $E_i(x)$ all depend Borel measurably on x, and that they satisfy the invariance relations:

$$\begin{align*}s(f^n(x)) = s(x) \, , \quad \chi_i(f^n(x)) = \chi_i(x) \, , \quad E_i(f^n(x)) = \mathcal{A}(x,n) E_i(x) \,. \end{align*}$$

According to the Multiplicative Ergodic Theorem of Oseledets, $\mu (\mathcal {R})=1$ for every f-invariant Borel probability measure $\mu $ on X. If $\mu $ is ergodic, then the Lyapunov exponent $\chi _i(x)$ is the same for $\mu $ -almost every x, and this common value is denoted $\chi _i(\mu )$ .

The notion of regularity as defined above is equivalent to Lyapunov-Perron regularity (or LP regularity). The latter requires that the cocycle $\mathcal A$ must be forward and backward regular. We refer the reader to [Reference Barreira and Pesin7] for these definitions. We note that our requirements (2.1) and (2.2) imply that every regular point is both forward and backwards regular; we stress that simultaneous forward and backward regularity does not imply regularity.

Here are some other properties that we will need: if x is forward regular, then

(2.3) $$ \begin{align} \lim_{n\to+\infty}\frac1n\log\|\mathcal{A}(x,n)\| &= \chi_1(x) \, , \quad \text{and}\end{align} $$

(2.4) $$ \begin{align} \lim_{n\to+\infty}\frac1n\log m(\mathcal{A}(x,n))&=\chi_{s(x)}(x) \, , \end{align} $$

where

$$\begin{align*}m(L) \mathrel{:=} \|L^{-1}\|^{-1} \end{align*}$$

is the co-norm of the linear map L. Furthermore, if $x \in \mathcal {R}$ ,

(2.5) $$ \begin{align} \lim_{n\to\pm\infty}\frac1n\log\|\mathcal{A}(x,n)|E_i(x)\| &= \chi_i(x) \, , \quad \text{and} \end{align} $$

(2.6) $$ \begin{align} \lim_{n\to\pm\infty}\frac1n\log m(\mathcal{A}(x,n)|E_i(x))&=\chi_i(x) \,. \end{align} $$

Given a regular point $x \in \mathcal {R}$ , the dimension $d_i(x)$ of the subbundle $E_i(x)$ is called the multiplicity of the Lyapunov exponent $\chi _i(x)$ . The collection

$$\begin{align*}\mathrm{Sp}(x) \mathrel{:=} \big\{(\chi_i(x),d_i(x)) \ : \ 1 \le i \le s(x)\big\} \end{align*}$$

is called the Lyapunov spectrum of the cocycle $\mathcal {A}$ at x. Sometimes we will also use the list of values of the Lyapunov exponent

$$\begin{align*}\widetilde\chi_1(x)\ge\cdots\ge\widetilde\chi_d(x)\end{align*}$$

in which distinct values $\chi _i(x)$ are repeated according to their multiplicities $d_i(x)$ , $1\le i\le s(x)$ . When we want to stress the dependence on the cocycle, we use more explicit notations such as $\mathrm {Sp}(\mathcal {A},x)$ , $\chi _i(\mathcal {A},x)$ , etc.

2.3 Cohomology

Let $\mathcal {A}$ and $\mathcal {B}$ be two cocycles over the same homeomorphism f, and of the same dimension. We say that they are continuously cohomologous if there exists a continuous conjugacy between them, that is, a continuous map $C \colon X \to \mathrm {GL}(d,\mathbb R)$ such that

$$\begin{align*}\mathcal{A}(x,n) = C(f^n x)^{-1} \mathcal{B}(x,n) C(x) \,. \end{align*}$$

Since C is continuous, it is tempered, and hence continuously cohomologous cocycles have the same set of Lyapunov–Perron regular points, and the same Lyapunov spectra.

2.4 The exterior power of cocycles

Consider the vector space of all alternating i-forms on $\mathbb R^d$ , that is, the space of all multilinear maps from $\mathbb R^d \times \cdots \times \mathbb R^d$ (with i copies) to $\mathbb R$ which are alternating. The dual of this vector space is denoted $(\mathbb {R}^d)^{\wedge i}$ . There exists a canonical alternating multilinear map

$$\begin{align*}(v_1, \dots, v_i) \in \mathbb R^d \times \cdots \times \mathbb R^d \mapsto v_1\wedge\cdots\wedge v_i \in (\mathbb{R}^d)^{\wedge i} \, , \end{align*}$$

called the exterior product. Given a linear operator L of $\mathbb {R}^d$ , we define the i-fold exterior power $L^{\wedge i}$ as the unique linear operator of $(\mathbb {R}^d)^{\wedge i}$ such that

$$\begin{align*}L^{\wedge i} (v_1\wedge\cdots\wedge v_i)=(Lv_1\wedge\cdots\wedge Lv_i) \,. \end{align*}$$

Let $\mathcal {A}$ be a cocycle over a homeomorphism f of a topological space X. For each $i=1,\dots ,n$ consider the cocycle $\mathcal {A}^{\wedge i} \colon X\times {\mathbb {Z}}\to \mathrm {GL}(n,\mathbb {R})^{\wedge i}$ defined by $\mathcal {A}^{\wedge i}(x,n) = \mathcal {A}(x,n)^{\wedge i}$ . We call $\mathcal {A}^{\wedge i}$ the i-fold exterior power cocycle of $\mathcal {A}$ .

One can show (see [Reference Arnold5, p. 211], [Reference Barreira and Pesin7, Section 3.1], or [Reference Raghunathan51]) that

1. 1. if $x\in X$ is a LP-regular point for the cocycle $\mathcal {A}$ , then it is also a LP-regular point for the exterior power cocycle $\mathcal {A}^{\wedge i}$ ;

2. 2. the list of values of the Lyapunov exponents of the cocycle $\mathcal {A}^{\wedge i}$ is

   (2.7) $$ \begin{align} \{\widetilde\chi_k(\mathcal{A}^{\wedge i},x)\} = \big\{\widetilde\chi_{j_1}(\mathcal{A},x)+\cdots+\widetilde\chi_{j_i}(\mathcal{A},x) \ : \ j_1>\cdots >j_i \big\}; \end{align} $$
   in particular,
   (2.8) $$ \begin{align} \widetilde\chi_1(\mathcal{A}^{\wedge i},x) = \widetilde\chi_{1}(\mathcal{A},x)+\cdots+\widetilde\chi_{i}(\mathcal{A},x) \,. \end{align} $$

2.5 Uniform splittings

Let f be a homeomorphism of a compact metric space X. Let $\mathcal {A}$ be a continuous linear cocycle over f. Suppose that for each $x \in X$ , we are given a splitting

(2.9) $$ \begin{align} \mathbb R^d=\bigoplus_{i=1}^s E_i(x) \end{align} $$

where each $\dim E_i(x)$ is independent of the point. Furthermore, assume that this splitting is $\mathcal {A}$ -invariant in the sense that $\mathcal {A}(x,n) E_i(x) = E_i(f^n(x))$ . We say that splitting is dominated if there exists $n_0>0$ such that for every point $x \in X$ and each i with $0 < i < s$ , we have:

(2.10) $$ \begin{align} m \big( \mathcal{A}(x,n_0)|E_i(x) \big)> 2 \big \| \mathcal{A}(x,n_0)|E_{i+1}(x) \big\| \,. \end{align} $$

Equivalently, for all choices of unit vectors $v_i\in E_i(x)$ , we have:

$$\begin{align*}\|\mathcal{A}(x,n_0)v_i\|> 2 \|\mathcal{A}(x,n_0)v_{i+1}\| \,. \end{align*}$$

We say that the subbundle $E_i$ dominates the subbundle $E_{i+1}$ .

Let us emphasize that domination is a uniform property: the amount of time $n_0$ needed for the validity of the inequalities above is the same for every point x. In fact, this assumption is strong enough to force continuity (see [Reference Bonatti, Díaz and Viana12, Section B.1] or [Reference Crovisier and Potrie19, Section 2.1]):

In particular, the angle between different subbundles of a dominated splitting is bounded below. Actually, for every partition of the set $\{1,\dots ,s\}$ into two nonempty (disjoint) subsets I, J, we have

$$\begin{align*}\inf_{x \in X} \measuredangle \left( \bigoplus_{i \in I} E_i(x) , \bigoplus_{j \in J} E_j(x) \right)> 0 \,. \end{align*}$$

Let us introduce two other uniform notions related to domination. Again, let $\mathcal {A}$ be a continuous linear cocycle over f, and suppose we are given an $\mathcal {A}$ -invariant splitting (2.9) into subspaces of constant dimensions. We say that the splitting is absolutely dominated if there exists $n_0>0$ such that for all pairs of points $x,y \in X$ and each i with $0 < i < s$ ,

$$\begin{align*}m \big( \mathcal{A}(x,n_0)|E_i(x) \big)> 2 \big \| \mathcal{A}(y,n_0)|E_{i+1}(y) \big\| \,. \end{align*}$$

In particular, the splitting is dominated.

Finally, an $\mathcal {A}$ -invariant splitting $\mathbb R^d = E^{\mathrm {u}}(x) \oplus E^{\mathrm {s}}(x)$ into two subbundles of constant dimensions is called uniformly hyperbolic if there exists $n_0>0$ such that for all point $x \in X$ ,

$$\begin{align*}m \big (\mathcal{A}(x,n_0)|E^{\mathrm{u}}(x) \big)> 2 > \frac{1}{2} > \big \| \mathcal{A}(x,n_0)|E^{\mathrm{s}}(x) \big\| \,. \end{align*}$$

Note that a uniformly hyperbolic splitting is always absolutely dominated and, in particular, continuous.

We will need the following characterization of uniform hyperbolicity:

Proposition 2.2. Let f be a homeomorphism of a compact metric space X. Let $\mathcal {A}$ be a continuous linear cocycle over f. For each $x \in X$ , define subspaces

If these two subbundles have constant dimensions and form a splitting $\mathbb R^d = E^{\mathrm {u}}(x) \oplus E^{\mathrm {s}}(x)$ at every point $x \in X$ , then this splitting is uniformly hyperbolic.

The statement above corresponds to Mañé [Reference Mañé41, Corollary 1.2] specialized to bundles of constant dimensions. We note that related results were obtained by Sacker and Sell [Reference Sacker and Sell54] and Selgrade [Reference Selgrade58]; see [Reference Chicone and Latushkin18, Chapters 1 and 6], [Reference Wen64, Chapter 6], and [Reference Blumenthal and Latushkin8] for more information.

2.6 Vector bundle setting

If $\mathcal {A}$ is a d-dimensional linear cocycle over $f \colon X \to X$ with generator A, we can consider the following transformation $T_{\mathcal {A}}$ of the product $X \times \mathbb R^d$ :

$$\begin{align*}T_{\mathcal{A}}(x , v) = (f(x), A(x) v) \,. \end{align*}$$

This is a skew-product over f which is linear on fibers. The transformation $T_{\mathcal {A}}$ completely determines the cocycle.

This motivates considering the more general situation where the product space $X \times \mathbb R^d$ is replaced with a vector bundle over X of fiber dimension d, and the skew-product $T_{\mathcal {A}}$ is replaced with a continuous vector bundle automorphism F that factors over f. The concepts discussed above can be extended to this setting in a straightforward manner.

The fiber bundle setting allows us, for example, to discuss derivative cocycles: in this case, f is a $C^1$ diffeomorphism and we consider the induced automorphism $F=Df$ on the tangent bundle.

For simplicity of presentation, in this paper we focus on linear cocycles (on trivial vector bundles $X \times \mathbb R^d$ ), but actually the statements remain correct for general vector bundles, unless noted otherwise.

2.7 Subadditivity

Recall that a sequence of functions $\varphi _n \colon X \to \mathbb R$ is called subadditive with respect to some dynamics $f \colon X \to X$ if, for all $n,m \ge 1$ ,

$$\begin{align*}\varphi_{n+m} \le \varphi_n \circ f^m + \varphi_m \,. \end{align*}$$

A standard example is the sequence $\varphi _n(x) = \log \|\mathcal {A}(x,n)\|$ , where $\mathcal {A}$ is cocycle over f.

Proposition 2.3 (Semi-uniform subadditive ergodic theorem [Reference Schreiber57, Theorem 1], [Reference Sturman and Stark59, Theorem 1.7]).

Let f be a continuous transformation of a compact metric space X. Consider a subadditive sequence of continuous functions $\varphi _n \colon X \to \mathbb R$ . Then:

(2.11) $$ \begin{align} \lim_{n \to \infty} \sup_{X} \frac{\varphi_n}{n} = \sup_{\mu \in \mathcal{M}_f} \lim_{n \to \infty} \int_X \frac{\varphi_n}{n} \, d\mu \, , \end{align} $$

where $\mathcal {M}_f$ denotes the set of f-invariant Borel probability measures. Furthermore, the supremum in the right hand side is attained at some ergodic measure.

We note that variations of this result have been found by several authors (see, e.g., [Reference Furman27, § 3]). The most complete version is [Reference Morris45, Theorem A.3], which shows that upper-semicontinuity is sufficient, and provides other characterizations of the quantity (2.11).

Corollary 2.4 follows directly from Proposition 2.3, or from the usual proof that unique ergodicity implies uniform convergence in the ergodic theorem.

3.1 Definition and basic properties

Let X be a compact metric space and $f:X\to X$ a homeomorphism. Let $\mathcal {A}\colon X\times \mathbb {Z}\to \mathrm {GL}(d,\mathbb {R})$ be a continuous cocycle over f. We say that $\mathcal {A}$ is completely regular if it satisfies the following two conditions:

1. 1. every point $x\in X$ is LP-regular;

2. 2. the Lyapunov exponents are independent of the point $x\in X$ ; more precisely, there are numbers $\chi _1>\cdots >\chi _s$ and positive integers $d_1,\dots ,d_s$ with $\sum _{i=1}^s d_i=d$ such that $\mathrm {Sp}(\mathcal {A},x)=\{(\chi _i,d_i), i=1,\dots ,s\}$ for every $x\in X$ .

Therefore, if a cocycle is completely regular, then at every x we have an Oseledets splitting

(3.1) $$ \begin{align} \mathbb R^d=\bigoplus_{i=1}^s E_i(x) \end{align} $$

where the number of subbundles and their dimensions are independent of the point. In general, the Oseledets splitting is only measurable and there is no a priori information on the speed of convergence in (2.5) and (2.6). However, under the assumption of complete regularity, we obtain better properties, as the following result shows.

Before proving the statement above in full generality, we establish first the case $s=1$ , where actually a weaker hypothesis suffices:

Proposition 3.2. Let $\mathcal {A}$ be a continuous linear cocycle over a homeomorphism f of a compact metric space X. Suppose that there exists a number $\chi \in \mathbb R$ such that for every f-invariant Borel probability measure $\mu $ on X, for $\mu $ -almost every $x\in X$ , all Lyapunov exponents at x are equal to $\chi $ . Then, for all $x \in X$ ,

$$\begin{align*}\lim_{n\to \pm \infty} \frac1n\log m(\mathcal{A}(x,n)) = \lim_{n\to \pm \infty} \frac1n\log\|\mathcal{A}(x,n)\| \end{align*}$$

and these limits are uniform with respect to x.

Proof. It is sufficient to consider positive times n, since negative times can be dealt with using the inverse cocycle. Consider the subadditive sequence of continuous functions $\varphi _n(x)=\log \|\mathcal {A}(x,n)\|$ . By assumption, the sequence $\frac {\varphi _n(x)}{n}$ converges pointwise to the constant $\chi $ . It follows from Proposition 2.3 that

$$\begin{align*}\lim_{n\to\infty}\sup_{x\in X}\frac{\varphi_n(x)}{n}=\chi. \end{align*}$$

Similarly, the sequence $\psi _n(x)=\log m(\mathcal {A}(x, n))$ is superadditive and $\frac {\psi _n(x)}{n}$ converges pointwise to the constant $\chi $ , implying that

$$\begin{align*}\lim_{n\to\infty}\inf_{x\in X}\frac{\psi_n(x)}{n}=\chi. \end{align*}$$

Since $\psi _n\le \varphi _n$ , we conclude that both limits $\frac {\varphi _n}{n}\to \chi $ and $\frac {\psi _n}{n}\to \chi $ are uniform.

Proof of Theorem 3.1.

Let $\mathcal {A}$ be a completely regular cocycle, with Oseledets splitting $\mathbb R^d=\bigoplus _{i=1}^s E_i(x)$ . Let $\chi _1>\cdots >\chi _s$ be the Lyapunov exponents, which are independent of $x \in X$ . The case $s=1$ is covered by Proposition 3.2, so assume that $s \ge 2$ .

Let us prove statement (1), that is, continuity of the subbundles $E_i(x)$ . Fix numbers $c_1,\dots ,c_{s-1}$ such that $\chi _i>c_i>\chi _{i+1}$ . For each $i\in \{1,\dots ,s-1\}$ , consider the cocycle $\widetilde {\mathcal {A}}_i=e^{-c_i}\mathcal {A}$ . This cocycle admits an invariant measurable splitting $E^{\mathrm {u}}_i(x)\oplus E^{\mathrm {s}}_i(x)$ , where

$$ \begin{align*}E^{\mathrm{u}}_i(x) = E_1(x)\oplus\cdots\oplus E_i(x), \quad E^{\mathrm{s}}_i(x) = E_{i+1}(x)\oplus\cdots\oplus E_s(x) \,. \end{align*} $$

Note that all Lyapunov exponents along $E^{\mathrm {u}}_i(x)$ (respectively, $E^{\mathrm {s}}_i(x)$ ) are positive (respectively, negative). An application of Proposition 2.2 shows that the splitting $E^{\mathrm {u}}_i \oplus E^{\mathrm {s}}_i$ is uniformly hyperbolic. In particular, the bundles $E_i^{\mathrm {u}}$ and $E_i^{\mathrm {s}}$ are continuous. It follows that $E_1 = E^{\mathrm {u}}_1$ and $E_s = E^{\mathrm {s}}_s$ are continuous. Furthermore, for each i, the bundle $E_i$ is the transverse intersection of $E_i^{\mathrm {u}}$ and $E_{i-1}^{\mathrm {s}}$ , and therefore, it is continuous as well. This completes the proof of statement (1).

Now we apply Proposition 3.2 (or actually its extension to the vector bundle setting) to the restrictions $\mathcal {A}(x, n)|E_i(x)$ . We then obtain the desired uniformity property, statement (2).

As an important consequence of Theorem 3.1, we obtain the property of absolute domination (defined in Section 2.5 above):

Proof. Let $\mathcal {A}$ be a completely regular cocycle with spectrum $(\chi _i,d_i)_{i=1,\dots ,s}$ . Assume that $s> 1$ . Fix a positive number $\varepsilon $ small enough so that $\chi _i-\varepsilon> \chi _{i+1} + \varepsilon $ for each i with $0< i < s$ . Let N be as in Statement (2) in Theorem 3.1. Then, for all $n>N$ , all $x,y \in X$ , and all i with $0< i < s$ , we have

$$\begin{align*}m \big( \mathcal{A}(x,n)|E_i(x) \big) \ge e^{(\chi_i - \varepsilon) n }> e^{(\chi_{i+1} + \varepsilon) n } \ge \big\| \mathcal{A}(y,n)|E_{i+1}(y) \big\| \,. \end{align*}$$

This proves absolute domination.

It is clear that complete regularity is invariant under continuous cohomology. Here is another basic property:

3.2 Examples

We present some examples of completely regular cocycles.

Example 3.10. Let f be a homeomorphism of a compact metric space X, $c_1> \cdots > c_s$ a list of distinct real numbers, and $d_1, \dots , d_s$ positive integers such that $d =d_1 + \dots + d_s$ . Suppose that $B : X \to \mathrm {GL}(d, \mathbb R)$ is a continuous block-diagonal matrix function

(3.3) $$ \begin{align} B(x) = \begin{pmatrix} B_1(x) & & 0 \\ & \ddots & \\ 0 & & B_s(x) \end{pmatrix}, \end{align} $$

where each diagonal block $B_i(x)$ has dimension $d_i \times d_i$ and has block-triangular form

(3.4) $$ \begin{align} B_i(x) = e^{c_i} \, \begin{pmatrix} U_{i,1}(x) & * & & * \\ & U_{i,2}(x) & \ddots & \\ & & \ddots & * \\ 0 & & & U_{i,\ell_i}(x) \end{pmatrix}, \end{align} $$

where each matrix $U_{i,j}(x)$ is orthogonal.

Consider the cocycle $\mathcal {B}(x,n)$ with generator $B(x)$ . Then, this cocycle is completely regular, with Lyapunov exponents $c_1, \dots , c_s$ , and respective multiplicities $d_1, \dots , d_s$ .

3.3 Hölder continuous cocycles over hyperbolic dynamics

We will discuss a setting where it is possible to obtain a concrete description of all completely regular cocycles. The result is stated in terms of vector bundle automorphisms (see Section 2.6) and continuous amenable reduction (see Example 3.11). Actually, we will work with Hölder continuous vector bundles and their automorphisms (see [Reference Kalinin and Sadovskaya37, § 2.2], [Reference Bochi and Garibaldi10, § 2.2–2.3]).

Proof. The implications (2) $\Rightarrow $ (1) $\Rightarrow $ (3) are automatic (and independent of f being Anosov or F being Hölder). We are left to prove that (3) $\Rightarrow $ (2).

So assume (3), that is, all periodic points have a common Lyapunov spectrum $(\chi _i,d_i)_{i=1}^s$ . As shown by Guysinsky [Reference Guysinsky29] and DeWitt–Gogolev [Reference DeWitt and Gogolev22, Theorems 1.2 and 3.10], the cocycle admits a dominated splitting $\mathcal {E}_1 \oplus \cdots \oplus \mathcal {E}_s$ where $\dim \mathcal {E}_i = d_i$ . In our setting, the subbundles $\mathcal {E}_i$ of a dominated splitting are Hölder continuous (see [Reference Brin and Katok14]). Consider the restriction of the automorphism F to one of these subbundles $\mathcal {E}_i$ . By a result of Kalinin and Sadovskaya [Reference Kalinin and Sadovskaya37, Theorem 3.9] (see also [Reference Sadovskaya and Damjanović56, § 13.7]), there exists a Hölder continuous metric (i.e., family of inner products) on $\mathcal {E}_i(x)$ and an invariant Hölder continuous field of flags

$$\begin{align*}\{0\} = \mathcal{E}_{i,0}(x) \subset \mathcal{E}_{i,1}(x) \subset \cdots \subset \mathcal{E}_{i,\ell_i}(x)= \mathcal{E}_i(x) \end{align*}$$

such that the action induced by F on the quotient bundle $\mathcal {E}_{i,j}/\mathcal {E}_{i,j-1}$ expands the (induced) metric by a constant factor $e^{\varphi _i(x)}$ , where $\varphi _i$ is a Hölder function on X. In our situation, $\varphi _i$ has the same integral with respect to all invariant probability measures supported on periodic orbits, namely $\chi _i$ . Therefore we can apply Livsic theorem and write $\varphi _i = \chi _i + \psi _i \circ f - \psi _i$ , where $\psi _i$ is a Hölder continuous function. We rescale the Riemannian metric inside $\mathcal {E}_i(x)$ by multiplying it by $e^{-\psi _i(x)}$ , so that the new expansion factor is the constant $e^{\chi _i}$ . Then we combine these metrics by declaring the subbundles $\mathcal {E}_i(x)$ to be orthogonal. This shows that F admits a Hölder continuous amenable reduction.

Let us note that in dimension $d=1$ , Theorem 3.13 is equivalent to Livsic theorem for the group $(\mathbb R,+)$ . Since Livsic theorem does not hold for functions that are merely continuous instead of Hölder (see [Reference Bousch and Jenkinson13, p. 215] or [Reference Kocsard38]), the regularity hypothesis on the automorphism is necessary for the validity of our statement.

On the other hand, we believe that Theorem 3.13 holds for (transitive) Anosov diffeomorphisms of class $C^1$ (instead of $C^{1+\delta }$ ), or even for (transitive) hyperbolic homeomorphisms (see, e.g., [Reference Ruelle53, Chapter 7], [Reference Alekseev and Yakobson4, Part II], [Reference Akin2, Chapter 11]).

We mention here the work [Reference Chen, Cao and Zou16] concerning the uniformity of the first Lyapunov exponent in a related setting.

3.4 Relation with Sacker–Sell theory

Let f be a homeomorphism of a compact metric space X. We assume that f is invariantly connected, that is, the only invariant clopen sets are $\emptyset $ and X. This condition is satisfied if f is transitive or X is connected.

Let $\mathcal {A}$ be a continuous linear d-dimensional cocycle over f. For each real number $\lambda $ , we define a cocycle

$$\begin{align*}\mathcal{A}_\lambda (x,n) \mathrel{:=} e^{-\lambda n} \mathcal{A}(x,n) \,. \end{align*}$$

The Sacker–Sell spectrum of $\mathcal {A}$ is the set

$$\begin{align*}\mathrm{SS}(\mathcal{A}) \mathrel{:=} \{ \lambda \in \mathbb R : \mathcal{A}_\lambda \text{ is not uniformly hyperbolic} \}. \end{align*}$$

(Here hyperbolic subbundles are allowed to be $\{0\}$ .) According to [Reference Sacker and Sell55, Theorem 2] (see also [Reference Johnson and Zampogni35, Theorem 2.4]), the Sacker–Sell spectrum is a compact subset of $\mathbb R$ with at most d connected components, that is,

(3.5) $$ \begin{align} \mathrm{SS}(\mathcal{A}) = [\alpha_1,\beta_1] \cup \cdots \cup [\alpha_s,\beta_s] \, , \quad (s \le d). \end{align} $$

Furthermore, there exists an absolutely dominated splitting (see Section 2.5 above) into s bundles,

(3.6) $$ \begin{align} \mathbb R^d = E_1(x) \oplus \cdots \oplus E_s(x) \,. \end{align} $$

As a consequence of domination (recall Proposition 2.1), the subbundles vary continuously with x. As shown by Johnson, Palmer, and Sell [Reference Johnson, Palmer and Sell34, Theorem 2.3] the boundary points $\alpha _i$ , $\beta _i$ of the connected components of the Sacker–Sell spectrum can be characterized in terms the Lyapunov exponents as follows. If we reindex the intervals in (3.5) so that $\beta _1\ge \alpha _1> \dots > \beta _s\ge \alpha _s$ , then, for every i we have

(3.7) $$ \begin{align} \beta_i &= \sup_{\mu} \widetilde{\chi}_1(\mathcal{A}|E_i, \mu) \, , \end{align} $$

(3.8) $$ \begin{align} \alpha_i &= \inf_{\mu} \widetilde{\chi}_{d_i}(\mathcal{A}|E_i, \mu) \, , \end{align} $$

where $\mu $ runs on the set of ergodic measures; furthermore, each sup and each inf are attained. Using Proposition 2.3, we can characterize these numbers as:

$$ \begin{align*} \beta_i &= \lim_{n \to \infty} \sup_{x\in X} \frac{1}{n} \log \| \mathcal{A}(x,n)|E_i\| \, , \\ \alpha_i &= \lim_{n \to \infty} \inf_{x\in X} \frac{1}{n} \log m(\mathcal{A}(x,n)|E_i) \,. \end{align*} $$

Note that the Sacker–Sell splitting (3.6) is actually the finest absolutely dominated splitting for the cocycle $\mathcal {A}$ , that is, the absolutely dominated splitting into the maximal number of subbundles.

The Sacker–Sell spectrum is a discrete set if and only if $\alpha _i = \beta _i$ for each $i = 1,\dots ,s$ . Cocycles with discrete Sacker–Sell spectrum were studied by Johnson and Zampogni [Reference Johnson and Zampogni35]. It turns out that this property coincides with complete regularity:

There is another classical object associated to a linear cocycle called the Mather spectrum (see [Reference Mather42], [Reference Pesin49, § 3.1]), which is directly related to the Sacker-Sell spectrum (see [Reference Chicone and Latushkin18, Corollary 6.45]).

4.1 Baire category

Recall that a subset A in a topological space X is called nowhere dense if its closure $\bar {A}$ has empty interior. A subset $A\subset X$ is said to be of the 1^st Baire category if it can be represented as a union of a countable collection of nowhere dense sets, and of 2^nd Baire category otherwise. Sets of 1^st Baire category are also called meager. Note that any subset of a meager set is a meager set.

A subset $A \subset X$ is called residual if its complement is meager. If X is a complete metric space, then residual sets are of 2^nd Baire category. In particular, X and every other dense $G_\delta $ set are of the second category.

Like the collection of sets of probability zero, the collection of meager sets contains the empty set, is closed under countable unions, is hereditary, and is proper. Therefore, one can think of meager sets as being ‘small’. See [Reference Oxtoby48] for a full discussion.

4.2 Dynamically defined meager sets

We now describe a general mechanism for the occurrence of meager sets. Let X be a complete metric space and $(\varphi _n)_{n>0}$ a sequence of real-valued continuous functions on X. Given numbers $\alpha ,\beta \in \mathbb {R}$ , define the following sets

$$ \begin{align*} I_\alpha &\mathrel{:=} \big\{x\in X\colon\liminf\varphi_n(x)<\alpha\big\} \, ,\\ S_\beta &\mathrel{:=} \big\{x\in X\colon\limsup\varphi_n(x)>\beta\big\} \,. \end{align*} $$

Proof. Fix numbers $\alpha < \alpha ' < \alpha " < \beta " < \beta ' < \beta $ . Letting

$$\begin{align*}\widetilde{I}_{\alpha'} \mathrel{:=} \bigcap_{m=1}^\infty\bigcup_{n=m}^\infty\{x\in X\colon \varphi_n(x)<\alpha'\} \, , \end{align*}$$

we have $I_\alpha \subset \widetilde {I}_{\alpha '} \subset I_{\alpha "}$ . The set $I_\alpha $ is dense, by assumption, and the set $\widetilde {I}_{\alpha '}$ is a $G_\delta $ . Therefore the set $I_{\alpha "}$ contains a dense $G_\delta $ , so it is residual. Similarly, $S_{\beta "}$ is residual. Thus, the set $I_{\alpha "} \cap S_{\beta "}$ is residual, and for any point in this set, $\lim _{n\to \infty }\varphi _n(x)$ does not exist.

Without trying to be comprehensive, let us illustrate how Theorem 4.1 can be applied to a simple dynamical setting. Suppose that $f : X \to X$ is a continuous transformation of a compact metric space. Given a (say) continuous function $\varphi : X \to \mathbb R$ , consider the sequence of Birkhoff averages

$$ \begin{align*} \varphi_n(x) \mathrel{:=} \frac{1}{n}\sum_{k=0}^{n-1}\varphi(f^k(x)) \end{align*} $$

and let $\mathcal {R}_\varphi $ be the set of points $x\in X$ for which $\lim _{n\to \infty }\varphi _n(x)$ exists. By Birkhoff ergodic theorem, we know that this set has full measure with respect to any invariant probability measure and hence, is ‘large’ from the measure-theoretical point of view. But what can we say about the Baire category of this set?

Following [Reference Akin, Auslander and Nagar3], a (perhaps noninvertible) continuous transformation f of a compact metric space X is called strongly transitive if the backward orbit $ \{f^{-n}(x) \colon n \ge 0\}$ of every point $x \in X$ is dense in X. Examples include any minimal homeomorphism, any transitive one-sided subshift of finite type, and the restriction of any rational map of $\bar {\mathbb {C}}$ to its Julia set (see [Reference Milnor44, Corollary 4.13]).

The literature contains many related results; we mention [Reference Johnson33, Theorem 4.3.(2)], [Reference Akin2, Theorem 11, p. 166], and [Reference Carvalho and Varandas15]. The general message is that the set of points for which Birkhoff ergodic theorem holds, while large from a measure-theoretic viewpoint, is often meager.

As it is reasonable to expect, similar phenomena for the multiplicative ergodic theorem of Oseledets occur in a variety of contexts: see for instance [Reference Abdenur, Bonatti and Crovisier1, Theorem 3.14], [Reference Tian60], [Reference Micena and de la Llave43], [Reference Varandas and Dias61], and [Reference Lin and Tian40, Corollary C]. We now turn to our investigation of this matter, where the previously discussed concept of complete regularity will naturally enter into the picture. Our goal is to establish sufficient conditions under which the set $\mathcal {R}$ of LP-regular points is meager, thereby ensuring that the set of irregular points is topologically large. We note that the set of irregular points may also be large in terms of entropy [Reference Chen, Zang and Zou17] or Lebesgue measure [Reference Kiriki, Li, Nakano and Soma39].

4.3 A criterion for generic irregularity

We now come back to the setting of continuous linear cocycles. We will formulate a sufficient condition for the set of LP-regular points to be meager. Our condition uses the notion of complete regularity studied above.

Given a homeomorphism f of a compact metric space X and a nonempty subset $Y\subset X$ , we define the stable set of Y as

$$ \begin{align*}W^{\mathrm{s}}(Y) = \big\{x\in X: d(f^n(x),Y)\to0 \text{ as } n\to+\infty \big\} \,. \end{align*} $$

Now we can state the following result:

The proof of this theorem will use Theorem 4.1 above and also the following lemma:

Lemma 4.4. Let $Y\subset X$ be a compact f-invariant subset such that the restriction of the cocycle $\mathcal {A}$ to Y is completely regular with Lyapunov exponents $\chi _1>\cdots >\chi _s$ . Then, for every $x\in W^{\mathrm {s}}(Y)$ ,

(4.1) $$ \begin{align} \lim_{n\to\infty}\frac1n\log\|\mathcal{A}(x,n)\|=\chi_1 \,. \end{align} $$

Proof. It follows from the definition of complete regularity and property (2.3) that the desired conclusion (4.1) holds for every point in Y. Actually, it follows from Theorem 3.1 that the limit in (4.1) is uniform over Y. In particular, given $\varepsilon>0$ , we can find $N>0$ such that, for all $y \in Y$ ,

$$\begin{align*}\frac{1}{N} \log\|\mathcal{A}(y,N)\| < \chi_1 + \varepsilon \,. \end{align*}$$

By continuity, if $d(x,Y)$ is sufficiently small, then

$$\begin{align*}\frac{1}{N} \log\|\mathcal{A}(x,N)\| < \chi_1 + 2\varepsilon \,. \end{align*}$$

Then, as a straightforward consequence of submultiplicativity of norms, we have

$$\begin{align*}\limsup_{n \to \infty }\frac{1}{n} \log\|\mathcal{A}(x,n)\| \le \chi_1 + 2\varepsilon \end{align*}$$

for all $x \in W^{\mathrm {s}}(Y)$ . Since $\varepsilon>0$ is arbitrary, the limsup above is actually $\le \chi _1$ for all $x \in W^{\mathrm {s}}(Y)$ .

In order to obtain an inequality in the reverse direction, fix $\varepsilon>0$ again. Using Theorem 3.1 we find $N>0$ such that, for all points $y \in Y$ and all unit vectors $u \in E_1(y)$ ,

(4.2) $$ \begin{align} \frac{1}{N} \log\|\mathcal{A}(y,N) u\|> \chi_1 - \varepsilon \,. \end{align} $$

Since the cocycle $\mathcal {A}$ restricted to the compact invariant set Y admits a dominated splitting with dominating bundle $E_1$ , we can define a family of cones $C(y) \subset \mathbb R^d$ depending continuously on $y \in Y$ with the following properties: for every $y \in Y$ , we have $E_1(y) \subset C(y)$ and

$$\begin{align*}\overline{\mathcal{A}(y,1)(C(y))} \subset \mathrm{int}(C(y)) \cup \{0\} \quad \text{(strict invariance).} \end{align*}$$

Such a cone field is easily constructed using an adapted metric: see [Reference Gourmelon28, Reference Crovisier and Potrie19]. The subbundle attracts vectors in the cone field in the following sense: for every $y \in Y$ and every unit vector $u \in C(y)$ ,

(4.3) $$ \begin{align} \measuredangle \big( \mathcal{A}(y,n)u , E_1(f^n(y)) \big) \to 0 \text{ as } n \to \infty, \end{align} $$

and this convergence is actually uniform with respect to y and u. We extend continuously the cone field to a neighbourhood U of Y so that the strict invariance property is satisfied for points in $U \cap f^{-1}(U)$ .

Fix $x\in W^{\mathrm {s}}(Y)$ . Replacing x by an iterate if necessary, we assume that $f^n(x) \in U$ for every $n \ge 0$ . Fix any unit vector $v \in C(x)$ . It follows from property (4.3) that there exist a sequence of points $y_n\in Y$ and a sequence of unit vectors $u_n \in E_1(y_n)$ such that

$$\begin{align*}d(f^n(x), y_n)\to 0 \quad \text{and} \quad \measuredangle(\mathcal{A}(x,n)v , u_n) \to 0 \quad \text{as } n \to +\infty \,. \end{align*}$$

Inequality (4.2) applies to each pair $(y,u) = (y_n, u_n)$ . For every sufficiently large n, the pair $\big (f^n(x),\frac {\mathcal {A}(x,n)v}{\|\mathcal {A}(x,n)v\|}\big )$ is close to $(y_n,u_n)$ , and therefore

$$\begin{align*}\frac{1}{N} \log \frac{\|\mathcal{A}(x, n+N) v \|}{\|\mathcal{A}(x, n) v \|}> \chi_1 - 2\varepsilon \,. \end{align*}$$

Now it is straightforward to check that

$$\begin{align*}\liminf_{n \to \infty }\frac{1}{n} \log\|\mathcal{A}(x,n) v\| \ge \chi_1 - 2\varepsilon \,. \end{align*}$$

Since $\varepsilon>0$ is arbitrary, the liminf above is actually $\ge \chi _1$ for all $x \in W^{\mathrm {s}}(Y)$ . The proof of the lemma is completed.

Proof of Theorem 4.3.

For each $j=1,2$ , let $\widetilde \chi _1^j\ge \cdots \ge \widetilde \chi _d^j$ be the list of values of the Lyapunov exponent of the cocycle $\mathcal {A}|X_j$ , where each value is repeated according to multiplicity. By assumption, these two lists are not identical.

We first assume that $\widetilde \chi _1^1\neq \widetilde \chi _1^2$ . For definiteness, assume that $\widetilde \chi _1^1> \widetilde \chi _1^2$ . Consider the sequence of continuous functions

(4.4) $$ \begin{align} \varphi_n(x)=\frac1n\log\|\mathcal{A}(x,n)\|, \quad x\in X \,. \end{align} $$

By Lemma 4.4, for each $j=1,2$ , we have $\lim \varphi _n(x) = \widetilde \chi _1^j$ for all $x \in W^{\mathrm {s}}(X_j)$ . Each of these stable sets is dense, by hypothesis. Applying Theorem 4.1 with $\alpha =\widetilde \chi _1^2$ and $\beta =\widetilde \chi _1^1$ , we obtain that the set of points for which $\lim _{n\to \infty }\varphi _n(x)$ exists is meager and hence, so is its subset $\mathcal {FR}$ of forward regular points (see [Reference Barreira and Pesin7]). This completes the proof of the theorem in the particular case when $\widetilde \chi _1^1 \neq \widetilde \chi _1^2$ .

We now proceed with the general case and show how to reduce it to the previous one by using exterior powers. As explained in Subsection 2.4, a regular point $x\in X$ for the cocycle $\mathcal {A}$ is also a regular point for the exterior power cocycle $\mathcal {A}^{\wedge i}$ . Now let i be the smallest index such that $\widetilde \chi _i^1\neq \widetilde \chi _i^2$ , that is, $\widetilde \chi _i(\mathcal {A}|X_1)\ne \widetilde \chi _i(\mathcal {A}|X_2)$ . Then, by formula (2.8), we obtain that

$$ \begin{align*}\widetilde\chi_1\big(\mathcal{A}^{\wedge i}|X_1\big) \ne \widetilde\chi_1\big(\mathcal{A}^{\wedge i}|X_2\big) \,. \end{align*} $$

Since the cocycle $\mathcal {A}$ is completely regular on subsets $X_1$ and $X_2$ , the exterior power cocycle $\mathcal {A}^{\wedge i}$ is also completely regular on these sets (see Proposition 3.4). By the previous case, the set of forward regular points for the cocycle $\mathcal {A}^{\wedge i}$ is meager, and hence, so is the set of forward regular points for the cocycle $\mathcal {A}$ . This completes the proof of the theorem.

4.4 Applications of the criterion

We present some corollaries of Theorem 4.3.

If p is a periodic point for f, let $O(p)$ denote the orbit of p. Given any linear cocycle $\mathcal {A}$ over f, the periodic point p is LP-regular, and its Lyapunov exponents can be computed from the eigenvalues of the matrix $\mathcal {A}(p,k)$ , where k is the period of p.

There are situations where Theorem 4.3 applies, but Corollary 4.5 does not. For example, suppose $\varphi ^t \colon M \to M$ is a transitive Anosov flow of a compact manifold. If $t_0>0$ is small enough, then $f \mathrel {:=} \varphi ^{t_0}$ has no periodic points, so we are out of the scope of Corollary 4.5. Now, suppose $x_1$ and $x_2$ are periodic points for the flow, and let $X_1$ and $X_2$ be the respective orbits. Then the restrictions $f|X_1$ and $f|X_2$ are conjugate to irrational rotations. If $\mathcal {A}$ is any continuous cocycle over f whose restrictions $\mathcal {A}|X_1$ and $\mathcal {A}|X_2$ are completely regular but have different Lyapunov spectra, then its set $\mathcal {R}$ of LP-regular points is meager in M. Indeed, the stable sets $W^{\mathrm {s}}(X_1)$ and $W^{\mathrm {s}}(X_2)$ are dense (see, e.g., [Reference Fisher and Hasselblatt26, Theorem 2.6.10]), so Theorem 4.3 applies.

On the other hand, there is a particular setting of the dynamics in which the denseness hypothesis in Corollary 4.5 is naturally satisfied, namely homoclinic classes. We recall the definition.

Let $f: M\to M$ be a $C^1$ diffeomorphism of a compact smooth Riemannian manifold M. Let also $p,q\in M$ be two hyperbolic periodic points. We say that p and q are homoclinically related and write $q\sim p$ if the sets of transverse intersection points $W^{\mathrm {u}}(O(p))\pitchfork W^{\mathrm {s}}(O(q))$ and $W^{\mathrm {s}}(O(p))\pitchfork W^{\mathrm {u}}(O(q))$ are both nonempty. This turns out to be an equivalence relation (see [Reference Newhouse47, Proposition 2.1]), and every equivalence class is an invariant subset. The homoclinic class of p is defined to be the closure of the equivalence class of p. (We note that two homoclinic classes may intersect without being identical.) Furthermore (see [Reference Newhouse47, Corollary 2.7]),

(4.5) $$ \begin{align} H_p=\overline{W^{\mathrm{u}}(O(p))\pitchfork W^{\mathrm{s}}(O(p))}. \end{align} $$

In particular, for any periodic point $q \sim p$ the set $W^{\mathrm {s}}(O(q))$ is dense in $H_p = H_q$ . Thus Corollary 4.5 implies the following result:

Of course, the most natural application of the result above is for the derivative cocycle restricted to the homoclinic class.

For the sake of completeness, let us describe here a related result by Tian [Reference Tian60]. Let f be a homeomorphism of a compact metric space X. We say that f has exponential specification property with exponent $\lambda> 0$ if for any $\delta> 0$ there is $N=N(\delta )> 0$ such that for any $k\ge 1$ , any k points $x_1,x_2,\dots ,x_k\in X$ , any integers $a_1\le b_1 <a_2\le b_2 <\ldots <a_k\le b_k$ with $a_{i+1}-b_i\ge N$ , one can find a point $y\in X$ such that

$$ \begin{align*}d(f^i(y),f^i(x_j))<\delta e^{-\lambda\min\{i-a_i,b_j-i\}} \end{align*} $$

for $a_j\le i\le b_j$ and $1\le j\le k$ .

A homeomorphism f with the exponential specification property may fail to have periodic orbits (see [Reference Tian60, Example 2.4.(2)]). Thus we cannot use Corollary 4.5 to derive Theorem 4.7. Note that Theorem 4.3 and its corollaries apply to all continuous cocycles. In contrast, Theorem 4.7 requires Hölder continuity of the cocycle, since its proof employs estimates from [Reference Kalinin36] that rely crucially on this stronger regularity assumption.

4.5 All-or-meager results

Here is a related result, this time following from Corollary 4.5 combined with Theorem 3.13:

Notice the sharp contrast between the two alternatives: either the set $\mathcal {R}$ is the whole space, or it is a meager subset. We call statements with this type of conclusion all-or-meager results. Another example is Theorem 4.2.

Let us look for other results of this type. Optimistically, one could ask: are the conclusions of Theorem 4.8 valid for any continuous linear cocycle over any homeomorphism of a compact space? The answer is negative, as the following example shows.

Example 4.9. Let $g : Y \to Y$ be a homeomorphism of a compact metric space Y and let $\mathcal {B}$ be a linear cocycle over g, with generator B. We assume two conditions: First, g admits a fully-supported Borel probability measure $\mu $ . Second, the set $\mathcal {R}_{\mathcal {B}}$ of LP-regular points is not the whole space Y. Examples satisfying these conditions can be found using Theorem 4.8, for instance. Note that $\mu $ -almost every point in Y has alpha- and omega-limit sets both equal to Y. Fix a point $y_0$ with these properties which is also an element of the full-measure set $\mathcal {R}_{\mathcal {B}}$ . Let $y_n \mathrel {:=} g^n(y_0)$ ; then, for any $N>0$ , the sets $\{y_n : n> N\}$ and $\{y_n : n < -N\}$ are both dense in Y. Now fix an increasing two-sided sequence $(t_n)_{n \in {\mathbb {Z}}}$ such that $t_n \to 0$ as $n \to -\infty $ and $t_n \to 1$ as $n \to +\infty $ . Define the following subsets of $Y \times [0,1]$ :

$$\begin{align*}Z \mathrel{:=} \{(y_n, t_n) : n \in {\mathbb{Z}} \} \quad \text{and} \quad X \mathrel{:=} (Y \times \{0,1\}) \cup Z \,. \end{align*}$$

Then $\overline {Z} = X$ and in particular X is compact. Also note that each point in Z has a neighbourhood disjoint from $Y \times \{0,1\}$ and in particular Z is a relatively open subset of X. Define a map $f \colon X \to X$ as follows:

$$\begin{align*}f(y,t) \mathrel{:=} (g(y),t) \text{ if } t=0,1, \quad f(y_n,t_n) \mathrel{:=} (y_{n+1},t_{n+1}) \,. \end{align*}$$

Then f is actually a homeomorphism of X, and the subset Z is the orbit of $(y_0,t_0)$ . Finally, define a cocycle $\mathcal {A}$ over f whose generator is $\mathcal {A}(y,t) \mathrel {:=} B(y)$ . Then the set $\mathcal {R}_{\mathcal {A}}$ of LP-regular points is not the whole space X, as its complement includes $(Y\setminus \mathcal {R}_{\mathcal {B}}) \times \{0,1\}$ . On the other hand, since the set $\mathcal {R}_{\mathcal {B}}$ contains the point $y_0$ , the set $\mathcal {R}_{\mathcal {A}}$ contains the point $(y_0,t_0)$ , as well as its orbit Z. Since Z is an open and dense subset of X, it follows that $\mathcal {R}_{\mathcal {A}}$ is not a meager set. Thus, we have exhibited an example where the all-or-meager dichotomy fails.

The example above is somewhat unsatisfactory because the map f has wandering points.

Returning to the case of transitive Anosov diffeomorphisms, it is an open problem whether Theorem 4.8 extends to cocycles that are merely continuous rather than Hölder continuous.

On the other hand, if we assume that the dynamics is minimal, then we have the following all-or-meager dichotomy that applies to all continuous linear cocycles:

The proof of this theorem is given in Section 4.7.

Theorem 4.10 extends some previous results such as [Reference Furman27, Theorem 4] and [Reference Hou, Lin and Tian32, Corollary 6.6].

Johnson and Zampogni asked a question in [Reference Johnson and Zampogni35, p. 104], which in our terminology is as follows: Assume that the dynamics f is minimal, and that for all points x and all nonzero vectors v, the forward Lyapunov exponent $\lim _{n \to \infty } \frac 1n \log \|\mathcal {A}(x,n)v\|$ is well-defined; does it follow that the cocycle is completely regular? Note that Theorem 4.10 provides a positive answer to this question under the stronger hypothesis that all points are LP-regular.

4.6 Singular values of cocycles

As a preparation for the proof of Theorem 4.10, we discuss the behaviour of the singular values of the cocycle. Furthermore, under assumption of minimality, we obtain new information about singular values in the absence of dominated splitting (Theorem 4.12).

The singular values of a linear operator $L \colon \mathbb R^d \to \mathbb R^d$ are the eigenvalues of $\sqrt {L^* L}$ , which we list with multiplicity as

$$\begin{align*}\sigma_1(L) \ge \cdots \ge \sigma_d(L) \,. \end{align*}$$

Each singular value $\sigma _k(L)$ depends continuously on L. The extreme singular values are the norm $\|L\| = \sigma _1(L)$ and the co-norm $m(L) = \sigma _d(L)$ . More generally, by the Courant–Fischer principle [Reference Horn and Johnson31, Theorem 7.3.8], for any $k = 1, \dots , d$ ,

$$\begin{align*}\sigma_k(L) &= \max \big\{ m(A|_E) \colon E \text{ is a subspace of dim. } k \big\} \\ &= \min \big\{ \|A|_F\|\ \ \ \colon F \text{ is a subspace of dim. } d-k+1 \big\} \,. \end{align*}$$

As a simple consequence, we have the following useful bounds:

(4.6) $$ \begin{align} m(L_3) \, \sigma_k(L_2) \, m(L_1) \le \sigma_k(L_3 L_2 L_1) \le \|L_3\| \, \sigma_k(L_2) \, \|L_1\| \,. \end{align} $$

The list of singular values of the exterior power $L^{\wedge r}$ coincides with the list $\sigma _{k_1}(L) + \cdots + \sigma _{k_r}(L)$ , where $1 \le k_1 < \cdots < k_r \le d$ , with repetitions being taken into account.

The Lyapunov exponents of a linear cocycle $\mathcal {A}$ are related to the singular values as follows: for every forward regular point x,

$$\begin{align*}\widetilde\chi_k(x) = \lim_{n \to +\infty} \frac{1}{n} \log \sigma_k(\mathcal{A}(x,n)) \,. \end{align*}$$

Dominated splittings (see Section 2.5) can be detected in terms of separation of singular values as follows:

Proposition 4.11 [Reference Bochi and Gourmelon11, Theorem A].

Let f be a homeomorphism of a compact metric space, and let $\mathcal {A}$ be a d-dimensional continuous linear cocycle over f. For every k with $0<k<d$ , the cocycle $\mathcal {A}$ admits a dominated splitting with a nontrivial dominating bundle of dimension k if and only if there exist positive constants c and $\varepsilon $ such that

$$\begin{align*}\frac{\sigma_{k}(\mathcal{A}(x,n))}{\sigma_{k+1}(\mathcal{A}(x,n)) }> c e^{\varepsilon n} \end{align*}$$

for all $x \in X$ and all $n>0$ .

In the result above, the ‘only if’ part is rather trivial. It is the ‘if’ part that is interesting: the exponential separation of singular values forces the existence of a dominated splitting.

Our next result can be seen as an improvement of Proposition 4.11 under the assumption of minimality:

Theorem 4.12. Let f be a minimal homeomorphism of a compact metric space, and let $\mathcal {A}$ be a d-dimensional continuous linear cocycle over f. Fix k with $0<k<d$ . Then the cocycle $\mathcal {A}$ does not admits a dominated splitting with a dominating bundle of dimension k if and only if there exists a residual subset G of X such that for every $x \in G$ ,

$$\begin{align*}\liminf_{n \to \infty}\frac{1}{n} \log \frac{\sigma_k(\mathcal{A}(x, n))}{\sigma_{k+1}(\mathcal{A}(x, n))} = 0 \,. \end{align*}$$

Note that the minimality assumption cannot be dropped, as Example 3.12 shows.

Proof of Theorem 4.12.

The ‘if’ part is a direct consequence of (the trivial half of) Proposition 4.11. So let us prove the ‘only if’ part. Assume that the cocycle $\mathcal {A}$ admits no dominated splitting with dominating bundle of dimension k. By Proposition 4.11, for every $i>0$ there exists $x_i \in X$ and $n_i> i$ such that

$$\begin{align*}\frac{1}{n_i} \log \frac{\sigma_{k}(\mathcal{A}(x_i,n_i))}{\sigma_{k+1}(\mathcal{A}(x_i,n_i))} < \frac{1}{i} \,. \end{align*}$$

Since X is compact and the cocycle is continuous, there exists $c>0$ such that

$$\begin{align*}\|\mathcal{A}(x,n)\| \le e^{c|n|} \end{align*}$$

for all $x\in X$ and $n \in {\mathbb {Z}}$ . Consider the matrix identity

$$\begin{align*}\mathcal{A}(f^j(x), n) = \mathcal{A}(f^{n}(x), j) \, \mathcal{A}(x, n) \, \mathcal{A}(f^j(x), -j) \,. \end{align*}$$

Using inequalities (4.6),

$$\begin{align*}\frac{\sigma_k(\mathcal{A}(f^j(x), n))}{\sigma_{k+1}(\mathcal{A}(f^j(x), n))} \le e^{4c|j|} \, \frac{\sigma_k(\mathcal{A}(x,n))}{\sigma_{k+1}(\mathcal{A}(x, n))} \,. \end{align*}$$

In particular, if $|j|< \sqrt {n_i}$ , then

Note that $\varepsilon _i \to 0$ as $i \to \infty $ .

Now define a sequence of open sets

$$\begin{align*}U_i \mathrel{:=} \left\{ x \in X \colon \frac{1}{n_i} \log \frac{\sigma_k(\mathcal{A}(x, n_i))}{\sigma_{k+1}(\mathcal{A}(x, n_i))} < \varepsilon_i \right\} \,. \end{align*}$$

By the previous observations, $U_i$ contains the orbit segment $\{f^j(x_i) : |j| < \sqrt {n_i}\}$ . Since $n_i \to \infty $ and f is a minimal transformation, there exists a sequence $\delta _i \to 0$ such that each set $U_i$ is $\delta _i$ -dense in X. It follows that, for each $m>0$ , the set $V_m \mathrel {:=} \bigcup _{i=m}^\infty U_i$ is open and dense. By Baire’s theorem, the set $G \mathrel {:=} \bigcap _{m=1}^\infty V_m$ is residual, and in particular dense.

For every point x in G, there exists a sequence $i_j \to \infty $ such that $x \in U_{i_j}$ , and so

$$\begin{align*}\lim_{j \to \infty}\frac{1}{n_{i_j}} \log \frac{\sigma_k(\mathcal{A}(x, n_{i_j}))}{\sigma_{k+1}(\mathcal{A}(x, n_{i_j}))} = 0 \,.\\[-42pt] \end{align*}$$

Let us highlight a consequence of Theorem 4.12 in the particular context discussed in Example 3.7, Section 3.2:

Note that the result above improves some of the conclusions of [Reference Furman27, Theorem 4].

Proof. For each x, let $\Lambda _x$ be the set of accumulation points of the sequence $\frac {1}{n}\log \|\mathcal {A}(x,n)\|$ . Then $\Lambda _x$ is a compact interval (see, e.g., [Reference Walters, Arnold and Wihstutz63, Lemma 2.5]). Note that $\Lambda _x \subset [0,\infty )$ because $|\det \mathcal {A}| \equiv 1$ , and $\Lambda _x \subset [0,\chi _1]$ due to Proposition 2.3. If $\chi _1 = 0$ , there is nothing to show. So, assume that $\chi _1$ is positive. Since the cocycle is not uniformly hyperbolic, by Theorem 4.12 there is a dense set of points x for which $\liminf \frac {1}{n}\log \|\mathcal {A}(x,n)\| = 0$ . On the other hand, there is another dense set of points x for which $\lim \frac {1}{n}\log \|\mathcal {A}(x,n)\| = \chi _1$ . The proof of Theorem 4.1 actually shows that if $0<\alpha "<\beta "<\chi _1$ , then there is a residual subset of points x such that

$$\begin{align*}\liminf \frac{1}{n}\log\|\mathcal{A}(x,n)\| < \alpha" < \beta" < \limsup \frac{1}{n}\log\|\mathcal{A}(x,n)\| \end{align*}$$

and in particular $\Lambda _x \supset [\alpha ", \beta "]$ . Therefore, the set of points x such that $\Lambda _x = [0,\chi _1]$ is residual.

4.7 Proof of Theorem 4.10

Recall that, given an ergodic measure $\mu $ , its Lyapunov exponents, repeated according to their multiplicities, are denoted $\widetilde {\chi }_1(\mathcal {A},\mu ) \ge \cdots \ge \widetilde \chi _d(\mathcal {A},\mu )$ .

Proof. As a first case, suppose that $\widetilde \chi _1(\mathcal {A},\mu ) = \widetilde \chi _d(\mathcal {A},\mu )$ for every f-ergodic Borel probability measure $\mu $ . Consider the following ‘normalized’ cocycle

(4.7) $$ \begin{align} \hat{\mathcal{A}}(x,n) \mathrel{:=} \left| \det \mathcal{A}(x,n) \right|{}^{-\frac{1}{d}} \mathcal{A}(x,n) \, , \end{align} $$

whose determinants are $\pm 1$ . Our assumption ensures that all Lyapunov exponents of $\hat {\mathcal {A}}$ with respect to all ergodic measures are zero. By Proposition 3.2, the cocycle $\hat {\mathcal {A}}$ is completely regular with zero spectrum.

Let $\varphi _n(x) \mathrel {:=} \log |\det \mathcal {A}(x,n)|$ , and let $\mathcal {D}$ be the set of $x \in X$ such that the following two limits exist and coincide:

(4.8) $$ \begin{align} \lim_{n\to \infty} \frac{\varphi_n(x)}{n} = \lim_{n\to -\infty} \frac{\varphi_n(x)}{n} \,. \end{align} $$

We claim that $\mathcal {R}$ (the set of LP-regular points for $\mathcal {A}$ ) equals $\mathcal {D}$ . Indeed, the inclusion $\mathcal {R} \subset \mathcal {D}$ always holds, as for LP-regular points the limit (4.8) will be the sum of the Lyaponov exponents. On the other hand, since $\hat {\mathcal {A}}$ is completely regular with zero exponents, it follows from (4.7) that

(4.9) $$ \begin{align} \log \|\mathcal{A}(x,n) v\| - \frac{\varphi_n(x)}{d} = o(|n|) \end{align} $$

for all $x \in X$ and all nonzero $v \in \mathbb R^d$ , showing the reverse inclusion $\mathcal {D} \subset \mathcal {R}$ .

Applying Theorem 4.2 to f and $f^{-1}$ , we see that the set $\mathcal {D}$ (that is, $\mathcal {R}$ ) is either meager or equal to X and, in the latter case, the limits (4.8) equal some constant $\chi $ . So, when $\mathcal {R}=X$ , (4.9) says that all Lyapunov exponents of the cocycle $\mathcal {A}$ are equal to $\chi $ and, in particular, the cocycle is completely regular.

Now consider the second case, where there exists an ergodic Borel probability measure $\mu $ such that $\widetilde \chi _1(\mathcal {A},\mu )> \widetilde \chi _d(\mathcal {A},\mu )$ . Then, for $\mu $ -almost every point y,

$$\begin{align*}\lim_{n \to \infty}\frac{1}{n} \log \frac{\sigma_k(\mathcal{A}(y, n))}{\sigma_{k+1}(\mathcal{A}(y, n))} = \widetilde\chi_k(\mu) - \widetilde\chi_{k+1}(\mu)> 0 \,. \end{align*}$$

By minimality of f, the measure $\mu $ has full support, so the formula above holds for a dense set of points y. On the other hand, by Theorem 4.12, there is another dense set where the formula fails, and actually the liminf of the sequence is $0$ . Now we can apply Theorem 4.1 and conclude that the set of points z for which

$$\begin{align*}\lim_{n \to \infty}\frac{1}{n} \log \frac{\sigma_k(\mathcal{A}(z, n))}{\sigma_{k+1}(\mathcal{A}(z, n))} \end{align*}$$

exists is meager in X. In particular, the set $\mathcal {R}$ of LP-regular points is meager, as we wanted to show.

At last, we can establish our general all-or-meager dichotomy for cocycles over minimal transformations:

5.1 All points being regular does not imply complete regularity

By Theorem 4.10, if the dynamics f is minimal, then Condition (1) in the definition of complete regularity does imply Condition (2), that is, if a cocycle has the property that all points are LP-regular, then the Lyapunov exponents do not depend on the point.

On the other hand, if f is non-minimal, then Condition (1) does not imply Condition (2). For example, let f be the automorphism of the two-torus $(\mathbb R/{\mathbb {Z}})^2$ induced by the matrix $\left ( \begin {smallmatrix} 1 & 1 \\ 0 & 1 \end {smallmatrix} \right )$ , that is,

(5.1) $$ \begin{align} f(x,y) \mathrel{:=} (x+y,y) \bmod {\mathbb{Z}}^2 \,. \end{align} $$

If we take a continuous cocycle of diagonal matrices, then every point will be LP-regular. But the Lyapunov exponents will not be constant, in general. This example was pointed out before in [Reference Johnson and Zampogni35, Example 4.1].

Let us exhibit an example where f is topologically transitive. Consider the group $G \mathrel {:=} \mathrm {SL}(2,\mathbb R)$ , the lattice $\Gamma \mathrel {:=} \mathrm {SL}(2,{\mathbb {Z}})$ , and the quotient $N\mathrel {:=}{_\Gamma }\backslash ^G$ , that is, the set of right cosets $\Gamma g$ . Then N is a Hausdorff but noncompact space with respect to the quotient topology (actually it is the unit tangent bundle of the modular surface). Let $X \mathrel {:=} N \cup \{\infty \}$ be the one-point compactification of N. Define $f \colon X \to X$ by $f(\infty ) \mathrel {:=} \infty $ and

$$\begin{align*}f(\Gamma g) \mathrel{:=} \Gamma g \left( \begin{smallmatrix} 1 & 1 \\ 0 & 1 \end{smallmatrix} \right) \,. \end{align*}$$

The map f is a homeomorphism and, like the previous example (5.1), is such the Birkhoff averages of every continuous function converge pointwise (see [Reference Dani21, Theorem 5.1]); furthermore, f is topologically transitive. Arguing as before, Condition (1) does not imply Condition (2) for this map f.

The following result provides a similar example with a derivative cocycle of a diffeomorphism.

Theorem 5.1. There exists a topologically transitive diffeomorphism $f: M\to M$ of a compact connected smooth Riemannian manifold M of dimension $11$ such that every point $x\in M$ is LP-regular for the derivative cocycle and the set of Lyapunov vectors

$$ \begin{align*}\big\{(\chi_1(x), \dots, \chi_{11}(x)) \in \mathbb R^{11} \ : \ x\in M\big \} \end{align*} $$

has the power of the continuum.

For our construction we need the following objects:

1. 1. The group $G \mathrel {:=} \mathrm {SL}(3,\mathbb {R})$ , which is a simple connected Lie group; we identify $\mathrm {SL}(2,\mathbb R)$ with the subgroup consisting of matrices of the form $\left ( \begin {smallmatrix} * & * & 0 \\ * & * & 0 \\ 0 & 0 & 1 \end {smallmatrix} \right )$ .

2. 2. A co-compact lattice $\Gamma \subset G$ such that $\Gamma \cap \mathrm {SL}(2,\mathbb R)$ is a co-compact lattice in $\mathrm {SL}(2,\mathbb R)$ (see [Reference Witte Morris65, Exerc. 4, §18.6]).

3. 3. The quotient $N\mathrel {:=}{_\Gamma }\backslash ^G$ , that is, the set of right cosets $\Gamma g$ ; this is a compact smooth manifold of dimension $8$ ; we denote by $m_N$ the normalized volume in N which is induced on N by the Haar measure on the group G.

4. 4. The one-parameter subgroup of unipotent matrices

   $$ \begin{align*} u^t \mathrel{:=} \begin{pmatrix} 1&t&0\\0&1&0\\0&0&1\end{pmatrix} \end{align*} $$
   and the corresponding unipotent flow
   $$ \begin{align*} \varphi^t \colon N &\to N \\ \Gamma g &\mapsto \Gamma g u^t \, , \end{align*} $$
   which preserves the volume measure $m_N$ .

5. 5. A $C^\infty $ strictly positive function $\tau :N\to \mathbb {R}$ whose choice will be specified later.

6. 6. The Anosov flow $\{\psi ^t\}$ which is a suspension flow with constant roof function $1$ over an Anosov automorphism of the two-dimensional torus $\mathbb {T}^2$ given by the matrix $A= \left ( \begin {smallmatrix} 2&1\\1&1 \end {smallmatrix} \right )$ ; we denote by S the $3$ -dimensional suspension manifold; the flow $\{\psi ^t\}$ preserves volume in S which we denote by $m_S$ .

Denote $M \mathrel {:=} N\times S$ . Given $\alpha>0$ , consider a skew-product map $f_\alpha : M\to M$ given by:

$$ \begin{align*}f_\alpha(x, y) \mathrel{:=} (\varphi^1(x), \psi^{\tau(x)+\alpha}(y)). \end{align*} $$

For every $\alpha $ , the map $f_\alpha $ is a $C^\infty $ diffeomorphism preserving a smooth measure on M. We will show that with an appropriate choice of the parameter $\alpha $ and the function $\tau $ , the map $f_\alpha $ has the desired properties.

For every point $(x,y)\in M$ we have the following splitting of the tangent space

(5.2) $$ \begin{align} T_{(x,y)}M=T_{(x,y)} (N\times \{y\})\oplus E^{\mathrm{s}}_{f_\alpha}(x,y)\oplus E^{\mathrm{u}}_{f_\alpha}(x,y)\oplus E^0_{f_\alpha}(x,y), \end{align} $$

where

(5.3) $$ \begin{align} E^{\mathrm{s}}_{f_\alpha}(x,y)=E^{\mathrm{s}}_{\psi^1}(y) \text{ and } E^{\mathrm{u}}_{f_\alpha}(x,y)=E^{\mathrm{u}}_{\psi^1}(y) \end{align} $$

are the one-dimensional stable and, respectively, the one-dimensional unstable subspaces and

(5.4) $$ \begin{align} E^0_{f_\alpha}(x,y)=\mathrm{Span}\left\{\tfrac{\partial}{\partial t}\right\}=E^0_{\psi^t}(y). \end{align} $$

By Ratner’s equidistribution theorem [Reference Ratner52, Theorem B], for every $x \in N$ , the future and past orbits $\{\varphi ^{\pm n} x\}_{n \ge 0}$ are equidistributed with respect to a probability measure $m_x$ on N, that is, for every continuous function $h \colon N \to \mathbb R$ , we have

(5.5) $$ \begin{align} \lim_{n \to \infty} \frac{1}{n} \sum_{j=0}^{n-1} h(\varphi^{\pm j} x) = \int_N h \, dm_x \,. \end{align} $$

Furthermore (see [Reference Ratner52, p. 255]), the measures $m_x$ are ergodic, and thus $\varphi $ is pointwise ergodic in the terminology of [Reference Downarowicz and Weiss23].

The desired properties of the map $f_\alpha $ follow immediately from the following three lemmas.

Lemma 5.2. There is a smooth strictly positive function $\tau : N\to \mathbb {R}$ such that the set

$$ \begin{align*}\Bigl\{\int_{N}\tau\,dm_x: x \in N\Bigr\} \end{align*} $$

has the power of continuum.

Lemma 5.4. For any choice of a smooth function $\tau $ and any $\alpha \in (0,1)$ the splitting (5.2) is continuous, invariant under $df_\alpha $ , and for every $(x,y)\in M$ ,

(5.6) $$ \begin{align} \lim_{n\to\pm\infty}&\frac1{n}\log\|Df_\alpha^n (v)\|=0 \quad \text{for every } v \in T_{(x,y)}(N\times\{y\}) \, ,\end{align} $$

(5.7) $$ \begin{align} \lim_{n\to\pm\infty}&\frac1{n}\log\|Df_\alpha^n|E^{\mathrm{s}}_{f_\alpha}(x,y)\| =-(\log\lambda)\int_{N}(\tau+\alpha)\,dm_x \, , \end{align} $$

(5.8) $$ \begin{align} \lim_{n\to\pm\infty}&\frac1{n}\log\|Df_\alpha^n|E^{\mathrm{u}}_{f_\alpha}(x,y)\| =\phantom{-}(\log\lambda)\int_{N}(\tau+\alpha)\,dm_x \, , \end{align} $$

(5.9) $$ \begin{align} \lim_{n\to\pm\infty}&\frac1{n}\log\|Df_\alpha^n|E^0_{f_\alpha}(x,y)\|=0 \, , \end{align} $$

where $\lambda>1$ is the top eigenvalue of the matrix A. In particular, every point $(x,y)\in M$ is LP-regular.

We proceed with proofs of these three lemmas.

Proof of Lemma 5.2.

Let $\widetilde {G}$ be the subgroup of G consisting of the matrices of the form $\left ( \begin {smallmatrix} * & * & 0 \\ * & * & 0 \\ 0 & 0 & 1 \end {smallmatrix} \right )$ , that is, the standard copy of $\mathrm {SL}(2,\mathbb R)$ in $\mathrm {SL}(3,\mathbb R)$ . Let $\pi \colon G \to N$ be the quotient map $\pi (g) \mathrel {:=} \Gamma g$ and let $N_0 \mathrel {:=} \pi (\widetilde G)$ . Since the unipotent subgroup $(u^t)$ is contained in $\widetilde G$ , the set $N_0$ is invariant under the unipotent flow $\{\varphi ^t\}$ . Recall that our lattice $\Gamma < G$ was chosen so that $\widetilde \Gamma \mathrel {:=} \Gamma \cap \widetilde G$ is a co-compact lattice in $\widetilde G$ . Therefore, the quotient $\widetilde N\mathrel {:=}{_{\widetilde \Gamma }}\backslash ^{\widetilde G}$ is a compact $3$ -manifold which is actually the unit tangent bundle of a compact hyperbolic surface. We define a unipotent flow on $\widetilde N$ by the formula $\widetilde {\varphi }^t (\widetilde \Gamma \widetilde g) \mathrel {:=} \widetilde \Gamma \widetilde g u^t$ ; this is actually the horocycle flow on the unit tangent bundle. Define a map $\widetilde {N} \to N_0$ by $\widetilde \Gamma \widetilde {g} \mapsto \Gamma \widetilde {g}$ . This map is a homeomorphism, and it conjugates the flows $\widetilde {\varphi }^t$ and $\varphi ^t|_{N_0}$ . By a theorem of Furstenberg, the horocycle flow $\widetilde {\varphi }^t$ is strictly ergodic (i.e., minimal and uniquely ergodic); furthermore, the time- $1$ map $\widetilde {\varphi }^1$ is also strictly ergodic: see [Reference Ellis and Perrizo24]. We conclude that the submanifold $N_0$ of N supports a unique $\varphi ^1$ -invariant measure, which we call $m_{N_0}$ .

Next, consider the one-parameter subgroup of G formed by the matrices of the form

$$\begin{align*}a^s \mathrel{:=} \begin{pmatrix} e^s & 0 & 0 \\ 0 & e^s & 0 \\ 0 & 0 & e^{-2s} \end{pmatrix} \, , \quad s \in \mathbb R \,. \end{align*}$$

Note that $a^s \widetilde {g} = \widetilde {g} a^s$ for every $\widetilde {g} \in \widetilde {G}$ , and in particular, $a^s u^t = u^t a^s$ . As a consequence, the flow

$$ \begin{align*} \zeta^s \colon N &\to N \\ \Gamma g &\mapsto \Gamma g a^s \,; \end{align*} $$

commutes with the unipotent flow, that is,

$$\begin{align*}\zeta^s \circ \varphi^t = \varphi^t \circ \zeta^s \,. \end{align*}$$

Therefore, the submanifold $N_s \mathrel {:=} \zeta ^s(N_0)$ supports a unique $\varphi ^1$ -invariant measure, namely ${m_{N_s} \mathrel {:=} (\zeta ^s)_*(m_{N_0})}$ . For every point x in $N_s$ , the measure $m_x$ defined by (5.5) coincides with $m_{N_s}$ .

We have constructed a continuous curve $s \mapsto m_{N_s}$ in the space of $\varphi ^1$ -invariant Borel probability measures on N (endowed with the weak topology). We will prove that this curve is nonconstant. Once we do that, the lemma will immediately follow, as it will be sufficient to take any $C^\infty $ strictly positive function $\tau $ for which the function $s \mapsto \int _{N}\tau \,dm_{N_s}$ is nonconstant.

Note that $a^s \not \in \widetilde G$ for all nonzero s. Since the projection $\pi \colon G \to N$ is a covering map and $\pi (\widetilde {G})$ is the compact set $N_0$ , it follows that $\pi (a^s) \not \in N_0$ for all sufficiently small $s \neq 0$ . But $\pi (a_s) \in N_s$ , and so $N_s \neq N_0$ for all sufficiently small $s \neq 0$ . Since $N_s$ is the support of the measure $m_{N_s}$ , this proves that the curve $s \mapsto m_{N_s}$ is nonconstant, as we wanted to show.

Proof of Lemma 5.4.

Equality (5.6) follows from the definition of the map $f_\alpha $ , and the following two facts: (a) all the Lyapunov exponents of $\varphi ^t$ are zero; and (b) the Lyapunov exponent of the Anosov flow in its flow direction is zero. Equality (5.9) follows from (5.4) and the fact that the function $\tau $ is bounded. To prove equality (5.7), observe that for every $n>0$ ,

$$\begin{align*}f^n_\alpha(x,y)=(\varphi^n(x), \psi^{n\alpha+\tau_n(x)}(y)), \end{align*}$$

where $\tau _n(x)=\sum _{k=0}^{n-1}\tau (\varphi ^k(x))$ . By property (5.5), for every $x \in N$ we have

$$\begin{align*}\tau_n(x)=n\int_{N}\tau \,dm_x + o(n). \end{align*}$$

The desired equality now follows from the fact that every $v\in E^{\mathrm {s}}(y)$ contracts under the Anosov flow by a constant factor $\lambda ^t$ . The proof of equality (5.8) is similar.

We proved the lemmas and, as explained before, this completes the proof of Theorem 5.1.

5.2 Agreement of Lyapunov spectra does not imply complete regularity

In this subsection we discuss the problem of whether Condition (2) alone is sufficient for complete regularity. Strictly speaking, as stated this condition subsumes the first and so the problem seems hollow. However, we may ask the following meaningful question: Suppose every two points that are LP-regular have the same Lyapunov spectrum. Does it follow that every point is LP-regular?

The answer is negative, as Example 3.12 illustrates. In that example, most points are wandering; in fact, there is no invariant Borel probability measure whose support is X. The next result provides a more interesting example:

In fact, the map f in our example is uniquely ergodic, and so, by Theorem 4.10, points that are not LP-regular for the cocycle $\mathcal {A}$ form a residual subset of X.

We do not know if there exist examples of cocycles over a hyperbolic base dynamics satisfying the conclusions of Theorem 5.5. If that is the case, the matrix maps cannot be Hölder continuous, by Theorem 3.13.

Another question is the following: Assuming Condition (1) in the definition of complete regularity, can Condition (2) be weakened to the following condition (2’)?

1. 2’. all ergodic measures have the same Lyapunov spectrum.

We now turn to the proof of Theorem 5.5.

If $f \colon X \to X$ is a homeomorphism of a compact metric space X, we say that f is a Veech-like map if it is strictly ergodic (i.e., minimal and uniquely ergodic) and $f^2$ is minimal but not uniquely ergodic. Veech-like maps were first constructed by Veech [Reference Veech62].

Walters [Reference Walters, Arnold and Wihstutz63] has shown that given a Veech-like map f, one can construct a nonuniformly hyperbolic cocycle with base f. We will construct a specific Walters cocycle with some additional properties which will be instrumental in our proof of Theorem 5.5. For this cocycle we need the base Veech-like map constructed in [Reference Bochi9, Sec. 4] (the idea of this construction is due to Scott Schmieding).

Proof of Theorem 5.5.

Let $e_1, e_2, \dots $ be words in the alphabet $\mathsf {A} \mathrel {:=} \{ \mathord {\uparrow }, \mathord {\downarrow } , 0 \}$ , defined in a recursive manner as follows:

(5.10) $$ \begin{align} \notag e_{1} &\mathrel{:=} \mathord{\uparrow} \mathord{\downarrow} \, , \\ e_{k+1} &\mathrel{:=} e_k^{k^2+1} \, 0 \, e_k \, \overline{e_k}^{k^2+1} \, 0 \, \overline{e_k} \, , \end{align} $$

where $\overline {e_k}$ denotes the word obtained from $e_k$ by switching the arrows. Let X be the subset of $\mathsf {A}^{\mathbb {Z}}$ formed by those bi-infinite sequences $\omega =(\omega _i)_{i\in {\mathbb {Z}}}$ such that for every $n> 0$ , the word $\omega _{-n}\omega _{-n+1}\cdots \omega _n$ occurs as a subword (i.e., a factor) of some $e_k$ . According to [Reference Bochi9, Theorem 4.1], X is a closed, shift-invariant subset of $\mathsf {A}^{\mathbb {Z}}$ , and if $f \colon X \to X$ is the restricted shift, then f is a Veech-like map. Define a continuous matrix valued map

$$\begin{align*}A(\omega) \mathrel{:=} \begin{pmatrix} 0 & e^{\varphi(\omega)} \\ e^{-\varphi(\omega)} & 0 \end{pmatrix} \quad \text{where} \quad \varphi(\omega) \mathrel{:=} \begin{cases} \phantom{-}1 &\text{if } \omega_0 = \mathord{\uparrow} \, , \\ -1 &\text{if } \omega_0 = \mathord{\downarrow} \, , \\ \phantom{-}0 &\text{if } \omega_0 = 0 \,. \end{cases} \end{align*}$$

Then [Reference Bochi9, Theorem 4.11] states that the corresponding cocycle $\mathcal {A}$ has nonzero Lyapunov exponents (with respect to the unique f-invariant measure), and it is not uniformly hyperbolic. Note that for any $\omega \in X$ and $n>0$ ,

$$ \begin{align*} \frac{1}{n}\log \left\|\mathcal{A}(\omega,n) \right\| &= \frac{1}{n} \left| \sum_{j=0}^{n-1}(-1)^j \varphi(f^j(\omega)) \right| \, , \\ \frac{1}{n} \log \left\|\mathcal{A}(\omega,-n) \right\| &= \frac{1}{n} \left| \sum_{j=-n}^{-1}(-1)^j \varphi(f^j(\omega)) \right| \,. \end{align*} $$

If $\omega $ is a LP-regular point, then the two sequences above must converge to the same limit as $n \to \infty $ . However, according to [Reference Bochi9, Lemma 4.12], this common limit, whenever it exists, is a number $c>0$ independent of $\omega $ . Therefore all LP-regular points of our Walters cocycle have the same Lyapunov exponents c and $-c$ .

At the same time, since f is uniquely ergodic and $\mathcal {A}$ is not uniformly hyperbolic, we must be in case (3) of Proposition 3.8. Consequently, $\mathcal {A}$ cannot be completely regular (even though, as we saw above, all LP-regular points have the same spectra).