real one-dimensional dynamics: real and complex methods

Real one-dimensional dynamics:real and complex methods

Sebastian van Strien, Imperial College

October 1, 2013

Sebastian van Strien, Imperial College Real one-dimensional dynamics

Throughout these talks we assume that N is an interval or a circleand that f : N → N is real analytic. For example:

f (x) = ax(1− x) or f (x) = x2 + c


One of the reasons real one-dimensional dynamics has been suchan exciting field is because

the theory is far from trivial, yet almost complete;

the theory can be considered as a model for what can happenin higher dimensions.

My first talk will be about theorems that can be obtained byreal tools.

The later talks will then discuss why one introduces complextools


Real methods: (for example) results about attractors

Notation: ω(x) is the set of accumulation points of the sequencex , f (x), f 2(x), . . . .

It would be great to describe all orbits of f , but it turns out to bemuch more fruitful to describe attractors and ergodic properties.

We say that a compact forward invariant set is a topological resp.metric attractor if

B(X ) = x ; ω(x) ⊂ X

is of second Baire category (i.e. countable intersection of openand dense) resp. has positive Lebesgue measure, and if for anyX ′ ( X , B(X ′) does not have this property.


Why complex methods: density of hyperbolicity and rigidity

The nicest maps are those where each attractor is a hyperbolicperiodic orbits. These maps are called the hyperbolic.

We will sketch a proof that - in some sense - most maps arehyperbolic. This problem goes back to Fatou (1930’s).

These latter results rely on constructing an extension of f to thecomplex plane.

This interplay of real and complex methods in interval dynamicswill be one of the main topics of these lectures.


First Lecture

Real Bounds and ErgodicProperties.


Description of attractors

Theorem

Each map has at least one and at most a finite number ofattractors. If X is an attractor then one of the following:

1 X is a periodic attractor;

2 X = ω(c) where c is a critical point of f so that ω(c) is aCantor set which is minimal and has zero Lebesgue measure;

3 X is equal to a finite union of intervals which contains acritical point (or equal to the entire space N)

Definition: An invariant set is called minimal if each forward orbitis dense in ω(c).

Corollary (Denjoy): An attractor of a circle diffeomorphism iseither the whole circle or a periodic orbit. (Iterates are either denseor converge to a periodic orbit.)


Examples

Examples of maps f : [0, 1]→ [0, 1]:

f (x) = 2x(1− x). Then x = 1/2 is an attracting fixed point:f (1/2) = 1/2 and f ′(1/2) = 0.

f (x) = 3x(1− x). Then x = 2/3 is a weakly attracting fixedpoint: f (2/3) = 2/3 and f ′(2/3) = −1.

f (x) = ax(1− x) is an infinitely renormalizable map wherethe parameter a ≈ 3.569.. is at the accumulation of perioddoubling: there exists a sequence of intervals Jn and integersp(n) so that

Jn, . . . , fp(n)−1(Jn) have disjoint interiors

andf p(n)(Jn) ⊂ Jn.

The resulting attractor is a Cantor set. It turns out this NOTthe only example of a Cantor attractor.


Examples

f (x) = 4x(1− x). In this case the attractor is [0, 1] and themap is conjugate to a tent map with slope ±2. There areinfinitely many periodic orbits (or all periods), but a.e. pointx ∈ [0, 1] has a dense orbit in [0, 1].


Historical Background

The proof of the previous theorem in the non-invertible casehas a long history:[Guc79, dMvS89, BL89, Lyu89, MdMvS92, vSV04]

Note that the objects in the classification in the theorem arethe same, regardless whether the attractor is a topological ora metric one.

Milnor posed the question whether a metric attractor is also atopological attractor (and vice versa). The answer turns outto be NO, as we will see.


Nice Intervals and first return maps

For simple maps such as f (x) = 4x(1− x) one describe the mapthrough a Bernoulli or Markov setting:

J1 → J1 ∪ J2, J2 → J1 ∪ J2

where J1 = [0, 1/2] and J2 = [1/2, 1]. However, for most maps thisis not possible.

Instead: use first return maps to so-called nice intervals.

Let I be an interval, and assume that there exists a (minimal)n > 0 so that f n(x) ∈ I . Then we denote the component off −n(I ) containing x by by Lx(I ).

An interval is called nice if no iterate of x ∈ ∂I ever getsmapped into the interior of I .

If I is nice then Lx(I ) ⊂ I whenever x ∈ I .

This makes it useful to work with first return domains.


If I is nice, then a pullback Lx(I ) is also nice.

Two pullbacks of I are either disjoint, or one is contained inthe other.

Nice intervals are easy to find.

Indeed, let’s say f is unimodal. Then take a periodic orbit,choose p in the orbit ‘closest to the critical point’. ThenI = [p, p′] is a nice interval where p′ so that f (p′) = p.

The first return map RI : Dom(I )→ I to I has (usuallyinfinitely many) diffeomorphic branches and a folding branch.

One of the main challenges is to control the distortion. If all thebranches were linear, then one knows essentially everything.


Notation and terminology

If T = [a− x , a + x ] is an interval in R and τ > 0 then

τ · T := [a− τx , a + τx ].

reminder: Lx(I ) is the component of f −n(I ) containing xwhere n is minimal so that f n(x) ∈ I .

J1, . . . , Jk have intersection multiplicity m if any point x iscontained in at most m of the intervals J1, . . . , Jk .

If f n(x) ∈ T where n is minimal, then the pullback of T isLx(T ) and the pullback chain is the collection of intervals

Lx(T ),Lf (x)(T ), . . . ,Lf n−1(x)(T ),Lf n(x)(T ) = T .

Let f have b critical points ci with critical order li . Then wesay that f has type b = (l1, . . . , lb).


Tools: Schwarzian derivative

The theorem require estimates on high iterates of a map. Since fis non-linear, as there are critical points, this is not so easy.

Schwarzian derivative: Define

Sf (x) =f ′′′(x)f ′(x)− (3/2)f ′′(x)

[f ′(x)]2.

Then S(f g) = Sf [g ′(x)]2 + Sg . Hence

Sf < 0 =⇒ Sf n < 0 for all n ∈ N.

Koebe: Then for δ > 0 there exists K so that the followingholds. If g : T → T ′ := g(T ) is a diffeomorphism and Sg < 0then for each x , y ∈ T so that g(x) ∈ (1− δ)g(T ) one has|Dg(x)|/|Dg(y)| ≤ K . See blackboard


Negative Schwarzian appear naturally:

Let f be a polynomial of degree ≥ 2 with real coefficients andassume that all zeros of Df are real. Then Sf < 0.

(Hint: By assumption Df (x) = A∏n

j=1(x − aj) where aj arereal. Then

Sf (x) = 2∑

i<j

1

(x − ai )(x − aj)− 3

2

[∑

i

1

(x − ai )

]2

.

It is not hard to see that this is negative for x real. (There isa more insightful way of showing this, which we will discussbriefly below.)


Many papers assumes that a map has negative Schwarzian. Thissimplifies because

each periodic attractor has a critical point in its immediatebasin

one has Koebe control on diffeomorphic branches.

It turns out that the assumption Sf < 0 (with extra work) canalways be replaced by assuming:

all periodic points of f are hyperbolic and repelling.


Cross-Ratio distortion

Schwarzian derivative is closely related to cross-ratio (there isa formula...)

When J ⊂ T ⊂ R are intervals, then C (T , J) :=|T ||J||L||R| where

L,R are the components of T − J.

If f : T → f (T ) is a continuous bijection then one canconsider the expansion of the cross-ratio:

C (fT , fJ)

C (T , J).


Cross-Ratio and Poincare metric

Expansion of cross-ratio corresponds to Sf < 0.

One can put the Poincare metric on CT = (C− R) ∪ T .Then C (T , J) is equal to the Poincare metric of J.

Another way of showing Sf < 0 for certain polynomials:

Take f : C→ C a real polynomial with only real critical pointsand so that f |T is a diffeomorphism.

Then define f −1 : Cf (T ) → CT by analytic continuation.

Example f (z) = z2, T = [1, 2], f (T ) = [1, 4], see blackboard.

The map f −1 : Cf (T ) → f −1(Cf (T )) is a conformal bijectionand therefore an isomorphism w.r.t. the Poincare metric onthese sets.

f −1(Cf (T )) ⊂ CT =⇒ f −1 : Cf (T ) → CT contracts thePoincare metric on these sets. This proves Sf < 0.


Distortion control due to disjointness

Many maps do not have negative Schwarzian. It turns out thatone can use distortion of cross-ratios instead, provided one hassome disjointness:

Theorem (Koebe in the case of disjoint intervals)

Assume that J ⊂ T and f n|T is a diffeomorphism, and theintersection multiplicity of T , . . . , f n−1(T ) is at most m.

Then Koebe holds (with bounds depending on m).


Real Bounds

One of the most basic tools in real one-dimensional dynamics arereal bounds

Theorem

[vSV04] Assume that I is a nice interval, x ∈ I and assume thatRI (x) /∈ Lx(I ). Then (1 + δ)L2

x(I ) ⊂ Lx(I ).

Let’s explain some of the ideas behind the proof and why this ishelpful.


The smallest interval argument

Let J, . . . , f n(J) be disjoint intervals. Then one of them, f k(J), isthe smallest. Assume the smallest is not the left or right mostinterval. Then the smallest f k(J) has two larger intervals f l(J)and f r (J) to its right and its left. Now take

T ′ = [f l(J), f r (J)] ⊃ f k(J)

and pullback T ′ ⊃ f n(J) to T ⊃ J.

the resulting chain has multiplicity ≤ 3;

(1 + δ)T ⊃ J.


The spiral structure argument

Let’s take x ∈ I and assume that x visits I at least n times. LetJ0, J1, J2, . . . be the return domains x visits consecutively.

Fix ρ > 0. For each n there exists i0 ≤ n so that

P1 (1 + ρ)Ji0 ⊂ I ;

P2 Ji0 has (at least) one ρ-small side and Ji0+1 is contained in aρ-small side of Ji0 ;

P3 For all i < i0 properties [P1] and [P2] do not hold (whichmeans that one has a spiral structure up to time i0) and theinterval Ji0+1 breaks the spiral structure;

P4 Properties [P1] and [P2] do not hold and the spiral structureholds until time n.

In each situation one obtains space, see blackboard.

Combining these ideas one obtains the proof of the real bounds.


Corollary (Absence of wandering intervals)

If W , f (W ), . . . are disjoint, then f n(W ) converges to anattracting periodic orbit.

Sketch of proof of Corollary (proof by contradiction):

By ‘surgery’ one can always assume that f has periodic orbits(and therefore has nice intervals), see blackboard.

Let x be an accumulation point of the orbit of W .


Sketch of the proof of absence of wandering intervals

Using (a small part of) the proof of real bounds, one can show thatthere exists a nested sequence of nice intervals In ⊃ I ′n 3 x so that

(1 + ρ)I ′n ⊂ Inthe first visit of W to In is contained in I ′nafter some further time W enters In+1.

Any pullback of In+1 which intersects I ′n is contained in I ′n.

This gives

(1 + ρ′)LW (I ′n) ⊂ LW (In) and LW (In+1) ⊂ LW (I ′n).

Combining this gives

(1 + ρ′)LW (In+1) ⊂ LW (In)

and therefore

(1 + ρ′)nW ⊂ (1 + ρ′)nLW (In+1) ⊂ LW (I0).

But since the length of (1 + ρ′)nW tends to infinity, this gives acontradiction.


Application of real bounds

Theorem [Koebe without disjointness]

Assume all periodic orbits of f are hyperbolic and repelling.

Then Koebe holds on diffeomorphic branches.


Examples of results one can obtain using real methods:Invariant measures

Definition: f has an absolutely continuous invariant probabiltiymeasure or an acip, if there exists a probability measure µ, so that

µ(f −1(B)) = µ(B) for each measurable set B and

µ(B) is small when B has small Lebesgue measure.

Theorem

The map f (x) = 4x(1− x) has an acip.

It is enough to show that ∀ε > 0 there exists δ > 0 so that foreach measurable set A ⊂ [0, 1] of Lebesgue measure < δ onehas f −n(A) has Lebesgue measure < ε for any n ≥ 0.

If A is an interval which does not contain 0 or 1, then thisimmediately follows from Koebe.

One can reduce the general case to this situation.


Invariant Measures

There is a long history of results on this (going back to the 50’s),with well-known results by Misiurewicz, Benedicks-Carleson,Collet-Eckmann, Nowicki-vS and others. The sharpest result is:

Theorem ([BSvS03] and [BRLSvS08])

Assume that f is real analytic and has no periodic attractors. Thenthere exists constant C (b) such that if

lim infn|(f n)′(f (c))| ≥ C

for each critical point c then f has an acip.

There is a remarkable sequel to this result, by Rivera-Lettelier and& Shen: one has superpolynomial decay of mixing in this case.


Method of Proof: decomposing pullbacks

Consider a set A of small size

Aim: estimate f −n(A).

Distinguish components of f −n(A).

A ‘good’ component J is one for where exists aneighbourhood T ⊃ J so that

f n|T is a diffeomorphismf n(T ) ⊃ (1 + ξ)f n(J) where ξ is large

The Lebesgue measure of all good components of f −n(A) isobtained in this way

Other components: decompose these branches and use aninductive estimate.


Wild attractors

Theorem (Existence of Wild attractors, [BKNvS96])

There exist maps of the form f (z) = zd + c with c ∈ R and deven (and large) with an invariant Cantor set which is a metric,but not a topological attractor.

Theorem (Non-existence of Wild attractors in the quadratic case,[Lyu94])

Assume that f is unimodal and has a quadratic critical point thenthe notions of topological and metric attractor coincide.

Note that wild attractors also exist for certain real polynomials ofhigher degree with only non-degenerate critical points.


Method of proof: Random Walks

One can decompose the space into disjoint intervals Jn

Each interval Jn maps diffeomorphically onto a countableunion of such intervals

One has sufficient control on non-linearity

Probabilistic proofs to show what happens with points, seeblackboard.


Intemingled attractors

A map with several critical points can have several attractorswhose basins are intermingled:

Theorem ([vS96])

There exists a polynomial f : [0, 1]→ [0, 1] with two critical pointswith two disjoint invariant Cantor sets Λi so that the basin of eachof these sets is dense and has positive Lebesgue measure.


Some questions about higher dimensional systems

Question (Wild attractors for two-dimensional diffeomorphisms)

Let M = S2. Are there diffeomorphisms f : M → M which havewild Cantor attractors? (That is, metric but not topologicalattractor.) It is well-known that Henon maps can have a Cantorset as an attractor, see [GvST89], [DCLM05]. Do these Cantorsets necessarily have to be of solenoidal type?

Question (Wandering domains for Henon maps)

Let H be a Henon map. Is it possible for H to have wanderingdomains, i.e. is it possible that there exists an open set U so thatU, f (U), . . . are all disjoint and so that U is not contained in thebasin of a periodic attractor?


Second Lecture

Second Lecture: Densityof Hyperbolicity andComplex Methods


Density of Hyperbolicity

The simplest situation is when the attractors of f are all hyperbolicperiodic orbits: such f called hyperbolic (also called Axiom A).It would be nice if every map can be approximated by a hyperbolicmap. This problem goes back in some form to

Fatou, who stated this as a conjecture in the 1920’s.

Smale gave this problem ‘naively’ as a thesis problem in the1960’s, see [Sma80].

Jakobson proved that the set of hyperbolic maps is dense inthe C 1 topology, see [Jak71];

The quadratic case x 7→ ax(1− x) was proved in a majorbreakthrough in the the mid 90’s by Lyubich [Lyu97] and alsoGraczyk and Swiatek [GS97].

Blokh and Misiurewicz [BM00] proved a partial result towardsthe density of hyperbolic maps in the C 2 topology.

Shen [She04] then proved the C 2 density of hyperbolic maps.


The general result is:

Theorem (Density of hyperbolicity for real polynomials, [KSvS07a])

Any real polynomial can be approximated by hyperbolic realpolynomials of the same degree.

The above theorem allows us to solve the 2nd part of Smale’seleventh problem for the 21st century. [Sma00]:

Theorem (Density of hyperbolicity for smooth one-dimensionalmaps, [KSvS07b])

Hyperbolic maps are dense in the space of C k maps of thecompact interval or the circle, k = 1, 2, . . . ,∞, ω.


For quadratic maps fa = ax(1− x), the above theorems assert thatthe periodic windows are dense in the bifurcation diagram.

The quadratic case turns out to be special, because in this casecertain return maps become almost linear. This special behaviourdoes not even hold for maps of the form x 7→ x4 + c .


Comments on the strategy of proof: local versus globalperturbations

It turns out that it is often not possible to perturb a map to ahyperbolic map by local methods (in the C k topology, k ≥ 2).

Instead one shows that a non-hyperbolic map is essentiallyuniquely determined by its conjugacy class: if f and g areconjugate then show they are quasi-symmetrically conjugate.This approach goes back to Sullivan.

In [KSvS07a] we showed that this rigidity holds for polynomialswith certain additional restrictions (e.g. all critical points real).

In fact, it holds in general:


Quasi-symmetric rigidity

Theorem (SvS)

Assume that f , g : [0, 1]→ [0, 1] are real analytic, topologicallyconjugate and that the topologically conjugacy is a bijectionbetween

the set of critical points and the order of correspondingcritical points is the same;

the set of parabolic periodic points.

Then the conjugacy between f and g is quasi-symmetric.

A homeomorphism h : [0, 1]→ [0, 1] is called quasi-symmetric ifthere exists K <∞ so that

1

K≤ h(x + t)− h(x)

h(x)− h(x − t)≤ K

for all x − t, x , x + t ∈ [0, 1].


Note that f and g can only have finitely many parabolicperiodic orbits (see [MdMvS92]

All conditions are necessary

Previous results:

Khanin and Teplinsky show this for critical circle maps(building on earlier work of de Faria, de Melo and Yampolsky).Levin + vS show that for covering maps with one inflectionpoint c , one can obtain a qs conjugacy restricted to ω(c),provided ω(c) is either minimal or every periodic orbit in ω(c)is repelling.Kozlovski-Shen-vS for real polynomials with only real criticalpoints.

In our proof complex methods are essential.

Interestingly, the proof even goes through to the C 3 category,(joint work with Trevor Clark).


Quasi-symmetric rigidity =⇒ density of hyperbolicity?

Let’s explain this for the family z2 + c.

Assume, by contradiction, that there exists a non-trivialinterval of parameters [cl , cr ] so that the corresponding mapare all non-hyperbolic.Hence, ∀c ∈ [cl , cr ] and for all n ≥ 0 one has f n

c (0) 6= 0.So all maps fc with c ∈ [cl , cr ] are topologically conjugate.Assume that [cl , cr ] is a maximal interval with this property(that this interval is closed when f is non-hyperbolic followsfrom kneading theory).By qs-rigidity Thm, fc , fc ′ are qs-conjugate ∀c , c ′ ∈ [cl , cr ].

Now assume that cl 6= cr . Then we will use quasiconformal mapsto obtain an open neighbourhood O ⊃ [cl , cr ] so that for allc , c ′ ∈ O the maps fc , fc ′ are also topologically conjugate.

This contradicts maximality of [cl , cr ]. Hence cl = cr , and densityfollows.

So let’s go to the complex plane!Sebastian van Strien, Imperial College Real one-dimensional dynamics

Quasiconformal homeomorphisms

An orientation preserving homeomorphism h is calledK -quasiconformal if

there exists a constant K <∞ such that for Lebesgue almostall x ∈ C

lim supr→0

sup|y−x |=r |h(y)− h(x)|inf |y−x |=r |h(y)− h(x)| ≤ K .

If K = 1 then h is conformal.

Such maps are, for example, Holder and Lebesgue almosteverywhere differentiable (as maps from C = R2 to C = R2).

(In general, a conjugacy cannot be C 1, because thenmultipliers at periodic points would be the same.)


In this case qs-rigidity =⇒ qc-rigidity.

Assume first that f (z) = z2 + cl and f (z) = z2 + cr areqs-conjugate.

Fact: Any qs-homeomorphism h on R can be extended to aK -quasiconformal-homeomorphism H on C.

Hence ∃ a qc map H so that H f = f H on R and near ∞.

Now define a sequence of lifts Hn inductively by H0 = H andf Hn+1 = Hn f . This can be done, see blackboard.

Hn is again K-qc for any n with the same K .

Hn+1 = Hn on ever larger sets.

Fact: The space of K -quasiconformal maps is compact.

Hence Hn converges to some K -qc homeomorphism H.

Therefore f H = H f .


f = H f H−1 for some qc-homeo H . So what?

Now we use the Measurable Riemann Mapping Theorem:

DH(z) exists for a.e. z .

So DH(z) sends ellipse based at z to circle based at H(z).

One can associate to this ellipse some numberµ(z) ∈ D = w ; |w | < 1 where |µ(z)| is the eccentricity ofthe ellipse.

By this theorem, associated to tµ(z) there is another qc mapHt with the same long and short axis and eccentricity t|µ(z)|.Normalize so that Ht(0) = 0 and Ht(x)/x → 1 as x →∞.

Since f = H f H−1 is holomorphic, the mapft = Ht f H−1

t is again conformal, see blackboard.

ft has a unique critical point, is holomorphic and thenormalisation implies that ft(z) = z2 + c(t).


What’s useful about ft = Ht f H−1t ?

Reminder: ft(z) = Ht f H−1t (z) = z2 + c(t)

H0 = id =⇒ f0 = H0 f H−10 = f = z2 + cl =⇒

c(0) = cl ;

f1 = H f H−1 = f = z2 + cr =⇒ c(1) = cr .

By the Measurable Riemann Mapping Theorem, t 7→ ft(0) isholomorphic. Hence t 7→ c(t) is holomorphic.

By construction, t 7→ c(t) is real and has no critical points.

Hence for t > 1, t ≈ 1 one has c(t) > cr and the map ft isstill conjugate to f .

=⇒ open neighbourhood of [cl , cr ] of conjugate maps.

Together this shows qs-rigidity =⇒ density of hyperbolicity.


qs-rigidity =⇒ density of hyperbolicity for realpolynomials with real critical points

If the two qs-conjugate polynomials only have real critical pointsthen one can generalise this argument:

use an inductive dimension reduction:

restrict to algebraic varieties of the form f ; f n(c1) = c2 oflower and lower dimension.


qs-rigidity =⇒ density of hyperbolicity for realpolynomials

If the two qs-conjugate polynomials f , f have non-real criticalpoints then f , f qs-conjugate 6=⇒ f , f are qc-conjugate.

Lifting f Hn+1 = Hn f not possible: one has no informationabout the orbits of the complex critical points.

Want all critical points to be captured (in hyperbolic basin).Let’s capture more and more critical points:

Step 1: Take any one-parameter families ft , t ∈ [−1, 1] ofregular maps: each neutral periodic orbit of ft has a criticalpoint in its basin. Assume this family so that f1 has morecaptured points than f0 and so that captured critical pointsfor f0 remain captured for ft ∀t ∈ [0, 1].

Step 2: Thm: ∃t ≈ 0 so that ft has new captured criticalpoints. Here use holomorphic motion and geometric controlfor certain complex box mappings. (Not soft...)


How to construct regular families?

Step 3: Approximate f by a polynomial f of the same degreewithout neutral periodic orbits and same captured critical pts.

Step 4: All maps C 3 near f are regular.

Step 5: Locally perturb f to a C 3 hyperbolic map g (here use‘complex bounds’)!!! Note f and g are not C∞ close at all.

Step 6: Approximate the smooth map g by a polynomial mapG of much higher degree.

Step 7: Consider the family ft = f + tG . By Step 4 this aregular family.

Step 8: Using Step 2: ∃t ≈ 0 so that ft has more capturedcritical points. However, ft has much higher degree.

Step 9: ft , f and f are all C 0 close on a large disc DR . Hence,using the so-called Straightening Theorem, ∃ a real pol. f

of the same degree as fconjugate to ft on DR/2

still close to f .


Density of families of entire families

Theorem (Hyperbolicity for entire maps (with Lasse Rempe))

Let f be an entire function with a finite number of critical valuesand either

f is bounded is on the real axis.

some sector condition is satisfied.

Then there exist orientation preserving homeomorphisms φ, ψarbitrarily close to the identity such that g := ψ f φ−1 is entireand hyperbolic.


Application: trigonometric polynomials

Consider generalized trigonometric polynomial Fµ : R/Z→ R/Z:

Fµ(t) = D·t+µ1+µ2m sin(2πmt)+m−1∑

j=1

(µ2j sin(2πjt)+µ2j+1 cos(2πjt)).

Note that if µ, µ′ ∈ R2m with µ1 − µ′1 ∈ Z, then fµ = fµ′ . Sochoose µ = (µ1, . . . , µ2m) ∈ ∆, where

∆ := µ ∈ R/Z×R2m−1 : µ2m > 0 and fµ is 2m−multimodal .

For example: the Arnol’d family x 7→ x + α + β sin(2πx). In this

case we have the following theorem:


Theorem (Density of hyperbolicity and rigidity in the trigonometricfamily, joint with Lasse Rempe)

Hyperbolic parameters in ∆ for which fµ are dense. Furthermore,

1 Consider the set [µ0] of parameters µ for which fµ istopologically conjugate to fµ0 by an order-preservinghomeomorphism of the circle. Then [µ0] has at most mcomponents.

2 If fµ0 has no periodic attractors on the circle, then eachcomponent of [µ0] is equal to a point.

This answers the conjectures posed by de Melo, Salomao andVargas.


Here

we need to pay attention to points that go repeatedly toinfinity and back again and show absence of line fields on thisset.

We also need to show that f , f are qs conjugate on the realline.

In the polynomial case this was not fully needed, but now wedo not have a straightening theorem.


Summary:

real method =⇒ real bounds =⇒

Koebecomplex bounds

complex method =⇒

quasiconformal mapsMeasurable Riemann Mapping Theorem

Holomorphic Motion


How to construct qs-symmetries?

One approach is to use Carleson box construction.

We shall use complex methods, namely a complex analogue of thenice interval (puzzle pieces) and then to use our

QC-Criterion: For any ε > 0 there exists a constant K with thefollowing property.Let φ : Ω→ Ω be a qc homeomorphism between two Jordandomains. Let X ⊂ Ω consist of pairwise disjoint topological discs(possibly infinitely many).Assume that the following hold

the components of X are topological discs with ε-boundedgeometry each of which ε-well-inside Ω (and the same holdsfor φ(X )).

φ is 1-qc on Ω− X .

Then there exists a new K -qc homeo ψ : Ω→ Ω which agrees withφ on ∂Ω.


Third Lecture

Third Lecture: ComplexBox Mappings and

Complex Bounds =⇒qc-Rigidity


Puzzle pieces for polynomial maps

Given a polynomial f , there exists a way of constructing nice sets,i.e. sets Pn so that no point on the boundary is ever mapped intothe interior of Pn.This construction uses external rays and equipotentials landing onperiodic orbits, see Misha’s lectures and blackboard. These curvescome from the Bottcher coordinates near ∞. The partitionelements are called Yoccoz puzzle pieces.


Existence of good rays

More precisely. One can prove

f has no neutral periodic orbits and J(f ) is connected =⇒∃ Jordan domain P which is strictly nice:

f n(x) /∈ P for all x ∈ ∂P and all n ≥ 0.


Complex box mappings

What we will get:

This is, an object analogous to a polynomial-like map F : U → V ,except that

there are several components of V ,

some components of V are also components of U

there may be infinitely many components of U intersectingthe orbit of the critical points.


Complex box mappings: formal definition

Definition (Complex Box Mappings)

F : U → V between open sets in C is a Complex Box Mapping if itis holomorphic and: following hold:

V is a union of finitely many pairwise disjoint Jordan discs;

every connected component V ′ of V is either a connectedcomponent of U or the intersection of V ′ and U is a union ofJordan discs with pairwise disjoint closures which arecompactly contained in V ′,

for each component U ′ of U, F (U ′) is a component of V .

F has finitely many critical points; in our setting eachcomponent of V contains precisely one critical point.

Even in the case of non-renormalizable polynomials, we obtain acomplex box mapping with each component of V containing onlyone critical point, only after obtaining complex bounds.


Complex bounds for box mappings

We will say that this comes with complex bounds if

bounds on the moduli of the relevant annuli

bounded geometry

That is, the assumptions of the QC-criterion hold.


Renormalisation of a complex box mapping F : U → V

Its Julia set is J(F ) = x ; F n(x) ∈ U for all n ≥ 0.The first return map to a component of U is again a complexbox mapping. This is called a renormalisation of F : U → V .

A component of F−n(V ) is called a puzzle piece ofF : U → V .

We fix a parametrisation of ∂V (say, coming from Botchercoordinates). This gives a boundary marking of the boundaryof the boundary of each puzzle piece.

F is renormalizable at c ⇐⇒ ∃s with F sn(c) ∈ LcV for alln ≥ 0.

If no first return map to any puzzle piece containing a criticalpoint is renormalizable, then we say that F (and the originalpolynomial) is non-renormalizable.

In the real case, renormalizable ⇐⇒ ∃ periodic interval.


QC-rigidity of non-renormalizable polynomials

A generalisation of the theorem of Yoccoz that Misha talked about:

Theorem (QC-rigidity of non-renormalizable complex box mappingswithout neutral points [KvS09])

Two such F : U → V and F : U → V are topol. conjugate, thenF , F qc-conjugate.

Theorem (QC-rigidity of non-renormalizable polynomials withoutneutral points [KvS09])

f , f topol. conjugate =⇒ f , f qc-conjugate.

The proof of Yoccoz for z → z2 + c does not work:

He uses that 2× 1/2 = 1; here 2 is because of combinatoricsand 1/2 because of the degree is z 7→ z2 + c .

z 7→ z4 + c gives 2× 1/4 < 1 which leads to failure.


Ingredients of the proof (for non-renormalizablepolynomials):

1 The spreading principle: allows one to extend these qc-mapsglobally

2 QC-criterion: geometric control gives qc maps defined onsmall puzzle pieces

3 The enhanced nest from [KSvS07a]: a way to choose suitablepuzzle pieces

4 A lemma of Kahn-Lyubich for estimating the moduli ofpullbacks of annuli.


Spreading Principle:

Assume

Take sets W , W containing all critical points consisting of aunion of puzzle pieces.

Assume that one has K -qc map h : W → W which respectsthe boundary marking.

Then

∃ K -qc map H : V → V

F H = H F outside W .

So it is clear what to aim for:Find a sequence of puzzle pieces Wn, Wn with the components ofWn, Wn shrinking to points and K -qc maps hn : Wn → Wn.

Then ∃ K -qc maps Hn : V → V with F Hn = Hn F outside Wn.Taking limits gives a K -qc conjugacy.


The QC-criterion

QC-Criterion [KSvS07a]: For any ε > 0 there exists a constantK with the following property.Let φ : Ω→ Ω be a qc homeomorphism between two Jordandomains. Let X ⊂ Ω consist of pairwise disjoint topological discs(possibly infinitely many).Assume

the components of X are topological discs with

ε-bounded geometryare all ε-well-inside Ω

(and the same holds for φ(X )).

φ is 1-qc on Ω− X .

Then there exists a new K -qc homeo ψ : Ω→ Ω which agrees withφ on ∂Ω.


The principal nest of puzzle pieces

Pick a critical point c .

One could choose inductively first return domains Lc(V ),L2c(V ) := Lc(Lc(V )), . . . and so on.

This is called the principal nest.

The reason we don’t use this is because the geometry of thesedomains gets unbounded, see blackboard.


The reluctantly or non-recurrent case

One possibility is that

there exists D ∈ Na puzzle piece J,

infinitely many puzzle pieces Jn containing a critical point andpn so that f pn : Jn → J are branched covering of degree ≤ D.

This is called the reluctantly recurrent case.

In this case, one can go from arbitrarily small scale with ‘boundeddistortion’ to large scale.

Then one can easily use the QC-criterion on these smaller andsmaller pieces, and using the spreading principle we are done.


The enhanced nest in the persistently recurrent case

Instead, we will choose inductively a sequence of puzzle pieces Ininductively with some suitable properties. So choose I0 = V .

In+1 is a pullback of In of bounded order: ∃ an integer pn sothat f pn : In+1 → In is a branched covering of degree ≤ Dwhere D is fixed.

there exists a dynamically defined annulus An ⊃ ∂In whichdoes not intersect ω(c).

some combinatorial properties, namely

the minimal amount of time it takes from In+1 to In+1 is atleast (1/3)pn

pn+1 ≥ 2pn.

Doing this carefully one obtains the enhanced nest.


Complex bounds in this setting

In [KSvS07a] we proved complex bounds for the enhancednest for real polynomials – by bare hand.

In [KvS09], we use a remarkable lemma due to Kahn &Lyubich, which allows us to prove the previous result fornon-renormalizable polynomials (not necessarily real).

The lemma by Kahn-Lyubich is hard to prove and hard tostate. Kozlovski and I have a short proof in the real case foran easier to state and somewhat sharper result.

I will state the result for the real setting on the next slide, butwon’t show how to apply it (this will use the properties of theenhanced nest from the previous slide).

The statement on the next slide will apply to a branchedcovering map G : U ′ → V of the form G = F n. Here

F : U → V is a real-symmetric complex box mapping.U ′ is a real-symmetric component of the domain of G := F n.


Proposition (Small Distortion of Thin Annuli)

∀K ∈ (0, 1) ∃ κ > 0. Let G : U ′ → V be as before, B areal-symmetric region in V and A a real-symmetric connectedcomponent of G−1(B). Write d = deg(G |A) = d andD = deg(G |U′).

mod(U ′ − A) ≥ KD

2dminκ,mod(V − B).

Corollary: For each D there exists κ > 0 so that if G : A→ B isunivalent then

mod(U ′ − A) ≥ (1/4) minκ,mod(V − B).

Improves on mod(U ′ − A) ≥ (1/D)mod(V − B) whenmod(V − B) ≈ 0, see blackboard: xn+1 ≥ min(κ, 2xn).


What about real analytic or C 3 maps?

For real analytic maps we do not have rays and equipotentials,let alone for C 3 maps. So what then?

For infinitely renormalizable maps we also cannot use theprevious method.

So we need to construct the complex box mappings – togetherwith the complex bounds – by hand.

Let’s assume, for now, that f : N → N is real analytic.Then ∃ a neighbourhood O of N on which f extends to a complexanalytic map f : O → C.However, there is not relation between O and f (O).


Poincare discs

Let’s take the Poincare metric defined on the slit region CJ . Theset of points with constant ≤ R to J w.r.t. Poincare distanceconsists of two circle segments. Let Dθ(J) be the Poincare discwith angle θ based on J.

470 CHAPTER VI. RENORMALIZATION

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J

Fig. 5.2: The set of points whose hyperbolic distance to J is equal to k.

Similarly, if T is as above but maps J dieomorphically to T (J) then

T (Dk(J)) Dk(T (J)) for all k > 0.

In the next lemma we will show that something similar holds for univalent

holomorphic maps which are just defined near J and not on all of CJ . This

lemma will only be used in Lemma 5.3 below. We should point out that Lemma

5.3 will only be used for maps in some Epstein class in which case the proof

is immediate. However, because Theorem 5.1 might also be true for any real

analytic map we have included a proof of Lemma 5.3 in this more general setting.

Given an interval J R, let DJ be the Poincare neighbourhood of J and J

the Poincare metric on CJ as above.

Lemma 5.2. Given a > 0 and r0 > 0 there exist K = K(r0, a) and l0 > 0 with

the following property. If satisfies the conditions:

1. is holomorphic and univalent on a Euclidean disc of radius a centered

at a point of an interval J R and Dr0(J) is contained in this Euclidean

disc;

2. maps the real axis into the real axis;

3. |J | l0.

Then, provided k r0,

(J)((x),(y)) (1 + K|J |)J(x, y)

for all x, y Dk(J); so in particular (Dk(J)) D(1+K|J|)k((J)).

Proof. Without loss of generality we may assume that (J) = J . Let B(x0; a)

be the Euclidean disc of radius a centered at a point x0 J which contains

Dr0(J) as above. Let us take x0 to be the middle point of J . Furthermore, let


Pullback of Poincare discs in the univalent case

If φ is a univalent map of CJ into Cφ(J) thenφ(Dθ(J)) ⊂ Dθ(φ(J)).

In particular, take a real polynomial f with only real criticalpoint and so that f |T is a diffeomorphism. Then defineφ : Cf (T ) → CT to be the analytic continuation off −1 : f (T )→ T .

Hence the pullback under f of Dθ(f (T )) is ⊂ Dθ(T ).

Take a real analytic map f : [0, 1]→ R.

It extends to an analytic map f : U → C without additionalcritical points.

Consider a very small interval J ⊂ [0, 1].

Find a much bigger disc D ⊃ J with D ⊂ U.

Then one can apply almost the same argument, see next slide.


Distortion of Poincare disc diffeomorphic situation

Lemma (Almost Schwarz Inclusion [dFdM99])

There exist K <∞, a0 > 0 and a function θ with θ(a)→ 0 asa→ 0 such that the following holds. Let F : D→ C be univalentand real-symmetric, and assume that I ⊂ R is an intervalcontaining 0 and |I | < a0. Let I ′ = F (I ). Then

(a) for all θ ≥ θ(|I |) (small θ),

F (Dθ(I )) ⊂ D(1−K |I |1+δ)θ(I ′),

where 0 < δ < 1 is a universal constant;

(b) for all θ ∈ (π/2, π) (lens shaped region),

F (Dπ−θ(I )) ⊂ Dπ−K |I |θ(I ′).


Distortion of Poincare disc under non-diffeomorphicsituation

Lemma

Let ` ≥ 2 be a natural number. Let P(z) = z`. Then there existsλ = λ(K , `) ∈ (0, 1) so that the following holds:

if ` is even, then

P−1(Dθ(−K , 1)) ⊂ Dλθ(−1, 1))

for all θ ∈ (0, π);

if ` is an odd integer, then

P−1(Dθ(−K `, 1)) ⊂ Dλθ(−K , 1))

for all θ ∈ (0, π).


So we need real bounds

See blackboard.

The enhanced nest will give us with some considerable care freespace.


What about the C 3 case: asymptotically conformalextensions

As before: ∂∂z = 1

2 ( ∂∂x + i ∂∂y ).

Definition: Let K be a compact subset of R2, U ⊃ K open andH : U → C be C 1. We say H is asymptotically holomorphic oforder t ≥ 1, on K ⊂ R2 if for every (x , y) ∈ K

∂

∂zH(x , y) = 0, and

∂∂z H(x , y)

d((x , y),K )t−1→ 0

uniformly as (x , y)→ K for (x , y) ∈ U − K .

In our application, K will be an interval contained in the real line.


Pullbacks of Poincare discs in the asymptoticallyholomorphic case

Proposition (Graczyk-Sands-Swiatek)

Let h : I → R be a C 3 diffeomorphism. Then there exist

a C 3 extension H of h to a complex neighbourhood of I , withH asymptotically holomorphic of order 3 on I .

K > 0 and δ > 0 such that if a, c ∈ I are distinct, 0 < α < πand diam(Dα(a, c)) < δ,. Moreover,

H(Dα(a, c)) ⊂ Dα(h(a), h(c)),

where α = α− K |c − a|diam(Dα(a, c)) and α < π.

This implies the previous inclusion lemma.


Summary

Done:

We constructed complex box mappings for non-renormalizablepolynomials without neutral periodic points.

We showed that because of the spreading principle andQC-criterion it is enough to obtain complex bounds.

We showed that one can get complex bounds byKahn-Lyubich’s lemma in this case.

To do:

We we need to show how to construct complex box mappingsfor C 3 and real analytic maps, and even for real infinitelyrenormalizable polynomials.

We will rely on real bounds and analysing pullbacks carefully.

Big bounds will not always be our friends.

When there are many critical points, we will need to fit all thistogether in order to obtain a global conjugacy.


Fourth Lecture

Fourth Lecture: ComplexBox Mappings andComplex Bounds

for real analytic (and C 3)maps


Reminder go goals and ingredients of the proof

We want to show that:Real analytic maps which are topologically conjugate areqs-conjugate

provided critical points and parabolic points are mapped eachother; critical points to critical points of the same order.

e.g. f (x) = 4x(1− x), f (x) = sin(πx) are qs-conjugate on [0, 1].

Ingredients of the proof:

1 The spreading principle: allows one to extend these qc-mapsglobally

2 QC-criterion: geometric control gives qc maps defined onsmall puzzle pieces

3 The enhanced nest from [KSvS07a]: a way to choose suitablepuzzle pieces

4 Complex bounds


Spreading Principle without Bottcher coordinates:

Assume F : U → V is a complex box mapping.

Take sets W , W containing all critical points consisting of aunion of puzzle pieces.

need to have a K -qc external map. H : V → V :H(U) = U and H F = F H on ∂U,(replacing boundary marking in polynomial case).

=⇒∃ K -qc map H : V → V

F H = H F outside W .

So it is clear what to aim for:Find a sequence of puzzle pieces Wn, Wn with the components ofWn, Wn shrinking to points and K -qc maps hn : Wn → Wn.

Then ∃ K -qc maps Hn : V → V with F Hn = Hn F outside Wn.Taking limits gives a K -qc conjugacy.


The QC-criterion

Define H(x) := lim infr→0

sup|y−x |=r |φ(y)− φ(x)|inf |y−x |=r |φ(y)− φ(x)| .

QC-Criterion [KSvS07a]: ∀ε,H > 0 ∃K <∞ with:Let φ : Ω→ Ω be a qc homeo between two Jordan domains. LetX ⊂ Ω consist of pairwise disjoint topological discs (possibly ∞many) and a set Z ⊂ Ω of zero Lebesgue measure.Assume

1 the components of X are topological discs with

ε-bounded geometryare all ε-well-inside Ω

(and the same holds for φ(X )).

2 H(x) ≤ H for x ∈ Ω− (X ∪ Z ).

3 H(x) <∞ for x ∈ Z .

Then ∃ new K -qc homeo ψ : Ω→ Ω which agrees with φ on ∂Ω.


Complex box mapping with complex bounds when ω(c) ispersistently recurrent

Lemma

Assume that f is real and non-renormalizable.f is persistently recurrent at c =⇒ ω(c) is minimalω(c) minimal =⇒ only finitely many domains of the first returnmap to any nice interval intersect ω(c).

We construct a generalised enhanced nest:

In is non-terminating then In+1 = f −pn(In) wheref pn : In+1 → In has degree ≤ D.If In is terminating (renormalisable), then In+1 = Lc(R(I )):

COMPLEX BOUNDS WORKING VERISON JULY 2012 9

minimal) and let us define the renormalization R(I) of I to be

R(I) := (↵, (↵)).

Notice that R(I) is not usually a periodic interval. In this case we let Yi denote the compo-nents of the I1 \f1(↵) and we will always take Y0 = R(I). We let Y denote the monotonebranch of the return map to I1 that contains , where 2 ↵, , ().

I

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R(I)

Figure 4. The figure shows two cases where I is terminating together withthe interval R(I) marked in dots.

Let us repeat the construction of the enhanced nest of [KSS] and extend it to the renor-malizable case. We will make use of the following combinatorially defined return time.

Lemma 2.2 ([KSS] Lemma 8.2). Let I 3 c be a critical puzzle piece. Then there exists apositive integer with f (c) 2 I such that the following holds. Let U0 = Compcf

(I) andUj = Compfj(c)f

(j)(I) for 0 j . Then

(1) #j : Uj \ Crit(f) 6= ;, 0 j 1 b2, and(2) U0 \ !(c) Compc(f

(Lf(c)f(I))).

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Lf(c)(I)

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A(I)

Figure 5. The sets A(I) and B(I)

For each critical puzzle piece, I 3 c0, we let = (I) be the smallest positive integer withthe properties specified by Lemma 2.2. We define

A(I) = Compc0f(Lf(c0)(I)),

B(I) = Compc0f(I),

(I) = the smallest successor of I.

So by construction and Lemma 2.2, A(I) B(I) and (B(I) \ A(I)) \ !(c0) = ;, giving amechanism for obtaining free space.

Next, let T = 5b where b is the number of critical points in !(c) and define

E(I) =

T BA(I) if I is non-terminating,Lc0(R(I)) if I is terminating.


Real bounds for the enhanced nest – when ω(c) ispersistently recurrent.

Real Bounds (a priori bounds): there exists ρ > 0 so that, e.g:

In non-terminating =⇒In+1 is ρ-nice: first return domains are ρ-well-inside In+1

In+1 is externally ρ-freeIf |In|/|In+1| < K then In+1 is ρ-internally free.

For any C ′ > 0 there exists C > 0 such thatIn non-terminating and∃x ∈ ω(c0) ∩ In with (1 + 2C )Lx(In) ⊂ In=⇒ (1 + 2C ′)In+1 ⊂ In : Big Bounds

If In−1 is terminating and In is non-terminating, then ∀C > 0∃ ρ > 0 so that |In|/|In+1| < C =⇒ In+1 is ρ-nice and ρ-free.

If In−1 and In are terminating (Feigenbaum case), then ....

Critical points of odd order cause a lots of troubles here.


Complex box mappings in the persistently recurrent

Using these, and similar real bounds:

Theorem (Trevor Clark, Sofia Trejo and SvS)

Assume that f is a real-analytic and ω(c) is persistently recurrent.Then ∃ρ > 0 so that the first return map to

In = ∪c ′Lc ′ In.

extends to a complex box mapping F : U → Vwith complex (a priori) bounds:the domains are ρ-well-inside and the range is ρ-free.

Provided we choose n large enough, ρ does not depend on f , onlyon the number and order of the critical points of f .


First we obtain quasi box mapping

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But there is a procedure to obtain from this a complex boxmapping. One can bound the domains by Poincare domains.

Then obtain

To construct the map H : V → V so that H(U) = U andH F = F U on ∂U is then easy.


Complex box mappings in the non-persistently recurrentcase

If ω(c) is non-persisently recurrent case it is possible that ∃infinitely many first return domains intersecting ω(c).

Again one there exists a complex box mapping to I . But nowthere may be infinitely many regions and they are lens shaped:

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Now to construct the map H : V → V so that H(U) = U andH F = F U on ∂U is less easy.


Construction of an external map H : V ,U → V , U

To construct a qc map H : V → V such that h(U) = U and sothat F h = h F on ∂U.

1 U has infinitely many components.

2 (V −U) ∩C+ is a quasi-circle: this can be done as for criticalcovering maps, see Levin and SvS, Inventiones 2000.

3 there exists a qc map from (V − U) ∩ C+ to (V − U) ∩ C+

which respects the dynamics. To do this we show in particular:

4 there exists a qc conjugacy (V − U) ∩ R→ (V − U) ∩ R. Todo this we will need to use another type of box mapping, theso-called touching complex box mappings G : B → A.


Touching complex box mappings

Touching complex box mappings G : B → A.

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I∗

Figure: A touching box mapping: the range contains critical points, butnot the domain.

∂B will contain the fixed points and parabolic periodic points(and a number of pre images of those).This is one reason why B and A will have tangenciesSo B − A is not a quasi-circle.For this reason it requires some work to show that G : B → Aand G : B → A are qc-conjugate.


Spreading

This is not enough for what we need, because one has to gluetogether this information around each of the critical points.Here we use a partial ordering on the set of critical points andversions of the spreading lemma and pullback arguments.For example, in the minimal case, there are many pre imagesof the fixed point (or other points which escape). These haveto be adjusted. This is again done using the touching boxmapping G : B → A.

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An application to localconnectivity

Theorem

The Julia set of a any real polynomial or rational maps with onlyreal critical points has a Julia set which is locally connected.

There were several previous papers dealing with special cases. Forexample, in a paper with Levin we deal with covering maps of thecircle with one critical points, but those methods break downcompletely with two odd critical points.


An application tomonotonicity of entropy

In the late 70’s, the following question attracted a lot of interest:does the topological entropy of the interval map x 7→ ax(1− x)depends monotonically on a ∈ [0, 4]? In the mid 80’s this wasquestion was solved:

Theorem

The topological entropy of the interval map x 7→ ax(1− x)depends monotonically on a ∈ [0, 4].


There are several proofs for this theorem.

In the 80’s this was proved using

Thurston’s rigidity theorem;by a geometric argument due to Milnor;Douady-Hubbard’s univalent parametrisation of hyperboliccomponents together with additional ray arguments;by a method due to Sullivan, see in [MT88].

All of these proofs rely on considering the map x 7→ ax(1− x)as a polynomial acting on the complex plane.

A somewhat different real method was used by Tsujii,[Tsu94]. He showed that periodic orbits never get destroyedusing a calculation on how the multiplier depends on theparameter. In hindsight this method turned out to besomewhat related to Adam Epstein’s work.


Milnor’s monotonicity conjecture

In the early 90’s, Milnor posed in [Mil92] the more general

Question

Monotonicity Conjecture. The set of parameters within a family ofreal polynomial interval maps, for which the topological entropy isconstant, is connected.

Milnor and Tresser proved this conjecture for cubic polynomials,see [MT00] (see also [DGMT95]). Their ingredients are planartopology (in the cubic case the parameter space istwo-dimensional) and density of hyperbolicity for real quadraticmaps.


The general solution of this conjecture

Take d ≥ 1, ε ∈ −1, 1 and the space Pdε of real polynomials

f : [0, 1]→ [0, 1] of fixed degree d with

f (0, 1) ⊂ 0, 1,all critical points in (0, 1)

first lap orientation preserving/reversing if ε = 1 resp. ε = −1.

Theorem (Monotonicity of Entropy, [BvS09])

For each d ≥ 1, ε ∈ −1, 1, h0 ≥ 0,

L(h0) := f ∈ Pdε ; htop(f ) = h0

is connected.

Theorem (Non-local connectivity of level sets, BvS13 )

For d ≥ 4, there exists a dense set of h0 ∈ (0, log(d − 1)) forwhich L(h0) is not locally connected.


Going beyond

Applying the methods from this talk we get

Theorem

The topological entropy of [0, 1] 3 x 7→ a sin(2πx) increases with a.

Even for polynomials with non-real critical points, much isunknown, but real degree four polynomials with one real criticalpoint and two non-real critical points, we have

Theorem (Davoud Cheraghi and SvS)

The set of maps in this space with constant entropy is connected.

Remark: we do not have a rigidity result in this space.


This motivates the following question

Question

Let H(x , y) = (1− ax2 + by , y) be the family of Henon maps. Itfollows from [KKY92] and [DGY+92] that if we fix b then the setof parameters a; htop(Ha,b) = c is not connected.

However, is it possible that the set (a, b); htop(Ha,b) = c areconnected?


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real one-dimensional dynamics: real and complex methods

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