ON THE UNIFORM HYPERBOLICITY OF SOME NONUNIFORMLY HYPERBOLIC SYSTEMS

JOSÉ F. ALVES, VÍTOR ARAÚJO, AND BENOÎT SAUSSOL

Abstract. We give sufficient conditions for the uniform hyperbolicity of certain nonuniformly hyperbolic dynamical systems. In particular, we show that local diffeomorphisms that are nonuniformly expanding on sets of total probability are necessarily uniformly expanding. We also present a version of this result for diffeomorphisms with nonuniformly hyperbolic sets.

Contents

1. Introduction 1 2. Preliminary lemmas 3 3. Proofs of main results 6 4. Final remarks 7 References 7

1. Introduction

Let f : M → M be a C^{1} local diffeomorphism of a manifold M endowed with a Riemannian metric which induces a norm | · | on the tangent space and a volume form that we call Lebesgue measure.

The map f is said to be uniformly expanding if there exist constants C > 0 and σ > 1 such that

(1)  |df_{x}^{n}(v)| ≥ Cσ^{n}|v| for all x ∈ M, v ∈ T_{x}M and n ≥ 1.

A rather complete description of the statistical properties of this kind of systems can be found in [7, 8, 11]. Recently several results have been obtained on the statistical properties of maps satisfying some weaker condition of expansion; see [3, 1]. We say that f is nonuniformly expanding at x ∈ M if condition (1) holds in average, meaning that

(2)  lim inf_{n→+∞} \frac{1}{n} ∑_{j=0}^{n−1} log |(df_{f^{j}x})^{−1}| < 0.

Date: October 31, 2001.

The authors were partially supported by PRODYN.

It is easy to see that if f satisfies condition (1) then it also satisfies condition (2) — in fact for every x ∈ M.

From [9] one knows that C^{2} local diffeomorphisms defined in one-dimensional manifolds are necessarily uniformly expanding if they are nonuniformly expanding almost everywhere, in the sense of Lebesgue measure. The situation is completely different in higher dimensions or with less regularity. Examples of maps which are nonuniformly expanding at Lebesgue almost every point but are not uniformly expanding are exhibited in [3].

As a by-product of our results we see that if condition (2) holds for every x ∈ M, then f has to be uniformly expanding. Actually, our first theorem gives a stronger result. Before we state it, let us recall that a Borel set in a topological space is said to have total probability if it has probability one for every f invariant probability measure.

Theorem A. Let f : M → M be a C^{1} local diffeomorphism defined in a compact Riemannian manifold. If f is nonuniformly expanding on a set of total probability, then f is uniformly expanding.

We emphasize that there are examples of unimodal maps satisfying all the conditions of the theorem except the local diffeomorphism assumption, for which the conclusion fails to hold; see [5, 6].

Corollary B. Let f be a C^{1} local diffeomorphism of the circle. If f has a positive Lyapunov exponent on a set of total probability, then f is uniformly expanding.

We have a version of these results for C^{1} diffeomorphisms f: M → M having invariant sets with some nonuniformly hyperbolic structure. Let Λ ⊂ M be an f invariant set with a df invariant continuous splitting of the tangent bundle over Λ

T_{Λ}M = E^{cs} ⊕ E^{cu}.

We say that f is nonuniformly expanding along the E^{cu} direction if, for Lebesgue almost all x ∈ Λ,

lim inf_{n→+∞} \frac{1}{n} ∑_{j=0}^{n−1} log |df^{−1}|E_{f^{j}x}^{cu}| < 0.

Similarly, we say that f is nonuniformly contracting along the E^{cs} direction if, for Lebesgue almost all x ∈ Λ,

lim inf_{n→+∞} \frac{1}{n} ∑_{j=0}^{n−1} log |df|E_{f^{j}x}^{cs}| < 0.

A special case is when Λ is a hyperbolic set, meaning that f has both uniform expansion in the E^{cu} direction and uniform contraction in the E^{cs} direction: there exist constants C > 0 and σ > 1 such that

(3)  |df_{x}^{n}(v^{u})| ≥ Cσ^{n}|v^{u}| and |df_{x}^{n}(v^{s})| ≤ Cσ^{−n}|v^{s}|

for all x ∈ M, v^{s} ∈ E^{cs}, v^{u} ∈ E^{cu} and n ≥ 1.

Examples of nonuniformly hyperbolic systems which are not uniformly hyperbolic can be found in [3, 4]. The works [3, 4, 13, 2] give substantial information about the statistical properties of systems with nonuniformly hyperbolic sets.

Theorem C. Let f : M → M be a C^{1} diffeomorphism with a positively invariant set Λ for which the tangent bundle has a continuous invariant splitting T_{Λ}M = E^{cs} ⊕ E^{cu}. If f is nonuniformly expanding (resp. contracting) along the E^{cu} (resp. E^{cs}) direction in a set of total probability in Λ, then f is uniformly expanding (resp. contracting) along the E^{cu} (resp. E^{cs}) direction.

In particular, if f has simultaneously nonuniform expansion along the E^{cu} direction and nonuniform contraction along the E^{cs} direction in a set of total probability in Λ, then Λ is a hyperbolic set.

Corollary D. Let f be a C^{1} surface diffeomorphism with a positively invariant set Λ for which the tangent bundle has a continuous invariant splitting T_{Λ}M = E^{cs} ⊕ E^{cu}. If f has a positive Lyapunov exponent in the E^{cu} direction and a negative Lyapunov exponent in the E^{cs} direction on a set of total probability, then Λ is a hyperbolic set.

Another version of these results can be given for partially hyperbolic sets with decomposition

T_{Λ}M = E^{s} ⊕ E^{c} ⊕ E^{u},

where E^{s} and E^{u} have uniform contracting/expanding behavior.

Corollary E. Let f : M → M be a C^{1} diffeomorphism on a compact Riemannian manifold with a positively invariant set Λ for which the tangent bundle has a continuous invariant splitting T_{Λ}M = E^{s} ⊕ E^{c} ⊕ E^{u}. If E^{c} has dimension 1 and f has a positive Lyapunov exponent in the E^{c} direction or a negative Lyapunov exponent in the E^{c} direction on a set of total probability, then Λ is a hyperbolic set.

Acknowledgements: We thank the participants in the workshop Concepts and Techniques in Smooth Ergodic Theory held at Imperial College, London, in July, 2001. Special thanks to the organizer, S. Luzzato, for his sponsoring of the discussions which prompted the ideias leading to this paper.

2. Preliminary lemmas

In this section we prove some general results for continuous dynamical systems on compact metric spaces. Let X be a compact metric space, f: X → X a continuous function, and φ: X → ℝ also continuous such that

(4)  lim inf_{n→+∞} \frac{1}{n} ∑_{j=0}^{n−1} φ(f^{j}x) < 0

holds in a set of points x ∈ X with total probability.

Lemma 2.1. If (4) holds in a set with total probability, then it holds for all x ∈ X.

Proof. Arguing by contradiction, let us suppose that there is some x ∈ X such that

lim inf_{n→+∞} \frac{1}{n} ∑_{j=0}^{n−1} φ(f^{j}x) ≥ 0.

Then for every k ∈ ℕ there must be an integer n_{k} for which

\frac{1}{n_{k}} ∑_{j=0}^{n_{k}−1} φ(f^{j}x) > −\frac{1}{k}.

It is no restriction to assume that n_{1} < n_{2} < ⋯ and we do it. We define the sequence of probability measures

μ_{k} = \frac{1}{n_{k}} ∑_{j=0}^{n_{k}−1} δ_{f^{j}x}, k ≥ 1,

where each δ_{f^{j}x} is the Dirac measure on f^{j}x. Let μ be a weak^{∗} accumulation point of this sequence when k → +∞. Taking a subsequence, if necessary, we assume that μ_{n_{k}} converges to μ. Standard arguments show that μ is f invariant. Since the function φ is continuous we have

∫ φ dμ = lim_{k→+∞} \frac{1}{n_{k}} ∑_{j=0}^{n_{k}−1} φ(f^{j}x) ≥ 0

by definition of μ and the way we have chosen the sequence (n_{k})_{k}.

However, since we are assuming that (4) holds in a set of total probability measure, we have that μ almost every y is such that

φ̃(y) = lim_{n→+∞} \frac{1}{n} ∑_{j=0}^{n−1} φ(f^{j}y) < 0.

On the other hand,

∫ φ̃ dμ = ∫ φ dμ ≥ 0

by the Ergodic Theorem. This gives a contradiction, thus proving the lemma. □

According to the statement of the previous lemma for each x ∈ X there are N(x) ∈ ℕ and c(x) > 0 such that

\frac{1}{N(x)} ∑_{j=0}^{N(x)−1} φ(f^{j}x) < −c(x).

Thus, by continuity, for each x ∈ X there is a neighborhood V_{x} of x such that for every y ∈ V_{x} one has

\frac{1}{N(x)} ∑_{j=0}^{N(x)−1} φ(f^{j}y) < −\frac{c(x)}{2}.

Since X is compact, there is a finite cover V_{x_{1}}, …, V_{x_{p}} of X by neighborhoods of this type. Let

(5)  \overline{N} = max\{N(x_{1}), …, N(x_{p})\} and c = min\{c(x_{1}), …, c(x_{p})\}.

We define for x ∈ X

N_{1}(x) = min{N(x_{i}): x ∈ V_{x_{i}}, i = 1, …, p}

and a sequence of maps N_{k} : M → \{1, …, \overline{N}\}, k ≥ 0, in the following way:

(6)  N_{0}(x) = 0, N_{k+1}(x) = N_{k}(x) + N_{1}(f^{N_{k}(x)}x).

(See that there is no conflict in the definition of N_{1}). For the proof of the next lemma it will be useful to take α = max_{x∈M} φ(x)

Lemma 2.2. There are C_{0} > 0 such that

∑_{j=0}^{m\overline{N}−1} φ(f^{j}x) ≤ −\frac{c}{2}m + C_{0}

for all x ∈ X and m ≥ 1.

Proof. Let us fix x ∈ X. Given m ≥ 1 we define

h = max\{k ≥ 1 : N_{k}(x) ≤ m\overline{N}\}.

We must have m\overline{N} − N_{h}(x) ≤ \overline{N}. It follows from (5) and (6) that for k ≥ 0

∑_{j=N_{k}(x)}^{N_{k+1}(x)−1} φ(f^{j}x) = ∑_{j=0}^{N_{1}(f^{N_{k}(x)})−1} φ(f^{N_{k}(x)+j}x)

≤ −\frac{c}{2}N_{1}(f^{N_{k}(x)}x)
≤ −\frac{c}{2}(N_{k+1}(x) − N_{k}(x))

Hence

∑_{j=0}^{m\overline{N}−1} φ(f^{j}x) = ∑_{j=N_{0}(x)}^{N_{1}(x)−1} φ(f^{j}x) + ⋯ + ∑_{j=N_{h−1}(x)}^{N_{h}(x)−1} φ(f^{j}x) + ∑_{j=N_{h}(x)}^{m\overline{N}−1} φ(f^{j}x)

≤ −\frac{c}{2}[(N_{1}(x) − N_{0}(x)) + ⋯ + (N_{h}(x) − N_{h−1}(x))] + α(m\overline{N} − N_{h}(x))
≤ −\frac{c}{2}N_{h}(x) + α\overline{N}
≤ −\frac{c}{2}m + α\overline{N}

(see that N_{h}(x) ≥ m\overline{N}/\overline{N} = m). Thus we just have to choose C_{0} = α\overline{N}. □

3. Proofs of main results

Here we use the results of the previous section to prove Theorems A and C. We start by assuming that f : M → M is a C^{1} local diffeomorphism nonuniformly expanding on a subset of total probability. The fact that f is a local diffeomorphism implies that the map

λ(x) = log |(df_{x})^{−1}|

is continuous on M. Moreover, the nonuniform expansion of f implies that λ is in the conditions of the previous section, namely it satisfies condition (4). Let \overline{N}, c and (N_{k})_{k} be defined as in (5) and (6) for this map λ. Take also α = max_{x∈X} λ(x).

Proposition 3.1. There are constants C > 0 and σ > 1 such that

|df_{x}^{n}(v)| ≥ Cσ^{n}|v|

for all x ∈ M, v ∈ T_{x}M and n ≥ 1.

Proof. Fix x ∈ M and 0 ≠ v ∈ T_{x}M. Observe that

|(df_{x}^{m\overline{N}})^{−1}| ≤ exp ( ∑_{j=0}^{m\overline{N}−1} λ(f^{j}x)).

Let C_{0} > 0 be as in Lemma 2.2. Taking K_{0} = e^{C_{0}} and ρ = e^{−c/2}, for all m ≥ 1 we have

|v| = |(df_{x}^{m\overline{N}})^{−1}(df_{x}^{m\overline{N}}) · v| ≤ Kρ^{m}|(df_{x}^{m\overline{N}}) · v|,

which is equivalent to

|(df_{x}^{m\overline{N}}) · v| ≥ \frac{1}{K} (ρ^{−1})^{m}|v|.

On the other hand, we have for all x ∈ M, v ∈ T_{x}M and n ≥ 1

|(df_{x}^{n}) · v| ≥ e^{−αn}|v|.

Let n ≥ 1 be given. There are m ≥ 1 and r ∈ \{0, …, \overline{N} − 1\} such that n = m\overline{N} + r, and so

|(df_{x}^{n}) · v| = |(df_{f^{m\overline{N}} x}^{r}) · (df_{x}^{m\overline{N}}) · v|
≥ e^{−rα} · |(df_{x}^{m\overline{N}}) · v|
≥ \frac{e^{−\overline{N}α}}{K} (ρ^{−1})^{m}|v|
= \frac{e^{−\overline{N}α}ρ^{r/\overline{N}}}{K} (ρ^{−1/\overline{N}})^{m\overline{N}+r}|v|
≥ Cσ^{n}|v|.

It is enough to take C = e^{−\overline{N}α}ρ/K > 0 and σ = ρ^{−1/\overline{N}} > 1. □

This proposition gives precisely the conclusion of Theorem A. For proving Theorem C it is enough to take

λ^{cu}(x) = log |df^{−1}|E_{x}^{cu}| or λ^{cs}(x) = log |df|E_{x}^{cs}|

in the place of λ(x). We use again the results of the previous section and the assumptions on f to derive the desired conclusions for df^{−1}|E_{x}^{cu} and df|E_{x}^{cs} exactly in the same way as in Proposition 3.1.

4. Final remarks

We finish this paper with several remarks related to the assumptions we made for the theorems we have proved.

Asymptotic expansion. First we present an example which illustrates the fact that having all Lypaunov exponents positive is not enough for assuring nonuniform expansion. Consider a period 2 orbit {p, q} for a local diffeomorphism f on a surface which, for a given choice of local basis at p and q, satisfies

df_{p} = ( \begin{matrix}1/2 & 0 \\ 0 & 3 \end{matrix} ) and df_{q} = ( \begin{matrix}3 & 0 \\ 0 & 1/2 \end{matrix} ) .

Then it is clear that both Lyapunov exponents at p or q are log(3/2) > 0 and the limit in (2) with x = p or q equals log 2 > 0.

Eventual expansion. Secondly we observe that the proof of Theorem A can be carried out under the assumption that f is eventually expanding at every point, meaning that for each x ∈ M there is N(x) ∈ ℕ such that

|(df_{x}^{N(x)})^{−1}| < 1.

This is essentially what we use in Lemma 2.2 and Section 3. A similar observation holds for Theorem C.

Continuous splitting. The final comment concerns the continuity of the splitting in Theorem C and Corollaries D and E. It is known [12, Exercice 4.1] that if a C^{1} map f admits an invariant compact set Λ with an invariant splitting T_{Λ}M = E^{cs} ⊕ E^{cu} satisfying the hyperbolicity conditions (3), then this splitting is continuous. Therefore the continuity assumption on the splitting for our theorems seems natural.

References

[1] J. F. Alves, V. Araújo, Random perturbations of nonuniformly expanding maps, preprint CMUP 2000.

[2] J. F. Alves, V. Araújo, Random perturbations of partially hyperbolic systems, in preparation.

[3] J. F. Alves, C. Bonatti, M. Viana, SRB measures for partially hyperbolic systems with mostly expanding central direction, Invent. Math. 140 (2000), 351-398.

[4] C. Bonatti, M. Viana, SRB measures for partially hyperbolic systems with mostly contracting central direction, Israel J. Math. 115 (2000), 157-193.

[5] H. Bruin, G. Keller, Equilibrium states for S-unimodal maps, Erg. Th. & Dyn. Sys. 18 (1998) 765-789.

[6] F. Hofbauer, G. Keller, Quadratic maps without asymptotic measure, Comm. Math. Phys. 127 (1990) 319-337.

[7] Y. Kifer, General random perturbations of hyperbolic and expanding transformations, J. Anal. Math. 47 (1986), 111-150.

[8] K. Krzyzewski, W. Szlenk, On invariant measures for expanding differentiable mappings, Stud. Math. 33 (1969), 83-92.

[9] R. Mañé, Hyperbolicity, sinks and measure in one-dimensional dynamics, Comm. in Math. Phys. 100 (1985) 495-524. Erratum, Comm. in Math. Phys. 112 (1987) 721-724.

[10] V. Pliss, On a conjecture due to Smale, Diff. Uravnenija 8 (1972), 262-268.

[11] D. Ruelle, The thermodynamical formalism for expanding maps, Com. Math. Phys. 125 (1989), 239-262.

[12] M. Shub, Global Stability of Dynamical Systems, Springer-Verlag, New York (1987).

[13] A. Tahzibi, Stably ergodic systems that are not partially hyperbolic, Preprint IMPA 2001.

Centro de Matemática da Universidade do Porto, 4099-002 Porto, Portugal

E-mail address: jfalves@fc.up.pt
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Centro de Matemática da Universidade do Porto, 4099-002 Porto, Portugal

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E-mail address: benoit.saussol@u-picardie.fr
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