Examples of $\mathcal{C}^r$ interval map with large symbolic extension entropy

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extension entropy

David Burguet

To cite this version:

David Burguet. Examples of

interval map with large symbolic extension entropy. Discrete and

Continuous Dynamical Systems - Series A, American Institute of Mathematical Sciences, 2010, 26 (3),

pp.873-899. �10.3934/dcds.2010.26.873�. �hal-00370669�

extension entropy

David Burguet, CMLS - CNRS UMR 7640

Ecole polytechnique 91128 Palaiseau Cedex France

Abstract : For any integer r ≥ 2 and any real > 0, we construct an explicit example of C^r interval map f with symbolic extension entropy hsex(f) ≥ _r−1^r logkf⁰k_∞− and kf⁰k_∞ ≥ 2.

T.Downarawicz and A.Maass [11] proved that for C^r interval maps with r > 1, the symbolic extension entropy was bounded above by _r−1^r logkf⁰k_∞. So our example prove this bound is sharp.

Similar examples had been already built by T.Downarowicz and S.Newhouse for diffeomorphisms in higher dimension by using generic arguments on homoclinic tangencies.

1 Introduction

1.1 Entropy of symbolic extensions

Let T be a dynamical system defined on a compact metrizable space X. We denote M(X, T) the set of invariant probability measures of (X, T) endowed with the weak star topology and Me(X, T)⊂ M(X, T) the subset of ergodic measures. A symbolic extension (Y, S) of (X, T) is an extension which is a subshift of a full shift over a finite alphabet. Given a dynamical sytem one can wonder if it admits a symbolic extension and how far this extension is from the initial system in the point of view of entropy. The symbolic extension entropy function estimates this defect.

Letπ: (Y, S)→(X, T) be a symbolic extension. We consider the fonctionh^π_ext :M(X, T)→R defined byh^π_ext(µ) := sup_π∗ν=µh(ν), wherehdenotes the usual Kolmogorov-Sinai entropy. Then the symbolic extension entropy functionhsex:M(X, T)→Ris defined as follows :

hsex(µ) := inf

π:(Y,S)→(X,T)h^π_ext(µ)

where the infinimum is taken over all symbolic extensions (Y, S) of (X, T) (when there is no symbolic extension, we puthsex≡+∞).

Finally the topological symbolic extension entropy hsex(T) is the infimum of the topological entropy of the symbolic extensions of (X, T) :

hsex(T) = inf

π:(Y,S)→(X,T)htop(T)

In fact the topological symbolic extension entropy hsex(T) is equal to the supremum of the symbolic extension entropy functionhsex[1].

M.Boyle and T.Downarowicz [1] reduce the problem of existence of symbolic extensions to the study of the convergence of the entropy computed at finer and finer scale. Let us explain more precisely their main result.

Leth_k :M(X, T)→Rbe the Katok entropy (cf Appendix) computed with precision_k where (_k)_k is a decreasing sequence converging to zero. One can define by induction the following transfinite sequence (for a real mapf onM(X, T) we denotefethe smallest upper-semicontinuous

1

function larger thanf ¹) :

• u0:= 0 ;

• ifαis a successor ordinal²:

u_α:= lim

k→+∞

u_α−1^+h−h_k

• ifαis a limit ordinal :

u_α:=sup^

β<α

u_β

The sequence (u_α)_αis stationary at some countable stepα^∗ (the ordinalα^∗is called the order of accumulation of (X, T)). The main result of [1] can then be stated in the following way :

hsex=h+uα^∗ (1)

1.2 Tail entropy

The tail entropy [15] of a dynamical system estimates the entropy appearing at arbitrarily small scales :

h^∗(T) = lim

→0lim

δ→0lim sup

n→+∞

1

nlog sup

x∈X

r(n, δ, B(x, n, ))

where B(x, n, ) :={y ∈X, ∀k= 0, ..., n−1, d(T^kx, T^ky)< } is the usual Bowen ball. In fact the tail entropy satisfies also a variational principle [9, 4] and can be also written in terms ofu₁ and the sequenceh_k :

h^∗(T) = lim

k→+∞kh−h_kk_∞= sup

µ∈M(X,T)

u₁(µ) (2)

A system is said to be asymptoticallyh-expansive if h^∗(T) = 0. It was proved by M. Boyle, D.

Fiebig, U. Fiebig [3] that any asymptotically h-expansive system satisfiesh_sex =h. Moreover it follows from Yomdin’s theory [7] thath^∗(T) = 0 forC^∞dynamical systems defined on a compact smooth manifoldM and thereforehsex=hfor such systems. In fact Yomdin’s theory provides us the following upper bound on the tail entropy forC^rsystems, which is due to J.Buzzi [7] :

h^∗(T)≤ dim(M)R(T)

r (3)

where R(T) := limn 1

nlogk(Tⁿ)⁰k∞ for any riemmanian metrickk onM. This upper bound (3) is known to be sharp [14],[7].

1.3 Existence of symbolic extensions for C

maps

It is still unknown if generalC^rdynamical systems admit symbolic extensions. But it was recently proved by A.Maass and T.Downarowicz in the case of interval maps [11]. Ifνis an ergodic measure of aC¹ interval mapf, one can define its Lyapounov exponentχ(ν) :=R

log|f⁰|dν. We consider χ⁺= max(χ,0) and we denoteχ⁺its harmonic extension onM([0,1], f) (the functionχ⁺is given by the formula χ⁺(µ) = limn→+∞1

n

R max(log|fⁿ|⁰(x),0)dµ(x) for allµ∈ M([0,1], f)). Observe thatχ⁺(µ)≤logkf⁰k_∞.

1iff is bounded, then the functionfecan be written in the following form : fe(µ) = lim sup_ν→µf(ν) ; iff is unbounded, then we putfe≡+∞

2in this case we denoteα−1 the ordinal precedingα

Theorem 1 [11] Let r >1. Letf : [0,1]→[0,1]be a C^r map, then for all ordinal αand for all µ∈ M([0,1], f),

uα(µ)≤χ⁺(µ)

r−1 (4)

Moreover for alln∈N,

un(µ)≤

n

X

k=1

χ⁺(µ)

r^k (5)

In particular (according to (1)),

h_sex(f)≤h_top(f) +logkf⁰k_∞ r−1 In higher dimension we conjecture :

Conjecture 1 Let r > 1. Let T : M → M is a C^r map, then for all ordinal α and for all µ∈ M(X, T),

uα(µ)≤ Pd

i=1χ⁺_i (µ) r−1 where (χi)i=1,...,d denote the dLyapounov exponents.

Moreover foralln∈N,

un(µ)≤

n

X

k=1

1 r^k

!



d

X

i=1

χ⁺_i(µ)





In particular,

hsex(T)≤htop(T) +logR(T)

r−1 (6)

1.4 Previous examples of higher dimensional diffeomorphisms with large symbolic extension entropy

S.Newhouse and T.Downarowicz [10] built examples ofC^r(r >1) diffeomorphism on any manifold of dimension ≥2 such that sup_µ∈M(X,T)hsex(µ) is equal to ^dim(M_r−1^)R(T)r. Therefore their exam- ples would prove the upper bound (6) is sharp. They also gave C¹ examples without symbolic extensions. Their examples are generic and the construction use homoclonic tangencies.

M.Boyle and T.Downarowicz [2] built explicitly a C^r example on a manifold of dimension 4 withhsex(T)> htop(T) by adapting an example ofC^rdiffeomorphism without measure of maximal entropy due to M.Misiurewicz [14].

1.5 Main statements

In the following paper we prove that Theorem 1 is sharp.

Theorem 2 Let r∈N^∗. There exists aC^r interval map fr: [0,1]→[0,1]fixing 0, such that for all integersn≥1 :

u_n(δ₀) =

n

X

k=1

1 r^k

!

logkf_r⁰k_∞>0 where δ₀∈ M([0,1], f_r) denotes the dirac measure at the point0.

In particular, ifω is the first ordinal with infinite cardinal, we have :

• if r >1, then u_ω(δ₀) =^log_r−1^kfr0k∞ ;

• if r= 1, then uω≡+∞ and thereforef1 does not admit symbolic extensions.

Recall thatχ⁺(µ)≤logkf_r⁰k_∞ for all invariant measureµ. Therefore the inequalities (4) and (5) of Theorem 1 are sharp forC^r interval maps.

Remark 1 One could also wonder if for all C^r (r >1) interval mapsf and for alln∈Nwe have supµ∈M([0,1],f)(un+1−un)(µ)≤ ^log_r^kfn+1^0k∞. In fact it is false : we explain in Section 3.5 how to modify the previous example to get a counter-example.

Recall that T.Downarowicz and A.Maass (Theorem 1) obtain the following upper bound on the topological symbolic extension entropy of aC^r (r >1) interval mapf :

hsex(f)≤htop(f) +logkf⁰k_∞

r−1 ≤ rlogkf⁰k_∞ r−1

By using the construction of the previous example we prove this upper bound is sharp in the following sense :

Theorem 3 Let r ≥ 2 be an integer. For any > 0, there exists a C^r interval map f_r, with kf_r,⁰ k_∞≥2 such that :

hsex(fr,)≥ rlogkf_r,⁰ k∞

r−1 −

But we do not know if our example can provide a new one satisfying h_sex(f_r) = h_top(f_r) +

logkf_r⁰k∞

r−1 .

Our examples are in the spirit of those of T.Downarowicz and S.Newhouse : we accumulate horseshoes at different small scales. The construction of such horseshoes is similar of examples due to J.Buzzi ofC^r interval maps without measures of maximal entropy [7],[17].

2 Sex entropy by accumulating small horseshoes

We recall first the main idea used by S.Newhouse and T.Downarowicz [10] to get a lower bound of the symbolic extension entropy by accumulating entropy at small scales. The following lemma is valid for general dynamical systems. Recall that hk denotes the Katok entropy at some scale k where (k)k is a decreasing sequence converging to zero. Also ifpis a periodic point we denote O(p) the orbit ofpandγp:= 1

]O(p) X

q∈O(p)

δq the periodic measure associated top.

Lemma 1 Let T :X →X be a continuous map defined on a compact metrizable space X. Letµ be an invariant probability measure.

We assume that for all k ∈N, there exists periodic points (p_{(i1,...,i2k+1)})_{(i1,...,i2k+1)∈N}2k+1 and invariant probability measures(µ_(i1,...,i2k))_{(i1,...,i2k)∈N}2k (we putN⁰={∅}andµ_∅=µ) such that :

1. for all (i1, ..., i2k) ∈ N^2k, the periodic measures γp_(i1,...,i

2k+1 ) are converging to µ(i₁,...,i_2k)

when i2k+1 goes to +∞;

2. for all(i₁, ..., i_2k+1)∈N^2k+1, the measuresµ_{(i1,...,i2k+2)}are converging to γ_p(i

1,...,i2k+1 ) when i2k+2 goes to+∞;

3. for allq∈N,

i_2klim→+∞hq(µ_(i1,...,i2k)) = 0

4. the limits lim

i1→+∞

lim

i2→+∞...

lim

i_2k→+∞h(µ_(i1,...,i2k))

...

exist.

Then for all n∈N: u_n(µ)≥

n

X

l=1

lim

i₁→+∞

lim

i₂→+∞...

lim

i_2l→+∞h(µ_(i1,...,i2l))

...

(7)

Proof : We prove (7) by induction onn. Assume the lemma for nand prove it for n+ 1. By definition, we have :

un+1(µ) = lim

q h−^hq+un(µ)

Then for allq∈Nwe get by using the first and the second hypothesis and by upper semi-continuity ofh−^h_q+u_n :

h−^hq+un(µ)≥lim sup

i₁

lim sup

i₂

(h−hq+un)(µ_(i1,i2))

Then according to the third hypothesis : h−^hq+un(µ)≥lim sup

i₁

lim sup

i₂

(h+un)(µ_(i1,i2))

and as the limits lim_i1 lim_i2h(µ_(i1,i2))

exist, we obtain : h−^h_q+u_n(µ)≥lim

i₁

lim

i₂ h(µ_(i1,i2))

+ lim sup

i1

lim sup

i2

u_n(µ_(i1,i2))

We apply finally the induction hypothesis to each measureµ(i₁,i₂)to get : un(µ_(i1,i2))≥

n

X

l=2

lim

i3→+∞

lim

i4→+∞...

lim

i2l→+∞h(µ_(i1,...,i2l))

...

We conclude that : u_n+1(µ)≥

n

X

l=1

lim

i₁→+∞

lim

i₂→+∞...

lim

i_2l→+∞h(µ_(i1,...,i2l))

...

In the following we also use the following equivalent version of the previous lemma :

Lemma 2 Let f :X →X be a continuous map defined on a compact metrizable space X with a fixed pointp.

We assume that for all k∈ N, there exists periodic points (p_{(i1,...,i2k+1)})_{(i1,...,i2k)∈N}2k and in- variant measures (µ_{(i1,...,i2k+1)})_{(i1,...,i2k+1)∈N}2k+1 such that :

1. for all(i1, ..., i_2k−1)∈N^2k−1, the periodic measuresγp_(i

1,...,i2k) are converging toµ_{(i1,...,i2k−1)} when i2k goes to+∞;

2. for all(i1, ..., i2k)∈N^2k, the measuresµ_{(i1,...,i2k+1)} are converging toγp_(i

1,...,i2k) wheni2k+1

goes to+∞(lim_nµ_n=δ_p fork= 0) ;

3. for allq∈N,

i_2k+1lim→+∞hq(µ_{(i1,...,i2k+1)}) = 0 4. the limits lim

i1→+∞

lim

i2→+∞...

lim

i2k+1→+∞h(µ_{(i1,...,i2k+1)})

...

exist.

Then for all n∈N: un(δp)≥

n−1

X

l=0 i₁→+∞lim

i₂→+∞lim ...

i_2l+1lim→+∞h(µ_{(i1,...,i2l+1)})

...

3 Our construction on the interval

3.1 Horseshoe for interval maps

The following notion of horseshoe for interval maps is due to M.Misiurewicz.

Definition 1 Let f be an interval map. A family J = (J1, ..., Jp) of closed disjoint intervals is called ap horseshoe ifJk ⊂f(Ji)for allj, k.

To simplify the notations we mean sometimes by J the union of the intervals defining the horseshoe J. Remark that any subfamily K of J is itself a horseshoe. If J = (J1, ..., Jp) is a p horseshoe ordered increasingly, i.e. ifi < j thenxi < xk for all (xi, xk)∈Ji×Jk, we denote by J⁰ thep−1 horseshoe (J1, ..., J_p−1).

Let us denote HJ := T

n∈ZTⁿJ and ({1, ..., p}^N, σ) the one sided shift with psymbols. The mapπ: (H_J, T)→(Σ⁺_p, σ) defined by (π(x))_k=qiff^k(x)∈J_q is a semi-conjugacy. In particular h_top(f) ≥logp. In fact horseshoes characterize entropy of continuous interval maps [13] : if f is a continuous interval map with entropy h_top(f) >0 then for all h < h_top(f) there exists a p horseshoe forf^N with log(p)/N > h.

In our construction we consider horseshoes of the following simple form.

Definition 2 Let f : [0,1]→[0,1] be a C^r interval map and let p and N be integers. A (p, N) quasi linear horseshoe(resp. a (p, N) linear horseshoe) forf is a phorseshoe ordered in- creasingly J= (J₁, ..., J_p)forf^N such that :

• |J1|=|J2|=...=|Jp|;

• f(J₁) =f(J₂) =...=f(J_p);

• f is increasing onJi wheni is odd andf is decreasing onJi wheni is even ;

• f is affine onJ_i for alli= 1, ..., p−1 (resp. for alli= 1, ..., p) ;

• there exists Ji< bi< Ji+1 such thatf^(l)(bi) = 0 forl= 1, ..., r andi= 1, ..., p−1 ;

• f_/f(J^N−1

1) is affine.

The slope of a quasi linear horseshoeJ is defined bys(J) :=k(f_/J^N

1)⁰k_∞.

We will writeH_J^N the compactf^N invariant set associated to a (p, N) (quasi-) linear horseshoe J = (J1, ..., Jp) forf, that isH_J^N :=T

n∈Nf^−nNJ, and we denoteHJthe compactf invariant set associated, that isH_J :=S

k=0,...,N−1f^k(H_J^N).

We will use the following technical lemma :

Lemma 3 Let f : [0,1]→ [0,1] be aC^r interval map and J = (J1, ..., Jp) a (p, N) quasi linear horseshoe forf with slopes(J)>1. Then there exists a sequence of periodic points(pn)_n∈N inJ⁰ with periods(N Pn)n∈N and a sequence of points(p⁰_n)n∈N inJ⁰ such that :

• the periodic measures γp_n converge to the measure of maximal entropy of HJ⁰ ;

• the sequence (f(pn))n∈N is monotone ;

• f^{N Pn} is increasing and affine on[p_n, p⁰_n] ;

• J_p⊂f^{N Pn}([p_n, p⁰_n]).

Proof : PutK:=f(J₁) =...=f(J_p). We assumef_/K^N−1is increasing (one can easily adapt the proof in the decreasing case).The mapπ : (H_J^N, f^N)→({1, ..., p}^N, σ) defined by (π(x))k =q if f^{N k}(x)∈Jq for allk∈Nis a semi-conjugacy. Asf^N is expanding on each element ofJ⁰(because s(J)>1) the restriction ofπonH_J^N0 is one-to-one and thereforeπ: (H_J^N0, f^N)→({1, ..., p−1}^N, σ) is a conjugacy. It is well-known that the periodic measures are dense inM(({1, ..., p−1}^N, σ)) : in particular there exists a sequence (qn)n∈Nof periodic points of ({1, ..., p−1}^N, σ) with periods (Pn)n∈Nsuch that the associated periodic measures converge to the measure of maximal entropyµ of ({1, ..., p−1}^N, σ). One can also clearly arrange this sequence such that for alln∈Nthe integer ]{k ∈ [0, Pn−1], (σ^kqn)0 is even} is even. We put pn = π⁻¹(qn) so that pn ∈J⁰ is a periodic point of f with periodN P_n. Moreover f^{N Pn} is increasing near p_n because we assume f_/K^N−1 is increasing and that]{k∈[0, P_n−1], (σ^kq_n)₀ is even} is even. By extracting a subsequence one can also assume that (f(p_n))_n∈Nis monotone.

The periodic measures γ_pn converge to _N¹ PN−1

k=0 f^∗kπ^∗−1µ∈ M([0,1], f) which is a measure of maximal entropy of HJ⁰. Indeed, as π is a conjugacy from (H_J^N0, f^N) to ({1, ..., p−1}^N, σ) we have h(π^∗−1µ, f^N) = h(µ, σ) = htop({1, ..., p−1}^N, σ) = htop(f^N, H_J^N0). Finally it is easily seen thath(_N¹ PN−1

k=0 f^∗kπ^∗−1µ, f) = _N¹h(π^∗−1µ, f^N) andh_top(f^N, H_J^N0) =N h_top(f, H_J⁰) so that h(_N¹ PN−1

k=0 f^∗kπ^∗−1µ, f) =htop(f, HJ⁰).

Observe now thatf^N is affine on each interval which is a connected component ofTPn−1

k=0 f^−kNJ⁰ becausef is affine on each element ofJ⁰. Moreover the image byf^{N Pn} of any such interval con- tains Ji for all i = 1, ..., p because J is a horseshoe for f^N. Let us denote [p⁰⁰_n, p⁰_n] the interval containingpn. Asf^{N Pn} is increasing nearpn and asJp stands at the right ofpn we conclude that Jp⊂f^{N Pn}([pn, p⁰_n]).

3.2 A model of C

interval maps with entropy of first order

The question of continuity of the entropy for smooth dynamical systems was studied early on.

M.Misiurewicz [14] gave the first examples of C^r diffeomorphisms defined on a compact mani- fold of dimension 4 without measures of maximal entropy. Then S.Newhouse [16] proved, using Yomdin’s theory, that the entropy function was upper semi-continuous forC^∞ systems. Counter- examples for interval maps appear much later. In his thesis [8] J.Buzzi built an example of C^r maps without measure of maximal entropy (see also [17]).

In Misiurewicz’s and Buzzi’s examples the stategy is the same : you construct ”smaller and smaller horseshoes” converging to a fixed point such that their entropies converge increasingly to the topological entropy. By a ”small” horseshoe J we mean that the orbit of the associated compact invariant setH_Jis contained in the-neighborhood of some periodic orbit for >0 small.

In this section we recall the main idea in the example of J.Buzzi, which will be a model of

”first order” in our example. We first begin with the following easy lemma, which will be useful in the next constructions :

Lemma 4 1. There exists a constant1≥M1>0 with the following properties.

Let a, b∈[0,1]. Letα∈R⁺ andc, d∈Rwith|c−d| ≤M1α|a−b|^r. Then there exists aC^∞ monotone mapf :R→Rsuch that :

• kfkr:= max_k=1,...,rkf^(k)k∞≤α;

• f(a) =c,f(b) =d;

• f^(k)(a) =f^(k)(b) = 0 fork= 1, ..., r.

2. There exists a constant 1 ≥ M2 > 0 with the following properties. Let a, b ∈ [0,1]. Let α∈R⁺,c∈Rand c⁰ ∈Rwith c⁰(b−a)≥0 and|c⁰| ≤M2α|a−b|^r−1. Then there exists a C^∞ monotone mapf :R→R such that :

• kfkr:= maxk=1,...,rkf^(k)k∞≤α;

• f(a) =c,|f(a)−f(b)| ≤α|a−b|^r ;

• f⁰(a) =c⁰,f⁰(b) = 0;

• f^(k)(a) =f^(k)(b) = 0 fork= 2, ..., r.

Proof : (1) We are easily reduce to the case a < b and c < d. Let F : R →R be aC^∞ non- decreasing map such thatF(0) = 0 andF(1) = 1 andF^(k)(0) =F^(k)(1) = 0 fork= 1, ..., r. Put M1:= min(_kFk¹

r,1). Fixa, b, c, d, αas in the statement (1) of the lemma. We definef as follows f :=|c−d|F(|a−b|⁻¹(.−a)) +c

Clearly f^(k)(a) = f^(k)(b) = 0 for k = 1, ..., r and f(a) = c, f(b) = d. Moreover kf^(k)k_∞ =

|a−b|^−k|c−d|kF^(k)k_∞≤αfor allk= 1, ..., r.

(2) We are easily reduce to the casea < bandc⁰>0. LetF :R→Rbe aC^∞ non-decreasing map such thatF(0) = 0,F(1) = 1,F⁰(0) = 1,F⁰(1) = 0 andF^(k)(0) =F^(k)(1) = 0 fork= 2, ..., r.

Put M2 := min(_kFk¹

r,1). Fixa, b, c, c⁰, α as in the statement (2) of the lemma. We definef as follows

f :=c⁰|a−b|F(|a−b|⁻¹(.−a)) +c

Clearlyf^(k)(a) =f^(k)(b) = 0 fork= 2, ..., randf(a) =c,f⁰(a) =c⁰. We putd:=f(b). Moreover kf^(k)k_∞ = c⁰|a−b|^−k+1kF^(k)k_∞ ≤ α for all k = 1, ..., r and |f(a)−f(b)| = |R

[a,b]f⁰(t)dt| ≤ c⁰|a−b|kF⁰k∞≤α|a−b|^r.

We can now explain our model :

Proposition 1 Let > 0, λ > 1, 0 ≤p < p⁰ < q⁰ < q ≤ 1 and let f : [0,1] → [0,1] be a C^r interval map, such thatpis a periodic point off of period P andf(q) =p,f^(k)(q) =f^(k)(q⁰) = 0 fork= 1, ..., r. We also assume there exists an integer S such that :

• f^S(p) =p, i.e. S is a multiple ofP ;

• f^S is increasing and affine on [p, p⁰]with slopeλ;

• q∈f^S([p, p⁰]).

Then there exists aC^rinterval mapgsuch thatf =goutside the interval]q⁰, q[,kf⁰k_∞=kg⁰k_∞ andkf−gkr≤. Moreover there exist a strictly increasing sequence of integers(Tn)_n∈N, a strictly increasing sequence of even integers(Nn)n∈N, a sequence of intervals([xn, yn])n∈Nand a sequence of linear horseshoes(Jⁿ)n∈N such that :

• f^(l)(x_n) =f^(l)(y_n) = 0for alln∈Nandl= 1, ..., r ;

• Jⁿ⊂[xn, yn]⊂[q⁰, q]for alln∈Nand lim

n→+∞xn= lim

n→+∞yn =q ;

• Jⁿ is a linear(Nn, STn+ 1)horseshoe forg with slope _n2(3n^λSTn2Nn)^r−1 ;

• lim

n→+∞h_top(Jⁿ) = lim

n→+∞

logNn

T_n = logλ r ;

• Invariant probability measures supported by HJⁿ converge to the periodic measure associated topwhenn goes to infinity.

x_n

Drawing 2 : Accumulation of small horsehoes

y_n x_n+1 q

y_n-1

Drawing 4

Drawing 5 graph of g near q

Proof :

Letn0be large enough such thatq−_n¹

0 > q⁰. For alln≥n0we putgequal to aNn zig zag of height _n2(2n¹_2N

n)^r on [x_n, y_n] := [q−_n¹, q−¹_n+_3n¹₂] as described on the above picture (Drawing 2).

More precisely for all i = 0, ..., Nn−1 the map g is affine on the interval J_iⁿ := [xn+ (i+

1

4)_3n2¹N_n, xn + (i+ ³₄)_3n2¹N_n] with slope _n2(3n⁽⁻¹⁾²N_nⁱ)^r−1 and g(J₁ⁿ) = ... = g(J_Nⁿ

n) = [an, an +

1

2n²(3n²N_n)^r] (we will specifyan later). Then according to Lemma 4 (2), one can extendg on the whole interval [x_n, y_n] such that :

• g^(k)(xn+i_3n2¹N_n) = 0 fork= 1, ..., r and fori= 0, ..., Nn ;

• k(g)/[x_n,y_n]kr≤ _M^4r−1

2n² ;

• g(y_n) =g(x_n)∈[a_n−_M ¹

2n²(3n²Nn)^r, a_n].

We chooseansuch thatf maps [an, an+_2n2(3n¹²N_n)^r] on the expanding part [p, p⁰] off^S during a time Tn and then comes back on [xn, yn], that is f^STn([an, an+ _2n2(3n¹²N_n)^r]) ⊃[xn, yn]. We chooseTn > T_n−1minimal for this property. This can be done becauseq∈f^S([p, p⁰]). In this way we obtain for all integersn > n0a linear horseshoeJⁿ = (J₁ⁿ, ..., J_Nⁿ

n) forg^STn+1. The condition onTn is :

1

n² ≤ λ^Tn_2n2(3n¹²N_n)^r ≤ λ1

n² (8)

and the condition on an is :

f^STn(an) =xn

that is :

λ^Tn(an−p) =xn−p We deduce from the inequality (8) that lim

n→+∞

logN_n Tn

= logλ

r . One can also replace N_n by Nn−1 to ensure Nn is even.

By using Lemma 4(1) one can now extend g in a C^r way on the whole interval [xn₀, q] such that for alln≥n0 :

kg/[y_n,x_n+1]kr ≤ 1 M1

g(y_n)−g(x_n+1) (yn−xn+1)^r

≤ 6^r M1

n^2r

a_n−a_n+1+ 1 M2n²(3n²Nn)^r

≤ 6^r M1

n^2r

λ^−Tn+ 1 M2n²(3n²Nn)^r

≤ 6^r(1 + 1/M₂) M1N_n^r

We extend gin aC^r way on [q⁰, x_n0] by putting for allx∈[q⁰, x_n0] : g(x) =f(q⁰) +g(xn₀)−f(q⁰)

f(q)−f(q⁰)

f

q−q⁰

x_n0−q⁰(x−q⁰) +q⁰

−f(q⁰)

One checks easily that g(q⁰) =f(q⁰) andg^(k)(q⁰) =g^(k)(x_n0) = 0 fork= 1, ..., r.

We conclude the proof by choosing n₀large enough such that ^6r(1+1/M_M ²⁾

1N_n^r

0

≤and k(f −g)/[q⁰,x_n0]kr≤.

3.3 Proof of Theorem 2

We build a collection ofC^r maps (gk)_{k∈N∪{∞}} defined on [0,1] fixing 0 and with first derivative bounded by 5. For alll∈Nand for all (i₁, ..., i_2l+2)∈N^2l+2there exist pointsp_i1,...,i2l+2, intervals [x_i1,...,i2l+1, y_i1,...,i2l+1], collections of disjoint closed intervalsJ_i1,...,i2l+1, integers P_i2l+2,T_i2l+1 and even integersN_i2l+1 such that for allk∈N∪ {∞}and all integers 0≤l≤kwe have :

• p_i1,...,i2l+2∈J_i⁰

1,...,i_2l+1 is a periodic point ofg_k^{Si1,...,i2l+1} of periodP_i2l+2 ;

• f^(m)(xi₁,...,i_2l+1) =f^(m)(yi₁,...,i_2l+1) = 0 form= 1, ..., r ;

• Ji₁,...,i_2l+1 ⊂[xi₁,...,i_2l+1, yi₁,...,i_2l+1]⊂[xi₁,...,i2l−1, yi₁,...,i2l−1]³;

• Ji₁,...,i_2l+1 is a quasi linear (Ni_2l+1, Si₁,...,i_2lTi_2l+1+ 1) horseshoe for gk and Ji₁,...,i_2k+1 is a linear horseshoe ;

• lim

i_2l+1→+∞kgk/[x_i1,...,i2l+1,y_i1,...,i2l+1]kr= 0.

3the last inclusion holds only forl6= 0

where the integers Si₁,...,i_m for m ≤ 2k+ 1 are defined inductively in the following way : Si₁ =Ti₁,Si₁,...,i_2l+1 =Si₁,...,i_2l×Ti_2l+1+ 1 andSi₁,...,i_2l+2 =Si₁,...,i_2l+1×Pi_2l+2. Remark that the integerSi₁,...,i_2l+2 is the period ofpi₁,...,i_2l+2 forgk.

These periodic points and horseshoes can also be arranged to satisfy the following properties so that one can apply Lemma 2 (to simplify the notations we writeHi1,...,i_2l+1 instead ofHJi1,...,i2l+1

andH_i⁰_1,...,i

2l+1 instead ofH_J⁰

i1,...,i2l+1

) :

for allk∈N∪ {∞}and for all integers 0≤l≤k,

1. the sequence of periodic measures (γp_i1,...,i2l+2)i_2l+2∈N converges to the measure of maximal entropy of H_i⁰_1,...,i

2l+1 wheni2l+2→+∞; 2. measures supported by H_i⁰

1,...,i_2l+1 converge to γ_pi

1,...,i2l wheni_2l+1 →+∞ (measures sup- ported byH_i⁰₁ converge toδ0 wheni1→+∞) ;

3. for all >0 there exists an integerIk such that :

∀i2l+1> Ik ∃x∈[0,1] s.t.H_i⁰_1,...,i2l+1 ⊂ \

n∈N

Bg_k(x, n, ) ;

4. lim

i₁→+∞... lim

i_2l+1→+∞h(H_i⁰_1,...,i2l+1, gk) =log 5 r^l+1.

One deduces easily from the above assertions 1-4 that the map g_∞ satisfies the assumptions 1-4 of Lemma 2. Then by applying this lemma for g_∞, we get for all integers n: u_n(δ₀) ≥

Pn k=1

1 r^k

logkg_∞⁰ k_∞ >0. The converse inequalitiesu_n(δ₀)≤ Pn k=1

1 r^k

logkg⁰_∞k_∞ follow from 5 of Theorem 1. This concludes the proof of Theorem 2 withf_r:=g_∞.

We explain now the construction of the sequence (gn)_n∈N and the mapg_∞. We first consider a C^r interval map g₋₁, such that g₋₁(0) = 0, g₋₁(¹₂) = 0, kg₋₁⁰ k_∞ = 5,g₋₁ is affine with slope λ= 5 on [0,¹₆] (¹₂ ∈g₋₁([0,¹₆]) = [0,⁵₆]) andg₋₁^(k)(¹₂) =g^(k)₋₁(¹₄) = 0 fork= 1, ..., r. We can assume moreover thatk(g₋₁)⁰_/[1

4,1]k_∞<4. See Drawing 3.

One can apply Proposition 1 to the map g₋₁ with= 1,S= 1,λ= 5,p= 0,p⁰ = ¹₆,q⁰ = ¹₄, q= ¹₂ and get a mapg₀ (with kg₀⁰k_∞ ≤5) which admits a sequence of horseshoes (H_i1)_i1∈Nand

sequences of periodic points (pi₁,i₂)i₁,i₂∈Nsatisfying all the above required conditions (1), (2), (3), (4) fork= 0.

Assume thatgk is already built and definegk+1.

The horseshoes Ji₁,...,i_2k+1 = (J1, ..., JN_i2k+1) are linear (Ni_2k+1, Si₁,...,i_2k+1) horseshoes for gk. To get gk+1 from gk we only change gk on [xi₁,...,i_2k+1, yi₁,...,i_2k+1] with i2k+1 large enough such that the modulus of therderivative ofg_k on [x_i1,...,i2k+1, y_i1,...,i2k+1] is less than ₂^M_k+r¹² . Let us con- sider one such horseshoe J_i1,...,i2k+1 and we denote itJ = (J₁, ..., J_N) (We also use the simplified notationsH:=H_J,H⁰:=H_J⁰, [x, y] := [x_i1,...,i2l+1, y_i1,...,i2l+1] andS :=S_i1,...,i2k+1).

First step : RecallH⁰ is a linear (N, S) horseshoe forgk. By applying Lemma 3 there exists a sequence of periodic points (pn)_n∈NinJ⁰ with periods (SPn)_n∈Nforgk and a sequence of points (p⁰_n)_n∈NinJ⁰ such that :

• the periodic measures γp_n converge to the measure of maximal entropy ofHJ⁰ ;

• the sequence (gk(pn))_n∈Nis monotone ;

• g_k^SPn is increasing and affine on [pn, p⁰_n] ;

• J_N ⊂g_k^SPn([p_n, p⁰_n]).

LetP denote the limit of (g_k(p_n))_n∈N. On the last branchJ_N ∈/ J⁰of the horseshoeJ we create a tangency of orderrwith the horizontal line{(x⁰, P), x⁰ ∈[0,1]}at the pointQ=g⁻¹_k (P)TJ_N by applying Lemma 4(1) to g_k on [b_N−1, Q] and on [Q, y]. We recall J_N−1 < b_N−1 < J_N and f^(l)(bN−1) = 0 for l = 1, ..., r. We get a new map uk. The norm kkr changed only on [b_N−1, y] in the following way : k(uk)_[bN−1,y]kr≤M₁⁻¹k(gk)_[bN−1,y]kr. Indeed|gk(b_N−1)−gk(Q)| ≤ kgkkr|b_N−1−Q|^rand|gk(y)−gk(Q)| ≤ kgkkr|y−Q|^r. AsN is even, the mapgk is non-increasing onJN (see Definition 2). Remark thatuk is again non-increasing onJN and the family of intervals (J1, ..., JN) is a quasi linear (N, S) horseshoe foruk.

P

Q Drawing 4 : first step

Drawing 5 graph of u_k

x b_N-1 y

Second step : Let us assume the sequence (gk(pn))_n∈N is converging non-increasingly. Let qn denote the point of JN such that uk(qn) = uk(pn) = gk(pn). Since uk is non-increasing on

JN, the sequenceqn is increasing. By extracting a subsequence one can assume that|qn−Q|<

2(|qn−qn+1|). Finally we putq0=b_N−1. We create tangencies of orderrwith the horizontal line {(x⁰, gk(pn)), x⁰∈[0,1]}at the point qn by applying again Lemma 4(1) to [a, b] = [qn, qn+1] and [d, c] = [uk(qn+1), uk(qn)] for all integers n. This can be done by preserving almost the normkkr

ofuk. In fact

|uk(qn+1)−uk(qn)| ≤ k(uk)[x,y]kr|qn−Q|^r<2^rk(uk)[x,y]kr|qn−qn+1|^r

We get a new map which is againC^r with horizontal tangencies of order rat each pointqn. We denote byvk this new map ; we havek(vk)/[x,y]kr<2^rM₁⁻¹k(uk)/[x,y]kr<2^rM₁⁻²k(gk)/[x,y]kr<

1/2^k.

v_k(q_n) = g_k(p_n)

Q Drawing 5 : second step

q_n graph of v_k

b_N-1 y

If the sequence (gk(pn))_n∈N converges increasingly, we can create in the same way horizontal tangencies of orderron [Q, y] accumulating onQ. In the following we assume always (gk(pn))_n∈N converges non-increasingly. The rest of the construction is completely similar in the increasing case.

Third step : According to Lemma 3 there exists p⁰_n such that v^SP_k ⁿ is affine on [pn, p⁰_n] with slope λequal to s(H)^Pn andJ_N ⊂v^SP_k ⁿ([p_n, p⁰_n]). By applying Proposition 1 with = ₂¹_k,

”S =SP_n”,λ=s(H)^Pn,p=p_n,p⁰ =p⁰_n,q⁰ =q_n andq=q_n+1, one can create small horseshoes accumulating onp_n for all integersnto get finallyg_k+1. We have created in this way a sequence of new horseshoes for eachJ =Ji₁,...,i_2k+1. Coming back to the initial notations this sequence of new horseshoes and their associated intervals are denotedJi₁,...,i_2k+3 and [xi₁,...,i_2k+3, yi₁,...,i_2k+3].

We also denote by Ti_2k+3 and Ni_2k+3 the integers such that Ji₁,...,i_2k+3 is a Ni_2k+3 horseshoe for g_k+1^{Ti2k+3Si1,...,i2k+2+1}. Finally p_n and P_n are respectively denoted by p_i1,...,i2k+2 and P_i2k+2. It follows easily from the construction that the new horseshoes J_i1,...,i2k+3 are (N_i2k+3, S_i1,...,i2k+3) linear horseshoes forgk+1and that the previous horseshoesJi₁,...,i_2l+1 forl < k are modified only on their last branch and therefore are again (Ni_2l+1, Si₁,...,i_2l+1) quasi linear horseshoes forgk+1. By Proposition 1 the slope of the horseshoeJi₁,...,i_2k+3 is related with the slope of the horseshoe Ji₁,...,i_2k+1 in the following way :

s(J_i1,...,i2k+3) =s(Ji1,...,i_2k+1)^Pi2k+2Ti2k+3

i²_2k+3(3i²_2k+3Ni_2k+3)^r−1 (9) Notice that the modifications to get g_k+1 from g_k are made only on the intervals [x, y] = [x_i1,...,i2k+1, y_i1,...,i2k+1] where the moduli of the derivatives of order≤rofg_k are less ₂^M_k+r¹² . There- fore