Dynamical Systems, Fractal Geometry and Diophantine ...

Fractal geometry and dynamical bifurcations Fractal geometry and dynamical bifurcations Markov and Lagrange spectra

Dynamical Systems, Fractal Geometry andDiophantine Approximations

Carlos Gustavo Tamm de Araujo Moreira(IMPA, Rio de Janeiro, Brasil)

International Congress of MathematiciansRiocentro - Rio de Janeiro - RJ - 03/08/2018


The theory of Dynamical Systems is concerned with theasymptotic behaviour of systems which evolve over time, andgives models for many phenomena arising from naturalsciences, as Meteorology and Celestial Mechanics.

Poincaré (late nineteenth century): study of the restrictedthree-body problem in Celestial Mechanics:• Qualitative theory of differential equations• Basic results of Ergodic Theory (e.g Poincaré’s recurrencelemma)


We will consider the dynamics of transformations f : X → X(discrete time)x → f (x)→ f 2(x) = f (f (x))→ . . .

and of flows X t : M → M, t ∈ R (typically given by solutions ofautonomous ordinary differential equations).

Usual goals:• Try to describe most orbits of most systems.• Characterize dynamical behaviors which are persistent insome sense.


A. Andronov, L. Pontryagin (30’s): Structural stability (flows inthe plane)

M. Peixoto (50’s): Typical flows in orientable surfaces arestructurally stable.

Morse-Smale systems:• Exist in any compact manifold• Are structurally stable(J. Palis, Ph.D. thesis in low dimensions; J. Palis and S. Smalein general).


Hyperbolic systems: introduced by Smale in the sixties, after aglobal example provided by Anosov, namely the diffeomorphismf (x , y) = (2x + y , x + y) (mod 1) of the torus T2 = R2/Z2 andthe following example, introduced by Smale himself.

Smale’s horseshoe

A B

C D

DC

A B


Let Λ ⊂ M be a compact subset of a manifold M.

We say that Λ is a hyperbolic set for a diffeomorphismϕ : M → M if ϕ(Λ) = Λ and there is a decompositionTΛM = Es ⊕ Eu of the tangent bundle of M over Λ such thatDϕ |Es is uniformly contracting and Dϕ |Eu is uniformlyexpanding.

We say that a diffeomorphism ϕ as above is hyperbolic if thelimit set of its dynamics is a hyperbolic set.


The importance of the notion of hyperbolicity is also related tothe stability conjecture by Palis and Smale, according to whichstructurally stable dynamical systems are essentially thehyperbolic ones (i.e. the systems whose limit set is hyperbolic).

• Proved by Mañé ([Ma88]) for diffeomorphisms in the C1

topology (after important contributions by Anosov, Smale, Palis,de Melo, Robbin and Robinson).

• Proved later by Hayashi ([Hay97]) for flows (also in the C1

topology).• Still open in the Ck topology for k ≥ 2.






topology).

• Still open in the Ck topology for k ≥ 2.






topology).• Still open in the Ck topology for k ≥ 2.


Homoclinic bifurcations, to be defined later, are perhaps themost important mechanism that creates complicated dynamicalsystems from simple ones.

“Rien n’est plus propre à nous donner une idée de lacomplication du problème des trois corps et en général de tousles problèmes de Dynamique..."(Poincaré, Les Méthodes Nouvelles de la Mécanique Céleste)


First natural problem related to homoclinic bifurcations:homoclinic explosions on surfaces

• One-parameter families ϕµ : M2 → M2, µ ∈ (−1,1).• ϕµ uniformly hyperbolic for µ < 0.• ϕ0 presents a quadratic homoclinic tangency associated to ahyperbolic periodic point (which may be isolated or belong to ahorseshoe.• The tangency unfolds for µ > 0 creating locally two transverseintersections between the stable and unstable manifolds of (thecontinuation of) the periodic point.A main question: what happens for (most) positive values of µ?


First natural problem related to homoclinic bifurcations:homoclinic explosions on surfaces• One-parameter families ϕµ : M2 → M2, µ ∈ (−1,1).• ϕµ uniformly hyperbolic for µ < 0.• ϕ0 presents a quadratic homoclinic tangency associated to ahyperbolic periodic point (which may be isolated or belong to ahorseshoe.

• The tangency unfolds for µ > 0 creating locally two transverseintersections between the stable and unstable manifolds of (thecontinuation of) the periodic point.A main question: what happens for (most) positive values of µ?


First natural problem related to homoclinic bifurcations:homoclinic explosions on surfaces• One-parameter families ϕµ : M2 → M2, µ ∈ (−1,1).• ϕµ uniformly hyperbolic for µ < 0.• ϕ0 presents a quadratic homoclinic tangency associated to ahyperbolic periodic point (which may be isolated or belong to ahorseshoe.• The tangency unfolds for µ > 0 creating locally two transverseintersections between the stable and unstable manifolds of (thecontinuation of) the periodic point.

A main question: what happens for (most) positive values of µ?


First natural problem related to homoclinic bifurcations:homoclinic explosions on surfaces• One-parameter families ϕµ : M2 → M2, µ ∈ (−1,1).• ϕµ uniformly hyperbolic for µ < 0.• ϕ0 presents a quadratic homoclinic tangency associated to ahyperbolic periodic point (which may be isolated or belong to ahorseshoe.• The tangency unfolds for µ > 0 creating locally two transverseintersections between the stable and unstable manifolds of (thecontinuation of) the periodic point.A main question: what happens for (most) positive values of µ?


Conjecture (Palis)Any diffeomorphism of a surface can be approximatedarbitrarily well in the Ck topology by a hyperbolicdiffeomorphism or by a diffeomorphism displaying a homoclinictangency.

• Proved by Pujals and Sambarino ([PS00]) in the C1 topology.

• General version: approximation (in arbitrary dimension) byhomoclinic tangency, heteroclinic cycle (a cycle given byintersections of stable and unstable manifolds of periodic pointsof different indexes) or hyperbolic.• Major advance: Crovisier and Pujals ([CP15]): homoclinictangency, heteroclinic cycle or essentially hyperbolic (finitenumber of hyperbolic attractors whose union of basins is openand dense) (C1).


Conjecture (Palis)Any diffeomorphism of a surface can be approximatedarbitrarily well in the Ck topology by a hyperbolicdiffeomorphism or by a diffeomorphism displaying a homoclinictangency.

• Proved by Pujals and Sambarino ([PS00]) in the C1 topology.• General version: approximation (in arbitrary dimension) byhomoclinic tangency, heteroclinic cycle (a cycle given byintersections of stable and unstable manifolds of periodic pointsof different indexes) or hyperbolic.• Major advance: Crovisier and Pujals ([CP15]): homoclinictangency, heteroclinic cycle or essentially hyperbolic (finitenumber of hyperbolic attractors whose union of basins is openand dense) (C1).


Homoclinic bifurcations→ complicated dynamical phenomena

• S. Newhouse ([N70], [N74], [N79]): Persistent homoclinictangencies and coexistence of infinitely many sinks (hyperbolictheory + new concepts in Fractal Geometry).

• L. Mora and M. Viana ([MV93]): any surface diffeomorphismpresenting a homoclinic tangency can be approximated by adiffeomorphism exhibiting a Hénon-like strange attractor (andsuch diffeomorphisms appear in any typical family goingthrough a homoclinic bifurcation).


Fractal dimensions, regular Cantor sets and homoclinic bifurcations

Fractal sets appear naturally in Dynamical Systems and fractaldimensions when we try to measure fractals. They areessential to describe most of the main results in thispresentation. The most important notion of fractal dimension ofa metric space is the Hausdorff dimension.

The Hausdorff dimension of a metric space X is

HD(X ) = infs > 0; infX⊂∪B(xn,rn)

∑r sn = 0.

It is a natural tool to measure fractal sets (as regular Cantorsets), and to compare subsets of zero Lebesgue measure ofthe real line.



Regular Cantor sets on the real line:• One-dimensional hyperbolic sets, defined by expandingmaps.• Some kind of self-similarity property: small parts of them arediffeomorphic to big parts with uniformly bounded distortion.

The usual ternary Cantor set is a regular Cantor set:

ψ:

0 2/3 11/3

1

Key question (in Dynamics and Diophantine Approximations):whether the arithmetic difference of two such sets hasnon-empty interior.



A horseshoe Λ in a surface is locally diffeomorphic to thecartesian product of two regular Cantor sets: the so-calledstable and unstable Cantor sets K s and K u of Λ, given byintersections of Λ with local stable and unstable manifolds ofsome points of the horseshoe.

• HD(Λ) = HD(K s) + HD(K u)→ plays a fundamental role inseveral results on homoclinic bifurcations associated to Λ.



From the dynamics side, in 1983, J. Palis and F. Takens provedthe following theorem about homoclinic bifurcations associatedto a hyperbolic set:

TheoremLet (ϕµ), µ ∈ (−1,1) be a family of diffeomorphisms of asurface presenting a homoclinic explosion at µ = 0 associatedto a periodic point belonging to a horseshoe Λ. Assume thatHD(Λ) < 1. Then

limδ→0

m(H ∩ [0, δ])

δ= 1,

where H := µ > 0 | ϕµ is persistently hyperbolic.



Central fact in the proof:

K1,K2 ⊂ R regular Cantor sets, HD(K1) + HD(K2) < 1 =⇒K1 − K2 = x − y | x ∈ K1, y ∈ K2 = t ∈ R|K1 ∩ (K2 + t) 6= ∅(the arithmetic difference between K1 and K2) is a set of zeroLebesgue measure (indeed HD(K1 − K2) < 1).

On the other side, we have the famous

Theorem (Marstrand)

Given a Borel set X ⊂ R2 with HD(X ) > 1 then, for almostevery λ ∈ R, πλ(X ) has positive Lebesgue measure, whereπλ : R2 → R is given by πλ(x , y) = x − λy .In particular, if K1 and K2 are regular Cantor sets withHD(K1) + HD(K2) > 1 then, for almost every λ ∈ R,K1 − λK2 has positive Lebesgue measure.



These facts inspired J. Palis to formulate, in 1983, the following

Conjecture (Palis)For typical pairs of regular Cantor sets (K s,K u),HD(K s) + HD(K u) > 1⇒ int(K s − K u) 6= ∅

Another motivation:• Newhouse’s (70’s): thickness τ(K ) of a regular Cantor set K ;τ(K ) · τ(K ′) > 1 =⇒ int(K − K ′) 6= ∅.• Open sets of diffeos with persistent homoclinic tangencies;infinitely many coexisting sinks (under dissipativity).• Every family of surface diffeomorphisms that unfold ahomoclinic tangency goes through such an open set.



We will discuss below the proof, in collaboration ofJean-Christophe Yoccoz, of a strong version of Palis’conjecture.

Jean-Christophe Yoccoz



M., Yoccoz, 2001Typically, HD(K s) + HD(K u) > 1⇒ K s − K u persistentlycontains intervals.

Indeed, we prove a more precise (and stronger) version of thisresult. We start with the following

Definition: Stable intersectionsTwo Cantor sets K1 and K2 have stable intersection if there is aneighbourhood V of (K1,K2) in the set of pairs of C1+-regularCantor sets such that (K1, K2) ∈ V ⇒ K1 ∩ K2 6= ∅.



Let Ω = (K1,K2)|K1,K2 are C∞ − regular Cantor sets andHD(K1) + HD(K2) > 1.The following result gives a (strong) positive answer for Palis’conjecture.

Theorem (M., Yoccoz)There is an open and dense set U ⊂ Ω such that if(K1,K2) ∈ U , then, ∀λ ∈ R∗, Is(K1, λK2) is dense in K1 − λK2and HD((K1 − λK2)\Is(K1, λK2)) < 1, whereIs(K ,K ′) := t ∈ R | K and (K ′) have stable intersection.

• Ω \ U has infinite codimension.• d-stable intersection, 0 < d < HD(K1) + HD(K2)− 1.





• Ω \ U has infinite codimension.

• d-stable intersection, 0 < d < HD(K1) + HD(K2)− 1.





• Ω \ U has infinite codimension.• d-stable intersection, 0 < d < HD(K1) + HD(K2)− 1.



Main techniques in M., Yoccoz, 2001:A recurrent compact set (of relative positions of limitgeometries) criterion for stable intersections (which impliesthat arithmetic differences persistently contain intervals).

The scale recurrence lemma, which provides a compactrecurrent set at the level of scales under a non-linearityhypothesis.An application of Erdos probabilistic method: a family ofC∞ small perturbations of a regular Cantor set (the secondCantor set is fixed) with a large number of parameterssuch that for most parameters there is a recurrent compactset for the corresponding pair of Cantor sets.



Main techniques in M., Yoccoz, 2001:A recurrent compact set (of relative positions of limitgeometries) criterion for stable intersections (which impliesthat arithmetic differences persistently contain intervals).The scale recurrence lemma, which provides a compactrecurrent set at the level of scales under a non-linearityhypothesis.

An application of Erdos probabilistic method: a family ofC∞ small perturbations of a regular Cantor set (the secondCantor set is fixed) with a large number of parameterssuch that for most parameters there is a recurrent compactset for the corresponding pair of Cantor sets.



Main techniques in M., Yoccoz, 2001:A recurrent compact set (of relative positions of limitgeometries) criterion for stable intersections (which impliesthat arithmetic differences persistently contain intervals).The scale recurrence lemma, which provides a compactrecurrent set at the level of scales under a non-linearityhypothesis.An application of Erdos probabilistic method: a family ofC∞ small perturbations of a regular Cantor set (the secondCantor set is fixed) with a large number of parameterssuch that for most parameters there is a recurrent compactset for the corresponding pair of Cantor sets.



In [MY10], still with J.-C. Yoccoz, we used these techniques inorder to prove the following

Theorem:Let (ϕµ), µ ∈ (−1,1) be a family of diffeomorphisms of asurface presenting a homoclinic bifurcation at µ = 0 associatedto a periodic point belonging to a horseshoe Λ. Assume thatHD(Λ) > 1. Then (under open and dense conditions)

lim infδ→0

m(T ∩ [0, δ])

δ> 0, where

T := µ > 0 | ϕµ presents persistent homoclinic tangencies.

Remark: In this setting we can still have positive density ofhyperbolicity ([M96]), but T ∪ H generically has full Lebesguedensity at µ = 0.



Generalizations of these methods gave related results in higherdimensions:•With J. Palis and M. Viana: Homoclinic explosions in arbitrarydimensions.

•With J. E. López Velázquez: projections of Cartesianproducts of regular Cantor sets.•With W. Silva: Geometry of horseshoes in arbitrarydimensions.•With A. Zamudio and H. Araújo: Conformal complex regularCantor sets.



Generalizations of these methods gave related results in higherdimensions:•With J. Palis and M. Viana: Homoclinic explosions in arbitrarydimensions.•With J. E. López Velázquez: projections of Cartesianproducts of regular Cantor sets.

•With W. Silva: Geometry of horseshoes in arbitrarydimensions.•With A. Zamudio and H. Araújo: Conformal complex regularCantor sets.



Generalizations of these methods gave related results in higherdimensions:•With J. Palis and M. Viana: Homoclinic explosions in arbitrarydimensions.•With J. E. López Velázquez: projections of Cartesianproducts of regular Cantor sets.•With W. Silva: Geometry of horseshoes in arbitrarydimensions.

•With A. Zamudio and H. Araújo: Conformal complex regularCantor sets.



Generalizations of these methods gave related results in higherdimensions:•With J. Palis and M. Viana: Homoclinic explosions in arbitrarydimensions.•With J. E. López Velázquez: projections of Cartesianproducts of regular Cantor sets.•With W. Silva: Geometry of horseshoes in arbitrarydimensions.•With A. Zamudio and H. Araújo: Conformal complex regularCantor sets.


Part II

Diophantine Approximations (andFractal Geometry):

The Markov and Lagrangespectra and generalizations


Diophantine approximations: the Markov and Lagrange spectra

Let α ∈ R \Q.

Dirichlet: The inequality |α− pq | <

1q2 has infinitely many rational

solutions pq .

Hurwitz, Markov: |α− pq | <

1√5q2 also has infinitely many

rational solutions pq for any irrational α. Moreover,

√5 is the

largest constant for which such a result is true for any α ∈ R \Q.


However, for particular values of α we can improve thisconstant:

|α− pq| < 1

q2

More precisely, we define k(α) := supk > 0 | |α− pq | <

1kq2 has

infinitely many rational solutions pq =

= lim supp,q→+∞ (q|qα− p|)−1.

We have k(α) ≥√

5, ∀α ∈ R \Q and k(

1+√

52

)=√

5.



|α− pq| < 1√

5q2


1kq2 has




5, ∀α ∈ R \Q and k(

1+√

52

)=√

5.



|α− pq| < 1

kq2


1kq2 has




5, ∀α ∈ R \Q and k(

1+√

52

)=√

5.


We will consider the set

L = k(α) | α ∈ R \Q, k(α) < +∞.

This set is called the Lagrange spectrum.

Hurwitz-Markov theorem determines the smallest element of L,which is

√5. This set L encodes many diophantine properties

of real numbers. It is a classical subject the study of thegeometric structure of L.


Markov (1879)

L ∩ (−∞,3) = k1 =√

5 < k2 = 2√

2 < k3 =

√2215

< . . .

where kn is a sequence (of irrational numbers whose squaresare rational) converging to 3

This means that the “beginning” of the set L is discrete. As wewill see, this is not true for the whole set L.


The elements of the Lagrange spectrum which are smaller than

3 are exactly the numbers of the form√

9− 4z2 where z is a

positive integer for which there are other positive integers x , ysuch that 1 ≤ x ≤ y ≤ z and (x , y , z) is a solution of theMarkov equation x2 + y2 + z2 = 3xyz.•(x , y , z) solution =⇒ (y , z,3yz − x), (x , z,3xz − y) solutions.

(1,1,1)

(1,1,2)

(1,2,5)

(1,5,13)

(1,13,34) (5,13,194)

(2,5,29)

(2,29,169) (5,29,433)


An important open problem related to Markov’s equation is theUnicity Problem, formulated by Frobenius about 100 years ago:for any positive integers x1, x2, y1, y2, z with x1 ≤ y1 ≤ z andx2 ≤ y2 ≤ z such that (x1, y1, z) and (x2, y2, z) are solutions ofMarkov’s equation we always have (x1, y1) = (x2, y2)?If the Unicity Problem has an affirmative answer then, for everyreal t < 3, its pre-image k−1(t) by the function k above consistsof a single GL2(Z)-equivalence class (this equivalence relationis such that

α ∼ aα + bcα + d

,∀a,b, c,d ∈ Z, |ad − bc| = 1.)


M. Hall proved in 1947 that if C(4) is the regular Cantor setformed by the numbers in [0,1] whose coefficients in thecontinued fractions expansion are bounded by 4, then one has

C(4) + C(4) = x + y ; x , y ∈ C4 = [√

2− 1,4(√

2− 1)].

This implies that L contains a whole half line (for instance[6,+∞)).


G. Freiman determined in 1975 the biggest half line that iscontained in L, which is [c,+∞), with

c =2221564096 + 283748

√462

491993569∼= 4,52782956616 . . . .

These last two results are based on the study of sums ofregular Cantor sets, whose relationship with the Lagrangespectrum will be explained below.


Sets of real numbers whose continued fraction representationhas bounded coefficients with some combinatorial constraintsare often regular Cantor sets, which we call Gauss-Cantor sets(since they are defined by the Gauss map g(x) = 1/x from(0,1) to [0,1)).

y = g(x) = 1

x


If the continued fraction of α is

α = [a0; a1,a2, . . . ]def= a0 +

1

a1 +1

a2 + . . .

.

then we have the following formula for k(α):

k(α) = lim supn→∞

(αn + βn),

where αn = [an; an+1,an+2, . . . ] and

βn = [0; an−1,an−2, . . . ,a1].


The previous formula follows from the equality

|α− pn

qn| =

1(αn+1 + βn+1)q2

n, ∀n ∈ N,

where pn/qn = [a0; a1,a2, . . . ,an],n ∈ N are the convergents ofthe continued fraction of α.

Remark: If∣∣α− p

q

∣∣ < 12q2 then p

q is a convergent pnqn

of thecontinued fraction of α.


This formula for k(α) implies that we have the followingalternative (dynamical) definition of the Lagrange spectrum L:

Let Σ = (N∗)Z be the set of all bi-infinite sequences of positiveintegers. If θ = (an)n∈Z ∈ Σ, we definef (θ) = α0 + β0 = [a0; a1,a2, . . . ] + [0; a−1,a−2, . . . ]. We have

L = lim supn→∞ f (σnθ), θ ∈ Σ

where σ : Σ→ Σ is the shift defined by σ((an)n∈Z) = (an+1)n∈Z.


The Markov spectrum M is the set

M = supn∈Z

f (σnθ), θ ∈ Σ.

It also has an arithmetical interpretation, namely

M = ( inf(x ,y)∈Z2\(0,0)

|f (x , y)|)−1,

f (x , y) = ax2 + bxy + cy2, b2 − 4ac = 1.

It follows from the dynamical characterization above that M andL are closed sets of the real line and L ⊂ M.


We have the following result about the Markov and Lagrangespectra:

TheoremGiven t ∈ R we have

HD(L ∩ (−∞, t)) = HD(M ∩ (−∞, t)) =: d(t)

and d(t) is a continuous surjective function from R to [0,1].Moreover:i) d(t) = min1,2D(t), whereD(t) := HD(k−1(−∞, t)) = HD(k−1(−∞, t ]) is a continuousfunction from R to [0,1).ii) maxt ∈ R | d(t) = 0 = 3.iii) There is δ > 0 such that d(

√12− δ) = 1.


In this work we also proved that:

• limt→+∞HD(k−1(t)) = 1

• L′ is a perfect set, i.e., L′ = L′′.

In collaboration with C. Matheus, we proved that0.53 < HD(M \ L) < 0.888.We also found the currently largest known element in M,namely[3; 2,2,2,1,2,3,3,2,2,2,1,2,2,1,2,1,2,1,1,2] +[0; 3,2,1,2,2,2,3,3] =

=7940451225305−

√3

2326589591051+−483 +

√330629

310= 3.70969985975 . . . .


Main tool from Fractal Geometry and Dynamical Systems:The scale recurrence lemma, which is the main technicallemma of the work with Yoccoz on Palis’ conjecture, can beused in order to prove the following dimension formula ([M16]):

Theorem: (Dimension formula)

If K and K ′ are regular Cantor sets of class C2 and K is nonessentially affine, then HD(K + K ′) = minHD(K ) + HD(K ′),1.

A version of this result was also proved by Hochman andShmerkin ([HS12]).


As we have seen, the sets M and L can be defined in terms ofsymbolic dynamics. Inspired by these characterizations, wemay associate to a dynamical system together with a realfunction generalizations of the Markov and Lagrange spectra asfollows:

DefinitionGiven a map ψ : X → X and a function f : X → R, we definethe associated dynamical Markov and Lagrange spectra asM(f , ψ) = supn∈Nf (ψn(x)), x ∈ X andL(f , ψ) = limsupn→∞f (ψn(x)), x ∈ X, respectively.Given a flow (ϕt )t∈R in a manifold X , we define the associateddynamical Markov and Lagrange spectra asM(f , (ϕt )) = supt∈Rf (ϕt (x)), x ∈ X andL(f , (ϕt )) = limsupt→∞f (ϕt (x)), x ∈ X, respectively.



DefinitionGiven a map ψ : X → X and a function f : X → R, we definethe associated dynamical Markov and Lagrange spectra asM(f , ψ) = supn∈Nf (ψn(x)), x ∈ X andL(f , ψ) = limsupn→∞f (ψn(x)), x ∈ X, respectively.

Given a flow (ϕt )t∈R in a manifold X , we define the associateddynamical Markov and Lagrange spectra asM(f , (ϕt )) = supt∈Rf (ϕt (x)), x ∈ X andL(f , (ϕt )) = limsupt→∞f (ϕt (x)), x ∈ X, respectively.



DefinitionGiven a map ψ : X → X and a function f : X → R, we definethe associated dynamical Markov and Lagrange spectra asM(f , ψ) = supn∈Nf (ψn(x)), x ∈ X andL(f , ψ) = limsupn→∞f (ψn(x)), x ∈ X, respectively.Given a flow (ϕt )t∈R in a manifold X , we define the associateddynamical Markov and Lagrange spectra asM(f , (ϕt )) = supt∈Rf (ϕt (x)), x ∈ X andL(f , (ϕt )) = limsupt→∞f (ϕt (x)), x ∈ X, respectively.


We will describe some results obtained in collaboration with S.Romaña, C. Matheus and A. Cerqueira.

Theorem (M., Romaña)

Let Λ be a horseshoe associated to a C2-diffeomorphism ϕsuch that HD(Λ) > 1. Then there is, arbitrarily close to ϕ adiffeomorphism ϕ0 and a C2-neighborhood W of ϕ0 such that,if Λψ denotes the continuation of Λ associated to ψ ∈W, thereis an open and dense set Hψ ⊂ C1(M,R) such that for allf ∈ Hψ, we have

int L(f ,Λψ) 6= ∅ and int M(f ,Λψ) 6= ∅,

where int A denotes the interior of A.


In a work in collaboration with A. Cerqueira and C. Matheus, weprove:

Lemma

Let (ϕ, f ) be a generic pair, where ϕ : M2 → M2 is adiffeomorphism with Λ ⊂ M2 a hyperbolic set for ϕ andf : M → R is C2. Let πs, πu be the projections of the horseshoeΛ to the stable and unstable regular Cantor sets K s,K u

associated to it (along the unstable and stable foliations of Λ).Given t ∈ R, we define

Λt =⋂

m∈Zϕm(p ∈ Λ|f (p) ≤ t),

K st = πs(Λt ),K u

t = πu(Λt ).

Then the functions ds(t) = HD(K st ) and du(t) = HD(K u

t ) arecontinuous and coincide with the corresponding boxdimensions.


The following result is a consequence of the scale recurrencelemma:

Lemma

Let (ϕ, f ) be a generic pair, where ϕ : M2 → M2 is adiffeomorphism with Λ ⊂ M2 a hyperbolic set for ϕ andf : M → R is C2. Then

HD(f (Λ)) = min(HD(Λ),1).

Moreover, if HD(Λ) > 1 then f (Λ) has persistently non-emptyinterior.

Using the previous lemmas we prove a generalization of theresults on dimensions of the dynamical spectra:


Theorem

Let (ϕ, f ) be a generic pair, where ϕ : M2 → M2 is aconservative diffeomorphism with Λ ⊂ M2 a hyperbolic set forϕ and f : M → R is C2. Then

HD(L(f ,Λ) ∩ (−∞, t)) = HD(M(f ,Λ) ∩ (−∞, t)) =: d(t)

is a continuous real function whose image is [0,min(HD(Λ),1)].

It is also possible to prove ([M17]) the following: let (ϕ, f ) be ageneric pair, where ϕ : M2 → M2 is a diffeomorphism withΛ ⊂ M2 a hyperbolic set for ϕ and f : M → R is C2. Thenmin L(f ,Λψ) = min M(f ,Λψ) = f (p), for a periodic point p ∈ Λψ,and is an isolated point in both L(f ,Λψ) and M(f ,Λψ).


In this context, in collaboration with D. Lima, we proved thefollowing result on the topological structure of typical dynamicalMarkov and Lagrange spectra:

Theorem (M., D. Lima)

Let Λ be a horseshoe associated to a conservative C2

diffeomorphism ϕ such that HD(Λ) > 1. Then there is,arbitrarily close to ϕ a diffeomorphism ϕ0 and a residual set Rin a C2-neighborhood W of ϕ0 such that, if Λψ denotes thecontinuation of Λ associated to ψ ∈ R, there is a residual setHψ ⊂ C1(M,R) such that for all f ∈ Hψ, we have

supt ∈ R|d(t) < 1 = inf int L(f ,Λψ) = inf int M(f ,Λψ) 6= ∅.


The classical Markov and Lagrange spectra can also becharacterized as sets of maximum heights and asymptoticmaximum heights, respectively, of geodesics in the modularsurface N = H2/PSL(2,Z).A small movie by Pierre Arnoux and Edmund Harriss

We extend the fact that these spectra have non-empty interiorto the context of negative, non necessarily constant curvatureas follows:

Theorem (M., Romaña)Let N provided with a metric g0 be a complete surface withfinite Gaussian volume and Gaussian curvature boundedbetween two negative constants, i.e., if KN denotes theGaussian curvature, then there are constants a,b > 0 such that

−a2 ≤ KN ≤ −b2 < 0.

SN is its unitary tangent bundle and φ its geodesic flow.


The classical Markov and Lagrange spectra can also becharacterized as sets of maximum heights and asymptoticmaximum heights, respectively, of geodesics in the modularsurface N = H2/PSL(2,Z).A small movie by Pierre Arnoux and Edmund HarrissWe extend the fact that these spectra have non-empty interiorto the context of negative, non necessarily constant curvatureas follows:

Theorem (M., Romaña)Let N provided with a metric g0 be a complete surface withfinite Gaussian volume and Gaussian curvature boundedbetween two negative constants, i.e., if KN denotes theGaussian curvature, then there are constants a,b > 0 such that

−a2 ≤ KN ≤ −b2 < 0.

SN is its unitary tangent bundle and φ its geodesic flow.


Theorem (M., Romaña, continuation)

Then there is a metric g close to g0 and a dense and C2-opensubset H ⊂ C2(SN,R) such that

int M(f , φg) 6= ∅ and int L(f , φg) 6= ∅

for any f ∈ H, where φg is the vector field defining the geodesicflow of the metric g.Moreover, if X is a vector field sufficiently close to φg then

int M(f ,X ) 6= ∅ and int L(f ,X ) 6= ∅

for any f ∈ H.

We proved analogous results for geometric Lorenz attractors incollaboration with M. J. Pacífico and S. Romaña.


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Muito obrigado!Muchas gracias!

Thank you very much!Merci beaucoup!

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