b = −β for some β > 0, and let → 0. The reason for that particular dependence for the endpoints is to determine the effect of the boundary condition at a, ignoring the effect from the other endpoint b. We show that we can choose C0 such that the limiting dynamics feels both the fluctuations produced by the noise and a deterministic drift repelling the motion from the left, which is interpreted as a soft wall effect. Under a further rescaling, we prove convergence to a Brownian motion reflected at the origin, which can be seen a hard wall.

2 Results and Strategy of Proofs

The above discussed results hold for a potential V having the stated properties, but in order to fix ideas and to be able to compute the exact coefficients in what follows, we choose

V (m) = 1 m2 − 1 2,

4

which attains its minimum at ±1 and yields m(x) = tanh(x).

For each > 0 we consider m( )(x, t ) the solution of (3) with boundary conditions m( )(−a, t ) = −1 and m( )(b, t ) = 1 and W (dx, dt ) a space-time white noise, that is, a centered Gaussian field with

E W (dx, dt )W (dx , dt ) = δ(x − x )δ(t − t )

for δ a Dirac delta. The usual way to give a precise meaning to the above is to write down an integral equation corresponding to (3) in terms of a heat kernel, to observe that then each term makes sense and to prove that the integral equation admits a solution continuous in both variables with probability one. (See for instance [12].) This is called a mild solution, and to that one we refer in what follows. For f a

Then:

(i) There exists an Ft -adapted real process X such that, for each θ , η > 0,

We would like to prove that the solution remains close to M for times of order of λ −1 , and to identify X (t ) satisfying (6) and (iii), but this cannot be obtained directly from (8). The strategy we follow is to consider iterations of the linearization as described, over consecutive intervals of length T = −γ , updating at the beginning of the k − t h interval the location zk of the interface mzk around which the linearization is performed. The zk that showed to be convenient is the “center”

Except for technicalities, the process X (t ) in the statement is given by the center of the solution m(t ). It is not difficult to obtain that the first order approx-

for t [Tk , Tk+1] as explained we obtain, after projecting both sides of an equation analogous to (8), an approximation for the difference of the centers of the solution on that interval:

where N LT are the non linear terms appearing between square brackets in (8). The next step is to sum in k, for k = 0, 1, . . . , [t −1+γ ], and to prove first that,

as → 0, the zk remain bounded, so basically (6) holds. Rough estimates follow from Gaussian estimates for the terms coming from the noise, conditioned on the value of the zk , as for instance in [2].

Second, we show that what we obtain after taking → 0 is a discrete version of the integral equation for Y (t ). To give an idea of why this is true, let us analyze briefly the terms in (9). To really accomplish the program (including controlling the errors behind the symbol ≈) involves very precise estimates on the linear operators H (zk ) in finite intervals whose extremes depend on as described. See [11] and [20].

It can be seen that the first term in the second line does not contribute to the limit, from estimates on the spectral gap for H (zk ) uniform in and from the definition of center.

That the non linear terms also do not contribute follows from a precise analysis of the sum in k of the corresponding terms. For the last term in (8) we have that

limiting equation. Finally, for the remaining term we have − 34 ϕzk − gT(zk )(ϕzk ),

sponding eigenvector of H (zk ). Since we can show λ0 ≈ 24 exp 4(a + zk ), after estimating Ψ0 and computing ϕzk it is possible to conclude that this term gives precisely the drift in (7).

Finally, to prove (iii), we show that the previous analysis can be extended for times of the order λ −1, and then, as the equation

suggests, we obtain convergence to a reflected Brownian motion. (See [18] for basic concepts).

As a final comment, let us say that the analysis can be extended to the symmetric interval [−a, b] with a = b = 14 log( −1), obtaining for the corresponding process Y (t ) describing the motion of the interface the equation

dY (t ) = 24 sinh(4Y (t ))dt + dB(t ),

where it appears a wall effect from both sides.

The result is used in [4] to show convergence ( as → 0) of the invariant measure for the solution of (3) with boundary conditions ±1 in the symmetric interval to a nontrivial limit related to the invariant measure for the limiting process Y (t ).

References

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3.L. Bertini, S. Brassesco, and P. Buttà, Soft and hard wall in a stochastic reaction diffusion equation. Arch. Ration. Mech. Anal. (to appear)

4.L. Bertini, S. Brassesco, and P. Buttà, Dobrushin states in the φ14 model. Arch. Ration. Mech. Anal. (to appear)

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J.Dyn. Differ. Equ. 1, 75–94 (1989)

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hktop(f ) “counts” the number of orbits starting from an arbitrary compact and smooth k-dimensional submanifold. We both recall known properties and establish new ones. We then recall the definition of entropy-expanding maps which generalize the complexity of interval dynamics with non-zero topological entropy. We also introduce a similar notion for diffeomorphisms:

Definition 1. A diffeomorphism of a d-dimensional manifold is entropy-hyperbolic if there are integers du, ds such that:

•hdtopu (f ) = htop(f ) and this fails for every dimension k < du;

•hdtops (f −1) = htop(f ) and this fails for every dimension k < ds ;

•du + ds = d.

We give simple sufficient conditions for entropy – expansion and entropy – hyperbolicity. Finally we announce some work in progress and state a number of questions.

We now recall some classical notions which may be found in [20]. In this paper, all manifolds are compact.

A basic measure of orbit complexity of a map f : M → M is the entropy. The topological entropy htop(f ) “counts” all the orbits and the measure-theoretic entropy (also known as Kolmogorov-Sinai entropy or ergodic entropy) h(f, μ) “counts” the orbits “relevant” to some given invariant probability measure μ. They are related by the following rather general variational principle. If, e.g., f is continuous and M is compact, then

htop(f ) = sup h(f, μ)

μ

where μ ranges over all invariant probability measures. One can also restrict μ to ergodic invariant probability measures.

This brings to the fore measures which realize the above supremum, when they exist, and more generally measures which have entropy close to this supremum.

As μ → h(f, μ) is affine, μ has maximum entropy if and only if almost every ergodic component of it has maximum entropy. Hence, with respect to entropy, it is enough to study ergodic measures.

Definition 2. A maximum measure is an ergodic and invariant probability measure

μ such that h(f, μ) = supν h(f, ν).

A large entropy measure is an ergodic and invariant probability measure μ such that h(f, μ) is close to supν h(f, ν).

The Lyapunov exponents for some ergodic and invariant probability measure μ are the possible values μ-a.e. of the limit limn→∞ n1 log Tx f n.v where · is some Riemannian structure and Tx f is the differential of f and v ranges over the non-zero vectors of the tangent space Tx M.

A basic result connecting entropy and hyperbolicity is the following theorem (proved by Margulis for volume preserving flows):

Theorem 3 (Ruelle’s inequality). Let f : M → M be a C1 map on a compact manifold. Let μ be an f -invariant ergodic probability measure. Let λ1(μ) ≥ · · · be its Lyapunov exponents repeated according to multiplicity. Then,

In good cases (with enough hyperbolicity), the entropy is also reflected in the existence of many periodic orbits:

Definition 4. The periodic points of some map f : M → M satisfy a multiplicative lower bound, if, for some integer p ≥ 1:

lim inf e−nhtop(f )#{x [0, 1] : f nx = x} > 0.

n→∞,p|n

Recall that many diffeomorphisms have infinitely many more periodic orbits (see [17, 18]).

The following type of isomorphism will be relevant to describe all “large entropy measures”.

Definition 5. For a given measurable dynamical system f : M → M, a subset S M is entropy-negligible if there exists h < supμ h(f, μ) such that for all ergodic and invariant probability measures μ with h(f, μ) > h, μ(S) = 0.

An entropy-conjugacy between two measurable dynamical systems f : M → M and g : N → N is a bi-measurable invertible mapping ψ : M \ M0 → N \ N0 such that: ψ is a conjugacy (i.e., g ◦ ψ = ψ ◦ f ) and M0 and N0 are entropy-negligible.

2 Low Dimension

Low dimension dynamical systems here means interval maps and surface diffeomor- phisms—those systems for which non-zero entropy is enough to ensure hyperbolicity of the large entropy measures.

2.1 Interval Maps

Indeed, an immediate consequence of Ruelle’s inequality on the interval is that a lower bound on the measure-theoretic entropy gives a lower bound on the (unique) Lyapunov exponent. Thus, invariant measures with nonzero topological entropy are hyperbolic in the sense of Pesin. One can obtain much more from the topological entropy:

Theorem 6. Let f : [0, 1] → [0, 1] be C∞. If htop(f ) > 0 then f has finitely many maximum measures. Also the periodic points satisfy a multiplicative lower bound.

98 Jérôme Buzzi

This was first proved by F. Hofbauer [15, 16] for piecewise monotone maps (admitting finitely many points a0 = 0 < a1 < · · · < aN = 1 such that f |]ai , ai+1[ is continuous and monotone). It was then extended to arbitrary C∞ maps in [5]. In

both settings, one builds an entropy-conjugacy to a combinatorial model called a Markov shift (which is a subshift of finite type over an infinite alphabet). One can

phism between the spaces of invariant probability measures which respects entropy and ergodicity.

Theorem 7. The natural extensions of C∞ interval maps with non-zero topological entropy are classified up to entropy-conjugacy by their topological entropy and finitely many integers (which are “periods” of the maximum measures).

The classification is deduced from the proof of the previous theorem by using a classification result [2] for the invertible Markov shifts involved.

The C∞ is necessary: for each finite r, there are Cr interval maps with nonzero topological entropy having infinitely many maximum measures and others with none.

Remark 8. These examples show in particular that Pesin hyperbolicity of maximum measures or even of large entropy measures (which are both consequences of Ruelle’s inequality here) are not enough to ensure the finite number of maximum measures.

2.2 Surface Transformations

As observed by Katok [19], Ruelle’s inequality applied to a surface diffeomorphism and its inverse (which has opposite Lyapunov exponents) shows that a lower-bound on measure-theoretic entropy bounds away from zero the Lyapunov exponents of the measure. Thus, for surface diffeomorphisms also, nonzero entropy implies Pesin hyperbolicity.

It is believed that surface diffeomorphisms should behave as interval maps, leading to the following folklore conjecture:

Conjecture 9. Let f : M → M be a C∞ surface diffeomorphism. If htop(f ) > 0 then f has finitely many maximum measures.

I would think that, again like for interval maps, finite smoothness is not enough for the above result. However counter-examples to this (or to existence) are known only in dimension ≥ 4 [22].

The best result for surface diffeomorphisms at this point is the following “approximation in entropy” [19]:

very reasonable definitions of Cr size), Fact 57, which is used to prove Lemma 18 below.

From now on, we fix some Cr size arbitrarily on the manifold M. We will later check that the entropies we are interested in are in fact independent of this choice.

Notations. It will be convenient to sometimes write φ instead of φ (Qk ), e.g., htop(f, φ) instead of htop(f, φ (Qk )).

3.2 Entropy of Collections of Subsets

Given a collection D of subsets of M, we associate the following entropies. Recall that the ( , n)-covering number of some subset S M is:

where Bf ( , n, x) := {y M : 0 ≤ k < n d(f k y, f k x) < } is the ( , n)- dynamic ball. The classical Bowen-Dinaburg formula for the topological entropy

of S M is htop(f, S) = lim →0 lim supn→∞ n1 log rf ( , n, S) and htop(f ) = htop(f, M).

Clearly htop(f, D ) ≤ Htop(f, D ). The inequality can be strict as shown in the following examples (the first one involving non-compactness, the second one involving non-smoothness).

Example 14. Let T : T2 → T2 be a linear endomorphism with two eigenvalues Λ1, Λ2 with 1 < |Λ1| < |Λ2|. Let L be the set of finite line segments. We have

0 < htop(T , L ) = log |Λ2| < Htop(T , L ) = log |Λ1| + log |Λ2|.

Example 15. There exist a C∞ self-map F of [0, 1]2 and a collection C of Cr curves with bounded Cr norm such that 0 < htop(F, C ) < Htop(F, C ). This can be de-

duced from the example with h1top(f × g) > max(htop(f ), htop(g)) in [6] by considering curves with finitely many bumps converging Cr to the example curve there,

which has infinitely many bumps.

3.3 Definitions of the Dimensional Entropies

We can now properly define the dimensional entropies. Recall that we have endowed M with a Cr size.

Definition 16. For each 1 ≤ r ≤ ∞, the standard family of Cr singular k-disks is the collection of all Cr singular k-disks. For finite r, the standard uniform family of Cr singular k-disks is the collection of all Cr singular k-disks with Cr size bounded by 1.

Definition 17. The Cr , k-dimensional entropy of a self-map f of a compact mani-

fold is:

hk,Ctop r (f ) := htop(f, Drk )

where Drk is a standard family of Cr k-disks of M. We write hktop(f ) for hk,C∞ (f )top and call it the k-dimensional entropy.

The Cr , k-dimensional uniform entropy Htopk,Cr (f ) is obtained by replacing

htop(f, Drk ) with Htop(f, Drk ) in the above definition where Drk is the standard uniform family. We write Htopk (f ) for Htopk,C∞ (f ) and call it the k-dimensional uniform

entropy.

Observe that hk,Ctop r (f ) and Htopk,Cr (f ) are non-decreasing functions of k and nonincreasing functions of r. Indeed, (1) Drk Dsk and Drk Dsk if r ≤ s; (2) for any

Lemma 18. Let f : M → M be a C∞ self-map of a compact manifold. We have:

Proof. We use one of the (simpler) ideas of Yomdin’s theory. For each n ≥ 1, we

≤Lip(f )−n.

2

It follows that rf ( , n, φ ∩ q) ≤ rf ( /2, n, φq ). Thus,

rf ( , n, Drk ) ≤ √kk ( /4)−k/r φ k/rCr Lip(f ) kr nrf ( /2, n, D∞k ).

Theorem 21 (Newhouse [23]). Let f : M → M be a C1+α , α > 0, smooth selfmap of a compact manifold. Then:

htop(f ) ≤ γ (f ).

Remark 22. More precisely, his proof gave

htop(f ) ≤ γ dcu (f )

where dcu is such that the variational principle htop(f ) = supμ h(f, μ) still holds when μ is restricted to measures with exactly dcu nonpositive Lyapunov exponents.

For C1+1-diffeomorphisms, deeper ergodic techniques due to Ledrappier and Young are available and Cogswell [11] has shown that, for any ergodic invariant probability measure μ, there exists a disk such that htop(f, μ) ≤ htop(f, Δ) ≤ γ (f, Δ). More precisely, the dimension of this disk is the number of positive Lyapunov exponents. For C∞ diffeomorphisms (more generally if there is a maximum measure) there exists a disk max such that

htop(f ) = htop(f, max) = γ (f, max).

I do not know if Newhouse’s inequality fails for C1 maps.

The proof of Newhouse inequality involves ergodic theory and especially Pesin theory. Indeed, this type of inequality does not hold uniformly:

Example 23. There exist a C∞ self-map F of a surface and a C∞ curve φ such that, for some sequence ni → ∞,

map such that: (i) f (0) = f (1) = 0; (ii) f (1/2) = 1; (iii) f |[0, 1/2] is increasing and f |[1/2, 1] is decreasing; (iv) f |[0, 1/2 − α] = 2(1 + α) and f |[1/2 + α] =

−2(1+α). As α is small, 1/2 has a preimage in [0, 1/2−α]. Let x−n be the leftmost preimage in f −n(1): x0 = 1, x−1 = 1/2, and for all n ≥ 2, x−n = 2−n(1 + α)−n+1. Let g : I → I be another C∞ map such that: (i) g(0) = 0; (ii) 0 < g < 1; (iii)

g(x−n) = x−n−1 for all n ≥ 0.

Consider the following composition of length 3n for some n ≥ 1:

Observe that after time 2n, the length of the curve gn ◦ f n is 2n · x−n = (1 + α)−n+1 whereas the number of ( , n)-separated orbits is less than −1n2n. After time 3n, the curve f n ◦ gn ◦ f n has image I with multiplicity 2n. It is therefore easy to analyze the dynamics of compositions of such sequences.

We build our example by considering a skew-product for which the curve will be a fiber over a point which will drive the application of sequences as above.

whenever its nth digit is 2. Thus we can specify the desired compositions of f and g just by picking θ S1 with the right expansion. We pick:

F Ni +2ni+1 ◦ φ1 has image [0, x−ni+1 ] with multiplicity 2 13 Ni × 2ni+1 . It follows that, setting ti := Ni + 2ni+1 ≈ 2ni+1,

as claimed.

Remark 24. The inequality in the previous example is obtained as the length is contracted after a large expansion. For curves, this is in fact general and it is easily shown that, for any C1 1-disk φ with unit length, for any 0 < < 1:

both quantities being defined by lim sup (this would fail using lim inf). However, one can find similarly as above, a C∞ self-map of a 3-dimensional compact manifold and a C∞ smooth 2-disk such that (2) fails though (3) seems to hold.

We ask the following:

Question 25. Let f : M → M be a C∞ self-map of a compact d-dimensional manifold. Is it true that, for any singular k-disk ψ (0 ≤ k ≤ d)

These might even hold for finite smoothness for all I know.

Conversely, entropy also provides some bounds on volume growth

Theorem 26 (Yomdin [27]). Let f : M → M be a Cr , r ≥ 1, smooth self-map of a compact manifold. Let α > 0. Then there exist C(r, α) < ∞ and 0(r) > 0 with the following property. Let φ : Qk → M be any Cr singular k-disk with unit Cr size for some 0 ≤ k ≤ d. Then, for any n ≥ 0,

vol(f n ◦ φ) ≤ C(r, α)Lip(f )( kr +α)n · rf ( 0, n, φ).

In particular,

γ k (f ) ≤ hktop(f ) + k lip(f ). r

Remark 27. The above extra term is indeed necessary as shown already by examples attributed by Yomdin [27] to Margulis: there is f : [0, 1] → [0, 1], Cr with htop(f ) = 0 and γ (f ) = lip(f )/r.

Remark 28. Yomdin’s estimate is uniform holding for each disk and each iterate. Its proof involves very little dynamics and no ergodic theory, in contrast to Newhouse’s inequality quoted above.

Corollary 29. Let f : M → M be a self-map of a compact manifold. If f is C∞, then

htop(f ) = γ (f ).

the volume, we have γ (f ) ≥ log ρ(f ). Hence, the following special case of the Shub Entropy Conjecture is proved:

Corollary 30 (Yomdin [27]). Let f : M → M be a self-map of a compact manifold. If f is C∞, then

log ρ(f ) ≤ htop(f ).

4.2 Resolution Entropies

The previous results of Yomdin and more can be obtained by computing a growth rate taking into account the full structure of singular disks. A variant of this idea is explained in Gromov’s Bourbaki Seminar [12] on Yomdin’s results. We build on [4].

Definition 31. Let r ≥ 1. Let φ : Qk → M be a Cr singular k-disk. A Cr -resolution R of order n of φ is a collection of Cr maps ψω : Qk → Qk , for ω Ω with Ω a

finite collection of words of length |ω| at most n with the following properties. For each ω Ω, let Ψω := ψσ |ω|−1ω ◦ · · · ◦ ψω(Qk ) (σ deletes the first symbol). We

require:

1.|ω|=n Ψω(Qk ) = Qk ;

2.ψω Cr ≤ 1 for all ω Ω;

3.f |ω| ◦ Ψω Cr ≤ 1 for all ω Ω.

The size |R| of the resolution is the number of words in Ω with length n.

Condition (2) added in [4] much simplifies the link between resolutions and entropy. It no longer relies on Newhouse application of Pesin theory and becomes straightforward:

{Ψω(t ) : t Qk and ω Ω with |ω| = n} is a ( , n)-cover of φ (Qk ).

On the other hand, the notion of resolution induces entropy-like quantities:

Definition 33. Let 1 ≤ r < ∞ and let f : M → M be a Cr self-map of a compact manifold. Let Rf (Cr , n, φ) be the minimal size of a Cr -resolution of order n of a Cr singular disk φ. The resolution entropy of φ is:

hR (f, φ) := lim sup 1 log Rf (Cr , n, φ).

n→∞ n

If D is a collection of Cr singular disks, its Cr resolution entropy is

hR,Cr (f, D ) := sup hR (f, φ)

φ D

and its Cr uniform resolution entropy is:

HR,Cr (f, D ) := lim sup 1 log Rf (Cr , n, D )

n→∞ n

where Rf (Cr , n, D ) := maxφ D Rf (Cr , n, φ). We set:

hk,CR r (f ) = hR,Cr (f, Drk ) and HRk,Cr (f ) = HR,Cr (f, Drk ).

The following is immediate but very important:

multiplicative:

Rf (Cr , n + m, Drk ) ≤ Rf (Cr , n, Drk )Rf (Cr , m, Drk ).

The key technical result of Yomdin’s theory can be formulated as follows:

Proposition 35. Let 1 ≤ r < ∞ and α > 0. Let f : M → M be a Cr self-map of a compact manifold. There exist constants C , C(r, α), 0(r, α) with the following property. For any Cr singular disk φ, any number 0 < < 0(r, α) and any integer n ≥ 1,

C k rf ( , n, φ) ≤ Rf (Cr , n, φ) ≤ C(r, α)Lip(f )( kr +α)nrf ( , n, φ).

Remark that the above constants depend on f . The first inequality follows from Fact 32. The second is the core of Yomdin theory, we refer to [4] for details.

5 Properties of Dimensional Entropies

We turn to various properties of dimensional entropies, most of which can be shown using resolution entropy and its submultiplicativity.

5.1 Link between Topological and Resolution Entropies

We start by observing that Proposition 35 links the topological and resolution entropies.

Corollary 36. For all positive integers r, k, any collection of Cr k-disks D and any Cr self-map f on a manifold equipped with a Cr size:

htop(f, D ) ≤ hR,Cr (f, D ) ≤ htop(f, D ) + k log Lip(f ) r

Htop(f, D ) ≤ HR,Cr (f, D ) ≤ Htop(f, D ) + k log Lip(f ). r

If the disks in D are C∞, then, for r ≤ s < ∞,

hR,Cr (f, D ) ≤ hR,Cs (f, D ) ≤ htop(f, D ) + k log Lip(f ). s

Letting s → ∞, we get:

Corollary 37. If f is C∞, then, for all 1 ≤ r < ∞, hR,Cr (f, D∞k ) = htop(f, D∞k ).

The same holds for uniform topological entropy.

5.2 Gap Between Uniform and Ordinary Dimensional Entropies

Yomdin theory gives the following relation:

Proposition 38. Let 1 ≤ r < ∞ and f : M → M be a Cr self-map of a compact d-dimensional manifold. For each 0 ≤ k ≤ d,

Proof. It is obvious that the uniform entropies dominate ordinary ones. By Fact 32, hk,Ctop r (f ) ≤ hk,CR r (f ) and Htopk,Cr (f ) ≤ HRk,Cr (f ). Therefore it is enough to show:

Let α > 0. Let 0 > 0 as in Proposition 35. This proposition defines a number C(r, α). By definition, for every φ Drk , there exists nφ < ∞ such that

k,Cr +

rf ( 0, nφ , φ) ≤ e(htop (f ) α)nφ .

We can arrange it so that this holds for all k-disks ψ in some C0 neighborhood Uφ of φ. We also assume nφ so large that C(r, α) ≤ eαnφ .

By Proposition 35, each such ψ admits a resolution with size at most

rf ( , nφ , ψ ) × C(r, α)Lip(f ) kr nφ ≤ e(hk,CR r (f )+ kr lip(f )+2α)nφ .

Drk is relatively compact in the C0 topology, hence there is a finite cover Drk Uφ1 · · · UφK . Let N := max nφj . It is now easy to build, for each n ≥ 0 and each ψ Drk a Cr resolution R of order n with:

Equation (4) follows by letting α go to zero.

5.3 Continuity Properties

Proposition 39. We have the following upper semicontinuity properties:

Proof. We prove (1). The sub-multiplicativity of resolution numbers observed in Fact 34 implies that: HRk,Cr (f ) = infn≥1 n1 log Rf (Cr , n, Drk ). For each fixed positive integer n, Rg (Cr , n, Drk ) ≤ 2k Rf (Cr , n, Drk ) for any g Cr -close to f (use a

linear subdivision). Thus f → HRk,Cr (f ) is upper semi-continuous in the Cr topology.

We deduce (3) from (1). Let fn → f in the Cr topology. By the preceding, HRk,Cr (f ) ≥ lim supn→∞ HRk,Cr (fn). By Proposition 38, Htopk (f ) ≥ HRk,Cr (f ) −

kr lip(f ).

(2) follows from (3) using Lemma 18.

On the other hand, f → Htopk (f ) fails to be lower semi-continuous except for interval maps for which topological entropy is lower semi-continuous in the C0 topology by a result of Misiurewicz. In every case there are counter examples:

Example 40. For any d ≥ 2 and 1 ≤ k ≤ d, there is a self-map of a compact manifold of dimension d at which hktop(f ) fails to be lower semi-continuous.

Let h : R → [0, 1] be a C∞ function such that h(t ) = 1 if and only if t = 0. Let Fλ : [0, 1]d → [0, 1]d be defined by

Fλ(x1, . . . , xd ) = (h(λ)x1, 4x1x2(1 − x2), x3, . . . , xd ).

Observe that if λ = 0, then h(λ) [0, 1) and Fλn(x1, . . . , xd ) approaches {(0, 0)} × on which Fλ is the identity. Therefore htop(Fλ) = 0. On the other hand htop(F0) = htop(x → 4x(1 − x)) = log 2. Now, Htopk (Fλ) ≤ htop(Fλ) = 0 for any

λ = 0 and Htopk (F0) ≥ hktop(f ) ≥ htop(F0, {1} × [0, 1] × {(0, . . . , 0)}) = log 2 for any k ≥ 1.

6 Hyperbolicity from Entropies

We now explain how the dimensional entropies can yield dynamical consequences. We start by recalling an inequality which will yield hyperbolicity at the level of measures. Then we give the definition and main results for entropy-expanding maps. Finally we explain the new notion of entropy-hyperbolicity for diffeomorphisms.

6.1 A Ruelle-Newhouse Type Inequality

One of the key uses of dimensional entropies is to give bounds on the exponents using the following estimate. This will give hyperbolicity of large entropy measure from assumptions on these dimensional entropies.

Theorem 41 ([7]). Let f : M → M be a Cr self-map of a compact manifold with r > 1. Let μ be an ergodic, invariant probability measure with Lyapunov exponents