# Robust vanishing of all Lyapunov exponents for iterated function systems

## Abstract

Given any compact connected manifold \(M\), we describe \(C^2\)-open sets of iterated functions systems (IFS’s) admitting fully-supported ergodic measures whose Lyapunov exponents along \(M\) are all zero. Moreover, these measures are approximated by measures supported on periodic orbits. We also describe \(C^1\)-open sets of IFS’s admitting ergodic measures of positive entropy whose Lyapunov exponents along \(M\) are all zero. The proofs involve the construction of non-hyperbolic measures for the induced IFS’s on the flag manifold.

## 1 Introduction

### 1.1 The hunt for (non-)hyperbolic measures

Since the Multiplicative Ergodic Theorem of Oseledets [19], the Lyapunov exponents of invariant probability measures are central in differentiable dynamics. As Oseledets reveals in the first paragraph of his celebrated paper, he was already interested in the dynamical implications of non-zero Lyapunov exponents. Many of these implications, at least in the case of volume-preserving dynamics, were discovered by Pesin during the mid-seventies (see e.g. [20]). Later, Katok [15] obtained strong consequences in the non-conservative case. Roughly speaking, the absence of zero Lyapunov exponents permits to recover many dynamical properties from uniformly hyperbolicity. We refer the reader to the book [5] for much information about the dynamics of systems without zero Lyapunov exponents, which are called *nonuniformly hyperbolic*.

However, nonuniform hyperbolicity is not necessarily ubiquitous. In the conservative situation, KAM theory gives rise to elliptic behavior which is robust in high regularity. In lower regularity, zero Lyapunov exponents can also occur generically (see [6]).

Outside the conservative setting, we consider the general problem of determining which are the possible Lyapunov spectra of the ergodic invariant probabilities of a given dynamical system.

Topological–geometrical properties of the dynamics impose restrictions on the Lyapunov exponents; to give an obvious example, if the system is uniformly hyperbolic (say, Anosov) then no zero Lyapunov exponents can occur.^{1} The converse of the implication above is false: there exist smooth systems whose Lyapunov exponents (with respect to all ergodic invariant measures) are uniformly bounded away from \(0\), but are not uniformly hyperbolic: see [4, 8, Remark 1.1]. Anyway, these examples seem to be very special, and it is natural to ask in what generality the lack of uniform hyperbolicity of a system forces the appearance of non-hyperbolic measures, that is, measures admitting at least one vanishing Lyapunov exponent. We are especially interested in the case that non-hyperbolic measures occur in a robust way, and we would like to understand further properties of those measures (e.g. multiplicity of zero exponents, support, approximation by periodic orbits, entropy, etc.).

An important result in this direction was obtained by Kleptsyn and Nalksy [16], who gave \(C^1\)-robust examples of diffeomorphisms having ergodic non-hyperbolic measures. Their examples exist on any compact manifold of dimension at least \(3\), and are partially hyperbolic with integrable circle fibers. The construction is based on their earlier paper joint with Gorodetski and Ilyashenko [11], which obtains similar results for iterated function systems (IFS’s) of the circle.

These ideas have been used in [10] to determine properties of homoclinic classes that imply the existence of non-trivial non-hyperbolic measures, however under \(C^1\)-generic assumptions. In [7], the construction was tuned to enlarge the supports of these measures: they can be taken as the whole homoclinic class.

The non-hyperbolic measures in all results above are obtained as limits of sequences of measures supported on periodic orbits whose central Lyapunov exponent converges to zero. Therefore the non-hyperbolicity of the system is detected by its periodic orbits. The general principle that periodic orbits carry a great amount of information about the dynamics has been successful in many occasions; see e.g. [21] in the uniformly hyperbolic context, [15] for nonuniformly hyperbolic context, and [1, 18] in the \(C^1\)-generic context. It is thus natural to reformulate the previous problems focusing on the simplest class of invariant measures, namely those supported on periodic orbits.

Another common feature of the non-hyperbolic measures from the results above is that they have only one vanishing Lyapunov exponent. Since there are open sets of diffeomorphisms with nonhyperbolic subbundles of any given dimension, one wonders if these systems have ergodic measures with multiple zero exponents. There is a clear difficulty in passing to higher dimensions: we lose the commutativity of the products of central derivatives, therefore also the losing the continuity of the exponents. Those properties were crucial in the constructions above.

Finally, we observe that all these non-hyperbolic measures have zero entropy.

- (a)
We construct \(C^2\)-open sets of IFS’s having ergodic measures with only zero exponents along \(M\), full support, and approximable in a strong sense by measures supported on periodic orbits.

- (b)
We construct \(C^1\)-open sets of IFS’s having ergodic measures with only zero exponents along \(M\), and positive entropy.

### 1.2 Precise statements of the main results

An *iterated function system*, or *IFS*, is simply a finite collection \(G=(g_0, \ldots , g_{\ell -1})\) of (usually continuous) self-maps of a (usually compact) space \(M\). Then we consider the semigroup generated by these transformations. An IFS can be embedded in a single dynamical system, the *1-step skew-product*\(\varphi _G :\ell ^\mathbb{Z }\times M\rightarrow \ell ^\mathbb{Z }\times M\) over the full shift \(\sigma \) on \(\ell ^\mathbb{Z }=\{0,\ldots ,\ell -1\}^\mathbb{Z }\), which is defined by \(\varphi _G(\omega , p)= (\sigma (\omega ), g_{\omega _0}(p))\).

*fibered Lyapunov exponents*, which are the values that can occur as limits

Our first result is as follows:

**Theorem 1**

Let \(M\) be a compact connected manifold without boundary. Then there exist an integer \(\ell \ge 2\) and an open set \(\mathcal U \) in \((\mathrm{Diff }^2(M))^\ell \) such that for any \(G = (g_0,\ldots ,g_{\ell -1})\in \mathcal{U }\) the \(1\)-step skew-product \(\varphi _G\) has an ergodic invariant measure \(\mu \) whose support is the whole \(\ell ^\mathbb{Z }\times M\) and whose fibered Lyapunov exponents all vanish. Moreover, the measure \(\mu \) is the weak-star limit of a sequence of \(\varphi _G\)-invariant measures \(\mu _n\), each of these supported on a periodic orbit.

As we will see, our strategy consists on proving a stronger version of Theorem 1, concerning IFS’s on flag manifolds – see Theorem 3.

Another main result is the following:

**Theorem 2**

- (a)
All Lyapunov exponents (tangent to \(M\)) of all invariant probabilities with support contained in \(\Lambda \) are zero.

- (b)
The restriction of \(\varphi _G\) to \(\Lambda _G\) has positive topological entropy.

The last assertion follows immediately from the Variational Principle.

Compared to Theorem 1, Theorem 2 improves the robustness class from \(C^2\) to \(C^1\). The non-hyperbolic measures produced by Theorem 2 have the additional property of positive entropy, but clearly do not have full support. Moreover, we do not know if those measures can be approximated by measures supported on periodic orbits.

Theorem 2 has a simpler proof than Theorem 1. The definition of the \(C^2\)-open set \(\mathcal{U }\) from Theorem 1 involves basically two conditions, “maneuverability” and “minimality”, while the \(C^1\)-open set \(\mathcal{V }\) from Theorem 2 only requires maneuverability. In particular, the sets \(\mathcal{U }\) and \(\mathcal{V }\) have nonempty intersection.

### 1.3 Questions

In view of our results extending [11], it is natural to expect a corresponding generalization of [16], that is, the existence of open examples of partially hyperbolic diffeomorphisms with multidimensional center so that there are measures all whose central exponents vanish.

We list other questions, mainly about IFS’s:

**Question 1**

Are there \(C^1\)- or \(C^2\)-robust examples with non-hyperbolic measures of full support and positive entropy?

**Question 2**

It is possible to improve Theorem 1 so that the set of measures that satisfies the conclusions is dense (or generic) in the weak-star topology?

Consider IFS’s of volume-preserving or symplectic diffeomorphisms. (See [17] for results and problems about such systems.) The proof of Theorem 2 can be easily adapted for the volume-preserving case.

**Question 3**

Does the analogue of Theorem 1 hold true in conservative contexts? A more interesting and difficult question is whether the measure \(\mu \) in the theorem can be taken of the form \(\mu = \mu _0 \times m\), where \(\mu _0\) is a shift-invariant measure and \(m\) is the volume on the fibers \(M\)?

## 2 Outlines of the proofs

We first outline the proof of Theorem 1, explaining the main ingredients and difficulties of it, and discussing the novelties in comparison with [11]. We also state a stronger result which implies Theorem 1.

Later, we will explain how the tools developed to prove Theorem 1 can be applied to yield the easier Theorem 2.

### 2.1 Ergodic measures as limit of periodic measures

The starting point is Lemma 2.1 below, which gives sufficient conditions for a sequence of invariant probability measures supported on periodic orbits to converge to an ergodic measure, and also permits to determine the support of the limit measure. Let us state this lemma precisely.

*shadows*\(\mathcal{O }\)

*during a proportion*\(1-\kappa \)

*of the time*if

**Lemma 2.1**

The lemma is just a rephrasing of Lemma 2.5 from [7], which in its turn is a refined version of Lemma 2 from [11].

### 2.2 The main difficulty with higher dimensions

We want to find a sequence \((\mathcal{O }_n)\) of periodic orbits for the skew-product map \(\varphi _G\) that fits in the situation of Lemma 2.1 and such that the resulting limit measure has the desired properties of zero (fibered) Lyapunov exponents and full support.

In the paper [11], which deals with the one-dimensional case (i.e., \(M\) is the circle), the sequence of periodic orbits is constructed in such a way that the Lyapunov exponent converges to zero. The construction is recursive: each new orbit \(\mathcal{O }_{n+1}\) is chosen in order to *improve* the previous one \(\mathcal{O }_{n}\), in the sense that the new Lyapunov exponent is closer to zero. It is easy to modify their construction so to ensure that each new orbit is denser in the ambient space, and thus, as we now know, obtain full support for the limit measure. There are practically no requirements on starting orbit \(\mathcal{O }_1\): it needs only to be attracting. We call this the *bootstrapping procedure*, because it starts from nothing and by successive improvements eventually achieves its goal. We will give more details about it later (Sect. 2.4).

With Lemma 2.1 we can guarantee ergodicity and full support of the limit measure, and so we are left to control its Lyapunov exponent. In the one-dimensional situation of [11], the Lyapunov exponent is given by an integral and so its dependence on the measure is continuous with respect to the weak-star topology. Since the Lyapunov exponent along the sequence provided by the bootstrapping procedure converges to zero, we obtain a limit measure with zero Lyapunov exponent, as desired.

However, if \(M\) has dimension \(d > 1\) then the Lyapunov exponents are no longer given by integrals. Worse still, they can indeed be discontinuous as functions of the measure; the best that can be said is that the top Lyapunov exponent is upper semicontinuous, while the bottom Lyapunov exponent is lower semicontinuous. So, even if all Lyapunov exponents along the orbit \(\mathcal{O }_n\) converge to zero as \(n \rightarrow \infty \), there is no guarantee that the limit measure will have zero Lyapunov exponents.

*Remark 2.2*

Usually, semicontinuity helps when we are trying to produce equal Lyapunov exponents (as e.g. in [6]). We could use semicontinuity here if we were able to apply Lemma 2.1 with \(1-\kappa _n\) arbitrarily small, but this is not the case. Incidentally, we can apply the lemma with \(\varepsilon _n\) arbitrarily small (see the proof of Theorem 4.1 in Sect. 5), but we make no use of this fact.

We overcome this difficulty by working with a skew-product on a larger space called the *flag bundle*. This permits us to recover continuity of Lyapunov exponents and thus prove Theorem 1 by the same bootstrapping procedure. Passing to the flag bundle, however, has a price: we lose one order of differentiability, and this is basically why our results need \(C^2\) regularity, as opposed to the \(C^1\) regularity required by [11].

### 2.3 Flag dynamics

*flag bundle*of \(M\), that is, the set of \((x, F_1, \ldots , F_d)\) where \(x \in M\) and \(F_1 \subset \cdots \subset F_d\) are nested vector subspaces of the tangent space \(T_x M\), with \(\mathrm{dim }F_i = i\). Such a sequence of subspaces is called a

*flag*on \(T_x M\). Then \(\mathcal{F }M\) is a compact manifold, and the natural projection \(\mathcal{F }M \rightarrow M\) defines a fiber bundle. Every \(C^r\) diffeomorphism \(g :M \rightarrow M\) can be lifted to a \(C^{r-1}\) diffeomorphism \(\mathcal{F }g :\mathcal{F }M \rightarrow \mathcal{F }M\) in the natural way, namely

**Theorem 3**

Let \(M\) be a compact connected manifold without boundary. Then there exist an integer \(\ell \ge 2\) and an open set \(\mathcal U \) in \((\mathrm{Diff }^2(M))^\ell \) such that for any \(G = (g_0,\ldots ,g_{\ell -1})\in \mathcal{U }\) the \(1\)-step skew-product \(\mathcal{F }\varphi _G\) has an ergodic invariant measure \(\nu \) whose support is the whole \(\ell ^\mathbb{Z }\times \mathcal{F }M\) and whose fibered Lyapunov exponents all vanish. Moreover, the measure \(\nu \) is the weak-star limit of a sequence of \(\mathcal{F }\varphi _G\)-invariant measures \(\nu _n\), each of these supported on a periodic orbit.

Theorem 1 follows; let us see why.

- (a)
The fibered Lyapunov exponents of \(\nu \) vary continuously with respect to \(\nu \), among ergodic measures: if \(\nu _n\) are ergodic measures converging to an ergodic measure \(\nu \), then the fibered Lyapunov exponents of \(\nu _n\) converge to those of \(\nu \).

- (b)
All the fibered Lyapunov exponents of (ergodic) \(\nu \) vanish if and only if all the fibered Lyapunov exponents of \(\mu \) vanish.

*Remark 2.3*

In fact, each fibered Lyapunov exponent of \(\mu \) is also a fibered Lyapunov exponent of \(\nu \) (see Example 3.9). Notice that there is no contradiction with the aforementioned discontinuity of the fibered Lyapunov exponents with respect to \(\mu \). Indeed, a convergent sequence of \(\varphi _G\)-invariant measures \(\mu _n\) whose limit is ergodic may fail (even after passing to a subsequence) to lift to a converging sequence of \(\mathcal{F }\varphi _G\)-invariant measures \(\nu _n\) whose limit is ergodic.

On one hand, property (b) makes Theorem 1 a corollary of Theorem 3. On the other hand, property (a) extirpates the difficulty explained before, and so allows us to prove Theorem 3 by the bootstrapping procedure, as we explain next.

### 2.4 The bootstrapping procedure on the flag bundle

An important feature of the bootstrapping procedure of [11] is that each periodic orbit must be hyperbolic attracting (along the fiber); this permits us to find each new orbit as a fixed point of a contraction. So let us see how to detect contraction.

Hence if \(x\) is a fixed point of diffeomorphism \(g :M \rightarrow M\) and the moduli of the eigenvalues of \(Dg(x)\) are all different and less than \(1\), then there is a unique flag \(\mathfrak{f }\) on \(T_x M\) such that \((x,\mathfrak{f })\in \mathcal{F }M\) is an attracting fixed point for \(\mathcal{F }g\). The converse is true: all hyperbolic attracting fixed points of \(\mathcal{F }g\) appear in this way.

Now let us sketch how to carry out a step in the bootstrapping procedure. Of course, some conditions are needed for the IFS \(G\); we will see along the way how these conditions should look like.

The orbit of \((\tilde{\omega }, \tilde{\xi })\) closely shadows the orbit of \((\omega , \xi )\) for most of the time.

The fibered Lyapunov exponents of the new orbit are still negative and different but closer to zero than those of the initial orbit.

The new orbit is “denser” in the ambient space than the previous orbit.

Take a small ball \(B_0\) around \(\xi \) in \(\mathcal{F }M\). Let \(n\) be very large. The ball \(B_0\) is mapped by \(h_1^n\) into a very small ball \(B_1\) around \(\xi \).

Then we select a long sequence of maps \(g_{s_1}, g_{s_2}\), ..., \(g_{s_m}\) in \(G\) such that the derivative of \(h_2 := \mathcal{F }g_{s_m} \circ \cdots \circ \mathcal{F }g_{s_1}\) at \(\xi \) is strongly expanding. This expansion, however, is not strong enough to compensate the previous contraction, so \(h_2\) sends \(B_1\) into a ball \(B_2\) much bigger than \(B_1\) but still much smaller than \(B_0\). (Actually the expansion factors must be chosen more carefully, but we will leave the details for later.) All this is possible if the set \(G\) has a property that we call

*maneuverability*. (See Sect. 4 for a precise definition.)- Next, we select maps \(\mathcal{F }g_{t_1}, \mathcal{F }g_{t_2}\), ..., \(\mathcal{F }g_{t_k}\), such that:The length \(k\) of this part must be large, but it will be much smaller than either \(n\) or \(m\), so there is plenty of space for \(B_3\) to fit inside \(B_0\). (Actually the tour must be made on \(\ell ^\mathbb{Z }\times \mathcal{F }M\), but this is not difficult to obtain.) This “
– the union of successive images of the ball \(B_2\) gets close to any point in \(\mathcal{F }M\) (we say that this orbit makes a “tour”);

– the last image, which is \(h_3(B_2)\) where \(h_3 := \mathcal{F }g_{t_k} \circ \cdots \circ \mathcal{F }g_{t_1}\) is contained in \(B_0\) (we say that the orbit “goes home”).

*tour and go home*” phase is possible if the IFS \(\mathcal{F }G\) is*positively minimal*on \(\mathcal{F }M\) (see Sect. 3.1). Since the composed map \(h_3 \circ h_2 \circ h_1^n\) sends the ball \(B_0\) inside itself, it has a fixed point \(\tilde{\xi }\). Using that the derivatives of the maps \(\mathcal{F }g_s\) are uniformly continuous, we are able to show that \(\tilde{\xi }\) is an attracting fixed point. Moreover, we can show that the \(h_2\) part has the effect of making the Lyapunov exponents closer to zero. The effect of \(h_3\) in the Lyapunov exponents is negligible, because the length \(k\), despite big, is much smaller the length \(pn+m\) of \(h_2 \circ h_1^n\).

So we find the desired periodic point \((\tilde{\omega }, \tilde{\xi })\), where \(\tilde{\omega }\) consists of infinite repetitions of the word \((\omega _0 \ldots \omega _{p-1})^n s_1 \ldots s_m t_1 \ldots t_k\).

We have sketched how the sequence of periodic orbits is produced. Of course, the actual construction is more quantitative, in order to guarantee that the Lyapunov exponents indeed converge to zero, and that the requirements of Lemma 2.1 are indeed fulfilled.

The last and relatively easy step of the proof of Theorem 3 is to show that there is a nonempty open subset of \((\mathrm{Diff }^2(M))^\ell \) (for sufficiently large \(\ell \)) where the prerequisites explained above (maneuverability and positive minimality on the flag manifold) are satisfied. This is done in Sect. 10.

In conclusion, the proof of Theorem 3 follows an strategy very similar to that of [11]. However, the control of the Lyapunov exponents is much more delicate because the derivatives do not commute. Here the flags come to our aid once again: there is a distinctive feature of flag manifolds that permits us to put all the derivatives in a standard *triangular* form, and therefore neutralize the non-commutativity effects.

### 2.5 A \(C^1\) construction with positive entropy but smaller support

Actually we impose some redundancy on the maneuverability property, which easily implies positive topological entropy.

### 2.6 Organization of the rest of paper

Section 3 contains the preliminary definitions and properties. In Sect. 4 we state explicit conditions (maneuverability and positive minimality on the flag manifold) on the IFS \(G\) that guarantee the existence of the measure \(\nu \) satisfying the conclusions of Theorem 3. In Sect. 5 we state Proposition 5.1, which makes precise the input and output of the recursive construction of periodic orbits; then, assuming this proposition, we prove that the bootstrapping procedure yields the desired results. In Sects. 6 and 7 we prove technical consequences of minimality and maneuverability for later use. In Sect. 8 we explain how maps on flag manifolds give rise naturally to triangular matrices, and why this is useful. Section 9 uses the material of all previous sections to prove the main Proposition 5.1. In Sect. 10 we prove that the existence of nonempty open sets of IFS’s satisfying the prerequisites of maneuverability and positive minimality on the flag manifold. Finally, in Sect. 11 we prove Theorem 2.

## 3 Preliminaries

In this section we collect definitions and basic properties about iterated function systems (IFS’s), flag manifolds and bundles, and the related Lyapunov exponents. Section 8 gives deeper extra information that is needed in the end of the proof.

### 3.1 Iterated function systems

If \(N\) is a compact metric space and \(h_0\), ..., \(h_{\ell -1}\) are homeomorphisms of \(N\), we denote by \(\langle H \rangle \) the semigroup generated by \(H\), i.e., the set of all maps \(h_{s_m} \circ \cdots \circ h_{s_1}\), where \(s_1\), ..., \(s_m \in \{0,\ldots ,\ell -1\}\). The concatenation \(w = s_1 \ldots s_m\) is called a *word* of *length*\(m\) on the *alphabet*\(\{0,\ldots ,\ell -1\}\); we then denote \(h_{[w]} = h_{s_m} \circ \cdots \circ h_{s_1}\).

An *iterated function system* (or *IFS*) is simply a semigroup \(\langle H \rangle \) with a marked set \(H\) of generators.

The *H-orbit* of \(x\in N\) is the set of the points \(h(x)\) where \(h\) runs on \(\langle H \rangle \). We say that \(H\) (or \(\langle H \rangle \)) is *positively minimal* if for every \(x\in N\) the \(H\)-orbit of \(x\) is dense in \(M\).^{2}

*1-step skew-product*

*Remark 3.1*

- (a)
If the IFS \(\langle H \rangle \) is positively minimal then \(\varphi _H\) is transitive on \(\ell ^\mathbb{Z }\times N\).

- (b)
The IFS \(\langle H \rangle \) is positively minimal if and only if for every point \(z=(\omega ,x) \in \ell ^\mathbb{Z }\times N\), the union of the positive iterates of the

*local strong unstable manifold*\(W_\mathrm{loc }^\mathrm{uu }(z) := \big \{ (\tilde{\omega }, x) ; \; \tilde{\omega }_i = \omega _i \text { for all } i<0 \big \} \) under \(\varphi _H\) are dense in \(\ell ^\mathbb{Z }\times N\).

If \(r, k\) are positive integers and \(s_{-r}, s_{-r+1}, \ldots , s_k\) are symbols in \(\{0,\ldots ,\ell -1\}\), then the *cylinder*\([\![s_{-r} \ldots ; s_0 \ldots s_k ]\!]\) is the set of \((t_n)_n \in \ell ^\mathbb{Z }\) such that \(t_n = s_n\) for all \(n\) with \(-r \le n \le k\). Cylinders \([\![s_{-r}\ldots ; ]\!]\) and \([\![;s_0\ldots s_k ]\!]\) are defined analogously.

### 3.2 Lyapunov exponents

It is convenient to consider Lyapunov exponents in the general setting of bundle automorphisms. Details can be found in the book [2].

Let \(X\) be a compact metric space. Let \(V\) be a vector bundle of rank \(d\) over \(X\), and let \(\pi :V \rightarrow X\) be the bundle projection. We endow \(V\) with a Riemannian norm, that is, a continuous choice of an inner product on each fiber.

Let \(T:X\rightarrow X\) be a continuous map. Let \(S :V \rightarrow V\) be a vector bundle morphism over \(T\), i.e., a continuous map such that \(\pi \circ S = T \circ \pi \) which is a linear map on each fiber. For \(x \in X\) and \(n \in \mathbb{N }\), the restriction of \(S^n\) to the fiber \(V_x := \pi ^{-1}(x)\) gives a linear map \(A^{(n)}_S(x) :V_x \rightarrow V_{T^n x}\). We write \(A_S(x) = A^{(1)}_S(x)\) and so \(A^{(n)}_S(x) = A_S(T^{n-1} x) \circ \cdots \circ A_S(x)\).

From now on, let us assume that each linear map \(A(x)\) is an isomorphism.

In the case that the vector bundle is trivial (i.e., \(V = X \times E\) where \(E\) is a vector space and \(\pi \) is the projection on the second factor) then the morphism \(S\) is also called a *linear cocycle*, and the map \(A = A_S :X \rightarrow \mathrm{GL}(E)\) is called the *generator* of the cocycle.

Sometimes, with some abuse of terminology, we also call a vector bundle morphism a *cocycle*.

^{3}By Oseledets Theorem, for \(\mu \)-almost every point \(x\in X\) and every vector \(v \in V_x \backslash \{0\}\), the

*Lyapunov exponent*

*Example 3.2*

Suppose that \(X = M\) is a smooth manifold of dimension \(d\), and \(V = TM\) is the tangent bundle of \(M\). Let \(T = g\) be a diffeomorphism of \(M\), and let \(S = Dg\) be the derivative of \(g\). This is sometimes called the “*derivative cocycle*”.

*nonlinear cocycle*. If \(\nu \) is a probability on \(X \times M\) that is \(\varphi \)-invariant and ergodic, then the

*fibered Lyapunov exponents*of the nonlinear cocycle \(\varphi \) with respect to \(\nu \) are the values that can occur as limits

*Example 3.3*

Consider the \(1\)-step skew-product \(\varphi _H\) on \(\ell ^\mathbb{Z }\times N\) defined in Sect. 3.1. If \(N\) is a smooth manifold and each generator \(h_s\) is a diffeomorphism then \(\varphi _H\) can be viewed as a nonlinear cocycle, and each ergodic invariant measure on \(\ell ^\mathbb{Z }\times N\) gives rise to fibered Lyapunov exponents.

Analogously, we can also consider “*nonlinear cocycles*” where the product \(X \times M\) is replaced by a fiber bundle over \(X\) with typical fiber \(M\) and structure group \(\mathrm{Diff }^1(M)\). For these nonlinear cocycles, we also consider the fibered Lyapunov exponents.

### 3.3 Flag manifolds and linearly induced maps

Let \(E\) be a real vector space of dimension \(d\). A *flag* on \(E\) is a sequence \(\mathfrak{f }= (F_i)_{i=1,\ldots ,d}\) of subspaces \(F_1 \subset F_2 \subset \cdots \subset F_d = E\) such that \(\mathrm{dim }F_i = i\) for each \(i\). If each subspace \(F_i\) is endowed with an orientation then we say that \(\mathfrak{f }\) is an *oriented flag*. The set of flags (resp. oriented flags) on \(E\) will be denoted by \(\mathcal{F }E\) (resp. \({\check{\mathcal{F }}}E\)), and called the *flag manifold* (resp. *oriented flag manifold*) of \(E\); indeed a differentiable structure is defined below.

Any flag (resp. oriented flag) \(\mathfrak{f }= (F_i)\) on \(E\) can be represented by a basis \((e_1, \ldots , e_d)\) of \(E\) such that for each \(i, (e_1, \ldots , e_i)\) is a basis (resp. positive basis) for \(F_i\). This representation is not unique. However, if one fixes an inner product on \(E\), then each *oriented* flag \(\mathfrak{f }\) on \(E\) has an unique orthonormal base that represents it; this basis will be denoted by \(\mathcal{O }(\mathfrak{f })\).

Thus one can endow the set \({\check{\mathcal{F }}}E\) with a structure of smooth manifold diffeomorphic to \(\mathrm{O}(d)\), the Lie group of \(d \times d\) orthogonal matrices. (More details are given in Sect. 8.1.) The disorientating mapping \({\check{\mathcal{F }}}E \rightarrow \mathcal{F }E\) is \(2^d\)-to-\(1\) covering map; its deck transformations are smooth, and therefore we can also endow \(\mathcal{F }E\) with a differentiable structure. The manifolds \(\mathcal{F }E\) and \({\check{\mathcal{F }}}E\) are compact and have dimension \(d(d-1)/2\); the former is connected, while the latter has \(2\) connected components.

*Example 3.4*

Suppose that \(E=E'=\mathbb{R }^d\) is endowed with the Euclidian inner product, and identify the isomorphism \(L\) with a \(d \times d\) invertible matrix. Suppose \(\mathfrak{f }_0\) is the canonical flag in \(\mathbb{R }^d\) (i.e., that \(\mathcal{O }(\mathfrak{f })\) is the canonical basis in \(\mathbb{R }^d\)). Consider the *QR decomposition* of \(L\), i.e., the unique factorization \(L=QR\) where \(Q\) is an orthogonal matrix and \(R\) is an upper triangular matrix with positive diagonal entries. (Those matrices are computed using the Gram–Schmidt process.) Then \(\mathcal{O }((\mathcal{F }L)(\mathfrak{f }_0))\) is the ordered basis formed by the columns of \(Q\), and \(M(L,\mathfrak{f }_0) = R\).

If \(\mathfrak{f }\) in a non-oriented flag then the entries of \(M(L, \mathfrak{f })\) are well-defined up to sign, and the diagonal entries are well-defined and positive.

### 3.4 Flag bundle dynamics

As is Sect. 3.2, let \(\pi :V \rightarrow X\) be a vector bundle of rank \(d\) over a compact metric space \(X\), endowed with a Riemannian metric, and let \(S :V \rightarrow V\) be a vector bundle morphism over a continuous map \(T :X \rightarrow X\) that is invertible in each fiber.

*flag bundle*associated to \(V\), that is, the fiber bundle over \(X\) whose fiber \((\mathcal{F }V)_x\) over \(x\in X\) is the flag manifold of \(V_x\). The vector bundle morphism \(S\) induces a fiber bundle morphism \(\mathcal{F }S\) of \(\mathcal{F }V\) also over \(T\). This is summarized by the following diagrams:Analogously we define the oriented versions \({\check{\mathcal{F }}}V\) and \({\check{\mathcal{F }}}S\).

*Remark 3.5*

The original proof of Oseledets Theorem relies on this construction to reduce the general case to the case of triangular cocycles: see [19, pp. 228–229], also [14, § 4], [5, § 3.4.2].

*Example 3.6*

Let us come back to the situation of Example 3.2. Consider the flag bundle \(\mathcal{F }(TM)\) associated to \(TM\); by simplicity we denote it by \(\mathcal{F }M\) and call it the *flag bundle* of the manifold \(M\). It is a compact manifold of dimension \(d(d+1)/2\), and it is connected if \(M\) is. If \(T = g\) is \(C^r\) diffeomorphism of \(M\), let \(S = Dg\) be the derivative of \(g\). We obtain an induced morphism \(\mathcal{F }(Dg)\) of the flag bundle \(\mathcal{F }(TM) = \mathcal{F }M\), which by simplicity we denote by \(\mathcal{F }g\). This morphism is a \(C^{r-1}\) diffeomorphism of \(\mathcal{F }M\) (and a homeomorphism if \(r=1\)). Analogously we define \({\check{\mathcal{F }}}M\) and \({\check{\mathcal{F }}}g\). If \(M\) is endowed with a Riemannian metric then \({\check{\mathcal{F }}}M\) can be naturally identified with the *orthonormal frame bundle*. The Riemannian metric on \(M\) induces Riemannian metrics on \(\mathcal{F }M\) and \({\check{\mathcal{F }}}M\), as explained in Sect. 8.2.

*Furstenberg vector*of \(S\) with respect to \(\nu \) as \(\vec {\Lambda }(S,\nu ) = (\Lambda _1,\ldots ,\Lambda _d)\), where

*Furstenberg–Khasminskii formulas*; see [2].

An obvious but important feature of the map \(\nu \mapsto \vec {\Lambda }(\nu )\) is that it is continuous with respect to the weak-star topology.

The next result relates the Furstenberg vector with the previously defined Lyapunov exponents:^{4}

**Proposition 3.7**

Suppose \(\nu \) is an ergodic probability measure for \(\mathcal{F }S\). Let \(\mu \) be the projection of \(\nu \) on \(X\) (thus an ergodic probability measure for \(T\)), and let \(\lambda _1\), ..., \(\lambda _d\) be the Lyapunov exponents of \(S\) with respect to \(\mu \). Then there is a permutation \((k_1,k_2,\ldots ,k_d)\) of \((1, \ldots , d)\) such that the Furstenberg vector \((\Lambda _1,\ldots ,\Lambda _d)\) is given by \(\Lambda _i = \lambda _{k_i}\).

*Proof*

Using (3.4), the proposition follows from corresponding results for grassmannians; see [2, pp. 265, 211]. \(\square \)

It follows from Proposition 3.7 that the Furstenberg vector is independent of the choice of the Riemannian metric on the vector bundle \(V\).^{5}

The next result, which will be proved in Sect. 8.3, relates the fibered Lyapunov exponents of the nonlinear cocycle \(\mathcal{F }S\) with the Furstenberg vector:

**Proposition 3.8**

*Example 3.9*

- (a)
regarding \(\mathcal{F }\varphi _G\) as a nonlinear cocycle over \(\varphi _G\), the fibered Lyapunov spectrum of \(\nu \) is \(\{\lambda _{k_i}-\lambda _{k_j} ; \; i< j\}\);

- (b)
regarding \(\mathcal{F }\varphi _G\) as a nonlinear cocycle over \(\sigma \), the fibered Lyapunov spectrum of \(\nu \) is \(\{\lambda _k\} \cup \{\lambda _{k_i}-\lambda _{k_j} ; \; i< j\}\).

## 4 Sufficient conditions for zero exponents

In this section we state explicit conditions on an IFS \(G = (g_0,\ldots ,g_{\ell -1}) \in (\mathrm{Diff }^1(M))^\ell \) that are sufficient for the existence of fully supported ergodic measures with zero exponents as those in Theorem 3.

The first condition is that the IFS \(\mathcal{F }G\) of induced homeomorphisms of the flag manifold \(\mathcal{F }M\) is positively minimal on \(\mathcal{F }M\). For conciseness, we say that the IFS \(G\) is *positively minimal on the flag manifold*.

*maneuverability*property if for every \((x,\mathfrak{f })\in \mathcal{F }M\) and for every sequence of signs \(t = (t_1,\ldots ,t_d)\in \{-1,+1\}^d, d=\mathrm{dim }M\), there is \(g\in G\) such that

Now we can state the following result:

**Theorem 4.1**

- (a)
Positive minimality on the flag manifold.

- (b)
Maneuverability.

- (c)
There is a map \(g\in \langle G \rangle \) with a fixed point \(x_0 \in M\) such that the eigenvalues of \(Dg(x_0)\) are all negative, simple, and of different moduli.

*Remark 4.2*

Notice that the hypotheses of the theorem are meaningful for \(C^1\) IFS’s. However, our proof requires \(C^2\) regularity. We do not know if the \(C^1\) result is true.

*Remark 4.3*

It is possible (by adapting arguments from Sect. 9) to show that Conditions (a) and (b) actually imply Condition (c), and therefore the latter could be removed from the statement of Theorem 4.1. As our ultimate goal is to show the existence of the robust examples from Theorems 1 and 3, we chose to sacrifice generality in favor of briefness.

The proof of Theorem 4.1 will take Sects. 5–9. In Sect. 10 we will prove that there exist nonempty \(C^2\)-open sets of IFS’s satisfying the hypotheses of the theorem (provided the number \(\ell \) of generators is large enough, depending on the manifold \(M\)). Since the measure \(\nu \) produced by Theorem 4.1 satisfies precisely the conclusions of Theorem 3, the latter follows. As we have seen in Sect. 3.4, Theorem 3 implies Theorem 1.

## 5 The bootstrapping procedure

As explained in the Introduction, the measure \(\nu \) in Theorem 4.1 will be obtained as the limit of a sequence of measures supported on periodic orbits, and this sequence is constructed recursively by a “bootstrapping procedure”. We state below Proposition 5.1, which gives the recursive step of the procedure. Then we explain how Theorem 4.1 follows from that proposition and Lemma 2.1. The proof of the proposition is given in Sect. 9.

To begin, we need a few definitions.

*projective*if \(\tau (t \vec {\lambda }) = \tau (\vec {\lambda })\) for all \(\vec {\lambda }\in \mathcal{C }\) and \(t>0\).

*stable flag*of \(z\) by \(\mathfrak{s }(z): =(S_1(z) \subset \cdots \subset S_d(z))\) where

We assume a Riemannian metric was fixed on the manifold \(M\). This induces a Riemannian metric on the flag manifold \(\mathcal{F }M\), as we will see in Sect. 8.

**Proposition 5.1**

(Improving a periodic orbit) Consider a finite set \(G=\{g_0,\ldots , g_{\ell -1}\}\) of \(\mathrm{Diff }^2(M)\) satisfying the hypotheses of Theorem 4.1. Then there exists a projective continuous function \(\tau :\mathcal{C }\rightarrow (0,1)\) such that the following holds:

- (a)The vector \(\vec {\lambda }(\tilde{z})\) belongs to \(\mathcal{C }\) and satisfies$$\begin{aligned} 0 < \Vert \vec {\lambda }(\tilde{z})\Vert < \tau (\vec {\lambda }(z)) \Vert \vec {\lambda }(z)\Vert \quad \text {and} \quad \measuredangle (\vec {\lambda }(\tilde{z}),\vec {\lambda }(z))<\theta . \end{aligned}$$(5.1)
- (b)
There is a positive number \(\kappa < \min ( 1, \Vert \vec {\lambda }(z) \Vert )\) such that the orbit of \((\tilde{z}, \mathfrak{s }(\tilde{z}))\) under \(\mathcal{F }\varphi _G \ \varepsilon \)-shadows the orbit of \((z,\mathfrak{s }(z))\) during a proportion \(1-\kappa \) of the time.

- (c)
The orbit of \((\tilde{z}, \mathfrak{s }(\tilde{z}))\) under \(\mathcal{F }\varphi _G\) is \(\delta \)-dense in \(\ell ^\mathbb{Z }\times \mathcal{F }M\).

We remark that the proposition is a multidimensional version of Lemma 3 from [11].

Next we explain how this proposition allows us to recursively construct the desired sequence of periodic measures whose limit is the measure sought after by Theorem 4.1. The other ingredients are Propositions 3.7 and 3.8, which allow us to pass the Lyapunov exponents to the limit, and Lemma 2.1, which gives the ergodicity and full support.

*Proof of Theorem 4.1*

- (a)The vector \(\vec {\lambda }(z_{n+1})\) belongs to \(\mathcal{C }\) and satisfies$$\begin{aligned} 0 < \Vert \vec {\lambda }(z_{n+1})\Vert < \tau (\vec {\lambda }(z_n)) \Vert \vec {\lambda }(z_n)\Vert \quad \text {and} \quad \measuredangle (\vec {\lambda }(z_{n+1}),\vec {\lambda }(z_n)) < \theta _n. \end{aligned}$$
- (b)
There is a positive number \(\kappa _n < \min ( 1, \Vert \vec {\lambda }(z_n) \Vert )\) such that the orbit of \((z_{n+1}, \mathfrak{s }(z_{n+1}))\) under \(\mathcal{F }\varphi _G \ \varepsilon _n\)-shadows the orbit of \((z_n,\mathfrak{s }(z_n))\) during a proportion \(1-\kappa _n\) of the time.

- (c)
The orbit of \((z_{n+1}, \mathfrak{s }(z_{n+1}))\) under \(\mathcal{F }\varphi _G\) is \(\delta _n\)-dense in \(\ell ^\mathbb{Z }\times \mathcal{F }M\).

Let \(\nu _n\) be the \(\mathcal{F }\varphi _G\)-invariant measure supported on the orbit of \((z_{n},\mathfrak{s }(z_{n}))\). To complete the proof, we will show that the sequence \((\nu _n)\) converges in the weak-star topology to a measure \(\nu \) with the desired properties.

Since \(\vec {\lambda }(z_n) \rightarrow \vec {0}\), it follows from Proposition 3.7 that the sequence of Furstenberg vectors \(\vec \Lambda (\nu _n)\) also converges to zero. Since the Furstenberg vector is continuous with respect to the weak-star topology, we have that \(\vec \Lambda (\nu ) = \vec {0}\). This implies that the fibered Lyapunov exponents of \(\nu \) are zero (recall Example 3.9), concluding the proof of Theorem 4.1. \(\square \)

## 6 Exploiting positive minimality

The aim of this section is to prove Lemma 6.3, a simple but slightly technical consequence of positive minimality, which will be used in Sect. 9 in the proof of Proposition 5.1.

We begin with the following lemma:

**Lemma 6.1**

(Go home) Let \(H=\{h_0, \ldots , h_{\ell -1}\}\) be a positively minimal set of homeomorphisms of a compact metric space \(N\). For every nonempty open set \(U \subset N\) there exists \(k_0 = k_0(U) \in \mathbb{N }^*\) such that for every \(x \in N\) there exists a word \(w\) of length at most \(k_0\) on the alphabet \(\{0,\ldots ,\ell -1\}\) such that \(h_{[w]}(x) \in U\).

*Proof*

Fix the set \(U\). By positive minimality, for every \(x \in N\) there is a word \(w(x)\) on the alphabet \(\{0,\ldots ,\ell -1\}\) such that \(h_{[w(x)]}(x) \in U\). By continuity, there is a neighborhood \(V(x)\) of \(x\) such that \(h_{[w(x)]}(V(x)) \subset U\). By compactness, we can cover \(N\) by finitely many sets \(V(x_i)\). Let \(k_0\) be the maximum of the lengths of the words \(w(x_i)\). \(\square \)

For the next lemma, recall from Sect. 3.1 the distance on \(\ell ^\mathbb{Z }\times N\) and the cylinder notation. Let us also use the following notation for segments of orbits: \(f^{[0,k]}(x) := \{x, f(x), f^2(x), \ldots , f^k(x)\}\).

**Lemma 6.2**

- (a)
for every \(\omega \in [\![; s_0 s_1 \ldots s_{k-1} ]\!]\), the segment of orbit \(\varphi _H^{[0,k]} (\omega ,x)\) is \(\delta \)-dense in \(\ell ^\mathbb{Z }\times N\);

- (b)
\(h_{[w]}(x) \in U\).

*Proof*

Let \(\delta \) and \(U\) be given. Choose a finite \((\delta /2)\)-dense subset \(Y \subset N\). Let \(m \in \mathbb{N }\) be such that \(2^{-m} \le \delta \) and let \(W\) be the set of all words of length \(2m+1\) on the alphabet \(\{0,\ldots ,\ell -1\}\). Enumerate the set \(Y \times W\) as \(\{(y_i, w_i) \; ; 1 \le i \le r \}\). For each \(i \in \{1,\ldots ,r\}\), let \(B_i\) be the open ball of center \(y_i\) and radius \(\delta /2\). Let \(w_i^-\) be the initial subword of \(w_i\) of length \(m\), and let \(U_i = \big (h_{[w_i^-]}\big )^{-1}(B_i)\). Define also \(U_{r+1} = U\).

\(x_{n_i} \in B_i\), that is, \(d(x_{n_i} , y_i) < \delta /2\), and in particular \(d(x_{n_i}, x^*) < \delta \).

\(\omega _{n_i} \in [\![s_0 \ldots s_{n_i-1} ; s_{n_i} \ldots s_{k-1} ]\!]\) and in particular \(d(\omega _{n_i},\omega ^*) \le 2^{-m} \le \delta \), because \(s_{n_i-m} \ldots s_{n_i+m} = w_i = s^*_{-m} \ldots s^*_{m}\).

The following is an immediate corollary of Lemma 6.2.

**Lemma 6.3**

- (a)
for every \((\omega ,x) \in [\![; s_0 s_1 \ldots s_{k-1} ]\!]\times B\), the segment of orbit \(\varphi _H^{[0,k]} (\omega ,x)\) is \(\delta \)-dense in \(\ell ^\mathbb{Z }\times N\);

- (b)
\(h_{[w]}(B) \subset U\).

*Proof*

Use Lemma 6.2 and continuity. \(\square \)

## 7 Exploiting maneuverability

The next lemma says that if an induced IFS on the flag bundle satisfies the maneuverability condition, then we can select orbits whose derivatives in the upper triangular matrix form (3.1) have approximately prescribed diagonals.

**Lemma 7.1**

*Proof*

Now let \((x,\mathfrak{f })\in \mathcal{F }M\) and \((\chi _1,\ldots ,\chi _d)\in [-c,c]^d\) be given. We inductively define the symbols \(s_0, s_1\), ..., \(s_{q-1}\) forming the word \(w\). The idea of the proof is simple: at each step we look at diagonal obtained so far, choose signs pointing towards the objective vector \((\chi _1,\ldots ,\chi _d)\), and apply uniform maneuverability to pass to the next step.

## 8 Triangularity

Estimating the size of products of matrices (and hence computing Lyapunov exponents) may be a difficult business. The task is much simpler if the matrices happen to be upper triangular: in that case the non-commutativity is tamed (see Proposition 8.5 and Lemma 8.6 below). Working in the flag bundle has the advantage of making all derivatives upper triangular, in a sense that will be made precise.

### 8.1 Linearly induced map between flag manifolds

We continue the discussion from Sect. 3.3. Here we will give geometrical information about diffeomorphisms \(\mathcal{F }L :\mathcal{F }E \rightarrow \mathcal{F }E'\) induced by linear maps \(L :E \rightarrow E'\).

*order*this basis as follows:

*canonical basis*or

*canonical frame*of \(\mathfrak so (d)\). Pushing forward by right translations, we extend this to a frame field on \(\mathrm{O}(d)\), called the

*canonical frame field*. We take on \(\mathrm{O}(d)\) the Riemannian metric for which the canonical frames are orthonormal.

^{6}

*canonical*. We endow \({\check{\mathcal{F }}}E\) with the Riemannian metric that makes these frames orthonormal.

Let \(L :E \rightarrow E'\) be an isomorphism between real vector spaces of dimension \(d\). Endow \(E\) and \(E'\) with inner products and consider on \({\check{\mathcal{F }}}E\) and \({\check{\mathcal{F }}}E'\) the associated canonical frame fields. For any \(\mathfrak{f }\in {\check{\mathcal{F }}}E\), let \(T(L,\mathfrak{f })\) denote the matrix of the derivative of the map \({\check{\mathcal{F }}}L :{\check{\mathcal{F }}}E \rightarrow {\check{\mathcal{F }}}E'\) with respect to the frames at the points \(\mathfrak{f }\) and \(({\check{\mathcal{F }}}L)(\mathfrak{f })\).

The following result is probably known, but we weren’t able to find a reference:

**Proposition 8.1**

*Proof*

Let \(R = M(L, \mathfrak{f })\). If \(Q \in \mathrm{O}(d)\) then \(\Phi (Q)\) is the unique matrix \(\hat{Q} \in \mathrm{O}(d)\) such that \(R Q = \hat{Q} \hat{R}\) for some upper triangular matrix \(\hat{R}\) with positive diagonal entries.^{7}

*up-to-sign frame*”. We push it forward by right translations and obtain a field of up-to-sign frames on \(\mathrm{O}(d)/H\), which is then pulled back to a well-defined field of up-to-sign frames on \(\mathcal{F }E\). There is are unique Riemannian metrics that make these up-to-sign frames orthonormal.

Now consider the diffeomorphism \(\mathcal{F }L :\mathcal{F }E \rightarrow \mathcal{F }E'\) induced by a liner isomorphism \(L :E \rightarrow E'\). Let \(T(L,\mathfrak{f })\) denote the up-to-sign matrix of the derivative of the map \(\mathcal{F }L\) with respect to the up-to-sign frames at the points \(\mathfrak{f }\) and \(({\check{\mathcal{F }}}L)(\mathfrak{f })\). Notice that the diagonal entries are well-defined. It follows from Proposition 8.1 that this “matrix” is upper-triangular and that its diagonal entries are the numbers (8.2).

As a consequence, we have the following fact:^{8}

**Corollary 8.2**

(Stable flag) Suppose that \(L :E \rightarrow E\) is a linear isomorphism whose eigenvalues have distinct moduli and are ordered as \(|\lambda _1| > \cdots > |\lambda _d|\). Consider the flag \(\mathfrak{s }= (S_i) \in \mathcal{F }E\) where \(S_i\) is spanned by eigenvectors corresponding to the first \(i\) eigenvalues. Then \(\mathfrak{s }\) is a hyperbolic attracting fixed point of \(\mathcal{F }L\).

*Remark 8.3*

The *QR algorithm* is the most widely used numerical method to compute the eigenvalues of a matrix \(A_0 \in \mathrm{GL}(d,\mathbb{R })\) (see [23, p. 356]).^{9} It runs as follows: starting with \(n=0\), compute the QR decomposition of \(A_n\), say, \(A_n = Q_n R_n\), let \(A_{n+1} := R_n Q_n\), increment \(n\), and repeat. Let us interpret the sequence of matrices \(A_n\) produced by the algorithm in terms of the diffeomorphism \(\mathcal{F }A_0 :\mathcal{F }\mathbb{R }^d \rightarrow \mathcal{F }\mathbb{R }^d\). If \(\mathfrak{f }_0\) is the canonical flag of \(\mathbb{R }^n\) then \(A_n\) is the matrix of \(A_0\) with respect to an orthonormal basis that represents the flag \(\mathfrak{f }_n := (\mathcal{F }A_0)^n(\mathfrak{f }_0)\). If the eigenvalues of \(A_0\) have different moduli then the sequence \((\mathfrak{f }_n)\) converges. (Actually \(\mathcal{F }A_0\) is a Morse–Smale diffeomorphism whose periodic points are fixed: see [22].) It follows that the sequence \((A_n)\) converges to upper triangular form. In particular, if \(n\) is large then the diagonal entries of \(A_n\) give approximations to the eigenvalues of \(A_0\). (In practice, the algorithm is modified in order to accelerate convergence and reduce computational cost.)

### 8.2 Geometry of the flag bundles

Now consider a compact connected manifold \(M\). We will discuss in more detail the flag bundles \({\check{\mathcal{F }}}M\) and \(\mathcal{F }M\), defined in Example 3.6.

The tangent space of the flag manifold \({\check{\mathcal{F }}}M\) at a point \(\xi = (x,\mathfrak{f })\in {\check{\mathcal{F }}}M\) has a canonical subspace called the *vertical subspace*, denoted by \(\mathrm{Vert }(\xi )\), which is the tangent space of the fiber \(({\check{\mathcal{F }}}M)_x = {\check{\mathcal{F }}}(T_x M)\) at \(\xi \).

*horizontal subspace*, denoted \(\mathrm{Horiz }(\xi )\), as follows: for each smooth curve starting at the point \(x\), consider the parallel transport of the flag \(\mathfrak{f }\), which gives a smooth curve in the manifold \({\check{\mathcal{F }}}M\); consider the initial velocity \(w \in T_\xi ({\check{\mathcal{F }}}M)\) of the curve. Then \(\mathrm{Horiz }(\xi )\) consists of all vectors \(w\) obtained in this form.

^{10}The tangent space of \({\check{\mathcal{F }}}M\) at \(\xi \) splits as

Since \(\mathrm{Vert }(\xi ) = T_\mathfrak{f }({\check{\mathcal{F }}}(T_x M))\), there is a well-defined canonical frame on \(\mathrm{Vert }(\xi )\) (see the previous subsection). On the other hand, there is a natural frame on \(\mathrm{Horiz }(\xi )\), namely, the unique frame sent by \(D \pi (\xi )\) to the frame \(\mathcal{O }(\mathfrak{f })\) of \(T_x M\). By concatenating these two frames (in the same order as in (8.4)), we obtain a frame of \(T_\xi ({\check{\mathcal{F }}}M)\), which will called *canonical*. So we have defined a canonical field of frames on \({\check{\mathcal{F }}}M\). We endow \({\check{\mathcal{F }}}M\) with the Riemannian metric that makes these frames orthonormal.

*Remark 8.4*

The triangularity property seem above can be summarized abstractly as follows: The fiber bundle \({\check{\mathcal{F }}}({\check{\mathcal{F }}}M) \rightarrow {\check{\mathcal{F }}}M\) has a special section which is invariant for under \({\check{\mathcal{F }}}({\check{\mathcal{F }}}g)\), for any \(g \in \mathrm{Diff }^2(M)\).

### 8.3 The Lyapunov exponents in the flag bundle

Using the description obtained in Sect. 8.1 for the derivatives of linearly-induced maps on flag manifolds, the proof of Proposition 3.8 will be reduced to the following standard result:

**Proposition 8.5**

*Proof*

See Lemma 6.2 in [14] (as mentioned there, the argument applies to any triangular cocycle over a compact metric space).^{11}\(\square \)

*Proof of Proposition 3.8*

We fix a continuous map \(T :X \rightarrow X\) of a compact metric space \(X\), a vector bundle \(V\) of rank \(d\) over \(X\) endowed with a Riemannian metric, and a vector bundle morphism \(S :V \rightarrow V\) over \(T\). that is invertible in each fiber.

First we will prove the corresponding statement of Proposition 3.8 for oriented flags. Let \(\pi :{\check{\mathcal{F }}}V \rightarrow X\) be the bundle projection. Consider the vector bundle \(W\) over \({\check{\mathcal{F }}}V\) whose fiber \(W_\xi \) over \(\xi \in \mathcal{F }V\) is the tangent space of the flag manifold \({\check{\mathcal{F }}}(V_{\pi (\xi )})\) at \(\xi \). Using the canonical frame field explained in Sect. 8.1, this vector bundle can be trivialized as \(W = ({\check{\mathcal{F }}}V) \times \mathbb{R }^{d(d-1)/2}\). The derivative of \({\check{\mathcal{F }}}S :{\check{\mathcal{F }}}V \rightarrow {\check{\mathcal{F }}}V\) induces a vector bundle automorphism \(U :W \rightarrow W\), which under the trivializing coordinates has a generator \(A :X \rightarrow \mathrm{GL}({d(d-1)/2},\mathbb{R })\) taking values on upper triangular matrices, with diagonals given by expressions as (8.2). Applying Proposition 8.5, the desired result follows.

The case of non-oriented flags follows easily from the oriented case and the following observation: any \(\mathcal{F }S\)-invariant ergodic probability on \(\mathcal{F }V\) can be lifted to a \({\check{\mathcal{F }}}S\)-invariant ergodic probability on \({\check{\mathcal{F }}}V\). \(\square \)

### 8.4 Products of triangular matrices

Proposition 8.5 indicates that off-diagonal entries of “random” products of triangular matrices are dominated by the diagonals. We will also need the following simple deterministic version of this fact:

**Lemma 8.6**

*Proof*

## 9 Improving a periodic orbit

The matrices in (8.5) that represent the derivatives are upper-triangular; to determine the Lyapunov exponents of the new periodic orbit we only need to know the matrix diagonals. In particular, since the “tour and go home” proportion of the orbit will be much smaller than the rest, it will be negligible for the estimation of Lyapunov exponents.

Nevertheless, we still will need to estimate norms of derivatives, since we will want to fit images of balls inside balls (recall Sect. 2.4). Lemma 8.6 allows us to basically disregard the off-diagonal elements. It is important to apply Lemma 8.6 only after multiplying together the derivatives along each segment of orbit provided by Lemma 7.1, because then the diagonal is controlled.

Let \(\vec {\lambda }=(\lambda _i)\) be the Lyapunov vector of the given periodic orbit. Let \((\tilde{\lambda }_i)\) denote the (still to be determined) new exponents, and let \((\chi _i)\) be the exponents along the correcting phase (corresponding to \(h_2\) in Sect. 2.4). So \(\tilde{\lambda }_i \simeq (1-\kappa _0) \lambda _i + \kappa _0 \chi _i\), where \(\kappa _0\) is the (still to be determined) approximate proportion of the correcting phase. We want the vectors \((\tilde{\lambda }_i)\) and \((\lambda _i)\) to form a small angle; so we take \(\chi _i = -a \lambda _i\), for some proportionality factor \(a>0\). The largest correcting exponent we can take is the number \(c\) given by Lemma 7.1. We take \(\chi _d = c\), and so we determine \(a = c/|\lambda _d|\).

- Let \(\gamma = \gamma (\vec \lambda )\) be the least gap in the sequence \(0 > \lambda _1 > \cdots > \lambda _d\), that is,It follows from the description of derivatives from (8.2) that the maximum expansion exponent (on \(\mathcal{F }M\)) around the original orbit is \(-\gamma (\vec \lambda )\). Analogously, the maximum expansion exponent along the correcting phase is \(\chi _d = c\). Since we want the ball \(B_2\) to have (much) smaller radius than \(B_0\), it is necessary that \((1-\kappa _0)(-\gamma ) + \kappa _0 c < 0\), that is, \(\kappa _0 < \gamma /(\gamma +c)\). We choose$$\begin{aligned} \gamma (\vec \lambda ) := \min \big \{ -\lambda _1, \lambda _1-\lambda _2, \lambda _2-\lambda _3, \ldots , \lambda _{d-1}-\lambda _d \big \}. \end{aligned}$$(9.1)where \(C > \max (\gamma ,c)\) is an upper bound for all expansions.$$\begin{aligned} \kappa _0 := \frac{\gamma }{2C} , \end{aligned}$$
- Finally, we estimate the factorWe choose \(\tau \) as something bigger than the right-hand side, e.g.:$$\begin{aligned} \frac{|\tilde{\lambda }_i|}{|\lambda _i|} \simeq (1-\kappa _0) + \kappa _0 (-a) \le 1 - a \kappa _0 = 1 - \frac{c\gamma }{2C |\lambda _d|}. \end{aligned}$$This is a continuous projective function, as required.$$\begin{aligned} \tau (\vec {\lambda }) := 1 - \frac{c}{C} \cdot \frac{\gamma (\vec \lambda )}{|\lambda _d|} , \end{aligned}$$(9.2)

*Proof of Proposition 5.1*

Suppose the set \(G = \{g_0, \ldots , g_{\ell -1}\} \subset \mathrm{Diff }^2(M)\) satisfies the assumptions (a), (b) and (c) of Theorem 4.1. Fix constants \(C>c>0\), where \(C\) satisfies (7.1) and \(c\) is given by Lemma 7.1. For any \(\vec {\lambda } \in \mathcal{C }\), let \(\gamma (\vec {\lambda })\) be the “gap” defined by (9.1). Define the function \(\tau :\mathcal{C }\rightarrow (0,1)\) by (9.2).

If \(\xi \in \mathcal{F }M\) and \(r>0\), let \(B(\xi , r) \subset \mathcal{F }M\) denote the ball of center \(\xi \) and radius \(r\), with respect to the Riemannian norm on \(\mathcal{F }M\) explained in Sect. 8.2.

**Claim 9.1**

*Proof of the claim*

**Claim 9.2**

\(r_j < \varrho '\) for each \(j=0,1,\ldots ,m\).

*Proof of the Claim 9.2*

Let \(B':=B(\xi _m,\varrho ')\) and find a corresponding word \(w' = s_0 s_1 \ldots s_{k-1}\) of length \(k \le k_1\) with the properties “group tour” (9.15) and “go home” (9.16).

This concludes the construction of the “improved” periodic orbit. The rest of the proof consists of checking that this orbit has the desired properties (a), (b), (c).

*Verifying property (a).*Consider the (periodic) orbit of \((\tilde{z}, \tilde{\mathfrak{f }})\) under \(\mathcal{F }\varphi _G\):

*Verifying property (b).*Write the original periodic orbit in \(\ell ^\mathbb{Z }\times \mathcal{F }M\) as

*Verifying property (c).* Since for \(j = np+qm\), the point \((\sigma ^j(\tilde{w}^\infty ), \tilde{x}_j, \tilde{\mathfrak{f }}_j)\) belongs to \([\![; s_0 s_1 \ldots s_{k-1} ]\!]\times B'\); therefore property (9.15) assures that the first \(k\) iterates of this point form a \(\delta \)-dense subset of \(\ell ^\mathbb{Z }\times \mathcal{F }M\). We have checked the last part of Proposition 5.1. The proof is completed.\(\square \)

## 10 Construction of the open set of IFS’s

The aim of this section is to show the existence of iterated function systems that \(C^2\)-robustly satisfy the hypotheses of Theorem 4.1, i.e., to prove the following:

**Proposition 10.1**

Let \(M\) be a compact connected manifold. There is \(\ell \in \mathbb{N }^*\) and a nonempty open set \(\mathcal{G }_0 \subset (\mathrm{Diff }^2(M))^\ell \) such that every \(G\in \mathcal{G }_0\) satisfies conditions (a), (b) and (c) of Theorem 4.1.

To begin, notice that if \(G_0=(g_0,\ldots ,g_{\ell _0-1}), G_1=(g_{\ell _0},\ldots ,g_{\ell _0+\ell _1-1})\) and \(G_2=(g_{\ell _0+\ell _1},\ldots ,g_{\ell _0+\ell _1+\ell _2-1})\) respectively satisfy conditions (a), (b) and (c), then \(G=(g_0,\ldots , g_{\ell -1})\) with \(\ell =\ell _0+\ell _1+\ell _2\) satisfies all three conditions. Therefore, to prove Proposition 10.1, one can prove independently the existence of open sets satisfying each of the three conditions.

Condition (c) is trivially nonempty and \(C^2\)-open (actually \(C^1\)-open).

*Proof that the maneuverability condition (b) is nonempty and open* This is an easy compactness argument, but let us spell out the details for the reader’s convenience.

In order to deal with the positive minimality condition (a), we start by proving the following criterion:

**Lemma 10.2**

For every \(i\) there exists a map \(h_i \in \langle H \rangle \) whose restriction to \(h_i^{-1}(V_i)\) is a (uniform) contraction.

The cover \(\{V_i\}\) has a Lebesgue number \(\delta \) such that the orbit of every point \(x\in N\) is \(\delta \)-dense in \(N\).

*Proof*

Since the cover \(\{V_i\}\) is finite, there exists \(\alpha \) with \(0<\alpha <1\) such that each restriction \(h_i | h_i^{-1}(V_i)\) is an \(\alpha \)-contraction.

**Claim 10.3**

If \(y \in N, r>0\), and \(\overline{B(y,r)} \subset V_i\) then \(h_i \big ( B ( h_i^{-1}(y) , \alpha ^{-1}r ) \big ) \subset B(y,r)\).

*Proof of the claim*

^{12}, the claim follows. \(\square \)

Now fix any point \(x \in N\), and assume the orbit of \(x\) is \(\varepsilon \)-dense for some \(\varepsilon \le \delta \). For any \(y \in N\), we can find some \(V_i\) containing \(\overline{B(y,\alpha \varepsilon )}\). By the \(\varepsilon \)-denseness of the orbit of \(x\), there is \(h \in \langle H \rangle \) such that \(h(x) \in B ( h_i^{-1}(y) , \varepsilon )\). It follows from Claim 10.3 that \(h_i \circ h (x) \in B(y,\alpha \varepsilon )\). Since \(y\) is arbitrary, this shows that the orbit of \(x\) is \(\alpha \varepsilon \)-dense. By induction, this orbit is \(\alpha ^n\varepsilon \)-dense for any \(n>0\); so it is dense, as we wanted to show. \(\square \)

If \(H\) is a finite set of *diffeomorphisms* of \(N\) satisfying the assumptions of Lemma 10.2, then these assumptions are also satisfied for sufficiently small \(C^1\)-perturbations of the elements of \(H\). In other words, the hypotheses of Lemma 10.2 are \(C^1\)-robust.^{13}

Therefore, to show that the positive minimality condition (a) has non-empty interior, we are reduced to show the following:

**Lemma 10.4**

Given any compact connected manifold \(M\), there is \(\ell \in \mathbb{N }^*\) and \(g_0,\ldots , g_{\ell -1}\in \mathrm{Diff }^2(M)\) such that the induced diffeomorphisms \(\mathcal{F }g_0\), ..., \(\mathcal{F }g_{\ell -1} \in \mathrm{Diff }^1(\mathcal{F }N)\) satisfy the hypotheses of Lemma 10.2.

*Proof*

This is another easy compactness argument.

For any \(x\in M\) and any flag \(\mathfrak{f }\) at \(x\), there is a diffeomorphism \(g\in \mathrm{Diff }^2(M)\) such that \(x\) is a hyperbolic attracting fixed point of \(g\) such that the eigenvalues of \(Dg(y)\) have different moduli, and moreover \(\mathfrak{f }\) is the stable flag of \(g\) at \(x\). Then \((x,\mathfrak{f })\) is a hyperbolic attracting fixed point of \(\mathcal{F }g\), and in particular there is a open neighborhood \(U{(x,\mathfrak{f })}\) on which \(\mathcal{F }g\) induces a uniform contraction. Denote \(V{(x,\mathfrak{f })}=(\mathcal{F }g)(U{(x,\mathfrak{f })})\). The sets \(V{(x,\mathfrak{f })}\) form an open cover of \(\mathcal{F }M\), so that one can extract a finite subcover \(V_0\), ..., \(V_{\ell _0-1}\). Let \(g_0\), ..., \(g_{\ell _0-1}\) be the corresponding diffeomorphisms as above.

Given two points \((x,\mathfrak{f }), (x',\mathfrak{f }') \in \mathcal{F }M\), it follows from the connectedness of \(M\) that there is \(g\in \mathrm{Diff }^2(M)\) such that \(\mathcal{F }g(x,\mathfrak{f })=(x',\mathfrak{f }')\). A simple compactness argument shows that there exist \(\ell _1\) and \(g_{\ell _0}\), ..., \(g_{\ell _0+\ell _1-1}\in \mathrm{Diff }^2(M)\) such that for every \((x,\mathfrak{f }) \in \mathcal{F }M\), the set \(\{{\mathcal{F }} g_{\ell _0}(x,\mathfrak{f }),\ldots , {\mathcal{F }} g_{{\ell }_0+{\ell }_1-1}(x,\mathfrak{f })\}\) is \(\delta \)-dense in \(\mathcal{F }M\), where \(\delta \) is a Lebesgue number of the cover \(\{V_0, \ldots , V_{\ell _0-1}\}\).

Denote \(\ell =\ell _0+\ell _1\). Now the action of \(H = \{\mathcal{F }g_0, \ldots \mathcal{F }g_{\ell -1}\}\) on \(N = \mathcal{F }M\) satisfies all the hypotheses of Lemma 10.2, which allows us to conclude. \(\square \)

This completes the proof of Proposition 10.1; as explained in Sect. 4, the main Theorems 1 and 3 follow.

*Remark 10.5*

It is possible to adapt the proof of Theorem 3 for the *oriented* flag bundle \({\check{\mathcal{F }}}M\), provided that the manifold \(M\) is non-orientable (basically because \({\check{\mathcal{F }}}M\) is then connected). However, if \(M\) is orientable, then the corresponding version of Theorem 3 is false. For example, since every diffeomorphism of \(M = \mathbb{C }P^2\) preserves orientation (see [12, p. 140]), the induced action on \({\check{\mathcal{F }}}M\) fixes each of the two connected components, and hence no \(1\)-step skew-product \({\check{\mathcal{F }}}\varphi _G\) can be transitive.

## 11 Positive entropy

*Proof of Theorem 2*

*bi-maneuverability property*if \(\ell \) is even and both IFS’s

The set \(\mathcal{V }\subset (\mathrm{Diff }^1(M))^\ell \) of the IFS’s with the bi-maneuverability property is nonempty and open, provided \(\ell \) is even and large enough; this follows immediately from the analogous statements for maneuverability that we proved in Sect. 10. We will prove that the set \(\mathcal{V }\) satisfies the conclusions of the theorem.

for each \(i \in \{1,\ldots ,d\}\), the number \(\log M_{i,i}(Dg_{\omega _n(\theta )} (x_n(\theta )),\mathfrak{f }_n(\theta ))\) is negative if \(\log M_{i,i}(Dg_{[\omega _{n-1}(\theta ) \ldots \omega _0(\theta )]} (x_0), \mathfrak{f }_0)\) is positive and positive otherwise.

\(\omega _n(\theta ) < \ell /2\) if and only if \(\theta _n = 0\).

This concludes the proof of Theorem 2. \(\square \)

*Remark 11.1*

The \(C^2\)-open sets of IFS satisfying the conclusions of Theorem 1 can be taken also satisfying the conclusions of Theorem 2: it suffices to replace maneuverability by bi-maneuverability in the construction.

*Remark 11.2*

As it is evident from its proof, Theorem 2 has a flag bundle version.

*Remark 11.3*

Artur Avila suggested an alternative proof of Theorem 2 using ellipsoid bundles instead of flag bundles, and obtaining compact sets \(\Lambda _G\) where derivatives along orbits are uniformly bounded away from zero and infinity.

## Footnotes

- 1.
A more sophisticated relation of this kind was obtained by Johnson et al. [14].

- 2.
Lemma 10.2 below gives a practical criterion for positive minimality of an IFS.

- 3.
All the measures we consider will be defined over the corresponding Borel \(\sigma \)-algebra.

- 4.
The continuous-time version of Proposition 3.7 is sometimes called the

*Liao spectrum theorem*; see e.g. [9]. - 5.
- 6.
This metric is obviously right-invariant. Actually, it is also left-invariant. Indeed, a calculation shows that \(\langle X, Y \rangle = - {{\mathrm{tr}}}XY/2\) for \(X, Y\in \mathfrak so (d)\), which is invariant under the adjoint action of the group, and therefore can be uniquely extended to a bi-invariant Riemannian metric. Another remark: this inner product is the Killing form divided by \(-2(d-2)\) (if \(d>2\)).

- 7.
Those familiar with the QR algorithm will recognize this equation; see Remark 8.3.

- 8.
Similar results are obtained in [22]; see Lemma 4.

- 9.
We thank Carlos Tomei for telling us about the QR algorithm.

- 10.
This field of horizontal subspaces is actually an Ehresmann connection on the principal bundle \({\check{\mathcal{F }}}M\).

- 11.
As we mentioned in Remark 3.5, Oseledets [19] reduced the proof of his theorem to the triangular case.

- 12.
Here we use that \(N\) is a connected Riemannian manifold (and not only a metric space).

- 13.
Using this, it is easy to establish the existence of \(C^1\)-robustly positively minimal finitely generated IFS’s on any compact connected manifold. More interestingly, Homburg [13] shows that two generators suffice.

## Notes

### Acknowledgments

We are grateful to the referee for some corrections.

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