Almost twenty years ago, Jean-Christophe Yoccoz gave several lectures (at ETH and Collège de France, for instance) on a proof of Jakobson theorem (based on the so-called Yoccoz puzzles), and he distributed hard copies of his lecture notes whenever they were requested (for example, he gave me such a copy when I became his post-doctoral student in 2007).
More recently, he gave a couple of lectures at Collège de France aiming to generalize his proof of Jakobson theorem to obtain some results improving upon the Wang-Young theory of rank one attractors.
Of course, his lectures (whose videos are available at his website) started by recalling several elements of his proof of Jakobson theorem and I decided to take the opportunity to write a series of posts about Jean-Christophe’s proof of Jakobson theorem (based on his hand-written lecture notes from 1997).
For the first installment of this series, we’ll review some aspects of the dynamics of the one-dimensional quadratic maps and, after that, we’ll state Jakobson’s theorem, describe the three main steps of Yoccoz proof of Jakobson theorem and implement the first step of the strategy.
The dynamics of one-dimensional affine maps is fairly easy understand. The change of variables provided by the translation transforms into . If , the choice shows that the affine map is conjugated to its linear part . If , is a translation.
In other words, the dynamics of polynomial maps is not very interesting when . On the other hand, we shall see below that quadratic maps are already sufficiently complicated to produce all kinds of rich dynamical behaviors.
2. Quadratic family
The quadratic family is where and .
Remark 1 This family is sometimes presented in the literature as or . As it turns out, these presentations are equivalent to each other: indeed, the affine change of variables converts into
The dynamics of near infinity is easy to understand: in fact, since for , one has that attracts the orbit of any with .
This means that the interesting dynamics of occurs in the filled-in Julia set:
Note that is totally invariant, that is, . Also, is a compact set because implies that and whenever , so that
Moreover, because it contains all periodic points of (i.e., all solutions of the algebraic equations , ).
Remark 2 is a full compact set, i.e., is connected: indeed, this happens because the maximum principle implies that a bounded open set with boundary must be completely contained in (i.e., ).
The dynamics of on is influenced by the behaviour of the orbit of the critical point . More precisely, let us consider the Mandelbrot set
Indeed, we shall see in a moment that there is a substantial difference between the dynamics of on depending on whether or .
Remark 3 If , then one can check by induction that for all . Thus, the Mandelbrot set is the compact set
Also, the maximum principle implies that is full (i.e., is connected).Furthermore, the Mandelbrot set is symmetric with respect to the real axis: if and only if .
Moreover, intersects the real axis at the interval : indeed, we already know that ; on the other hand, the critical orbit is trapped between the fixed point (solving the equation ) and its preimage under for , so that ; finally, the fixed point moves away from the real axis for and this allows for the critical orbit to escape to infinity, so that .
Let us analyze the dynamics of on . For this sake, let us take a sufficiently large open disk centered at the origin (and containing the critical value ). The preimage is a topological open disk whose closure is contained in , and is a degree two covering map.
By induction, we can define inductively . We have two possibilities:
- (a) if the critical value belongs to , then is a topological disk whose closure is contained in , and is a degree two covering map;
- (b) if for some , then is the union of two topological disks and whose closures are disjoint, and and are univalent maps with images ; in particular, by Schwarz lemma, , , are (uniform) contractions with respect to the Poincaré (hyperbolic) metrics on and , the filled-in Julia set
is a Cantor set, and the dynamics of on is conjugated to a full unilateral two-shift , , via , where .
Remark 4 Observe that, in item (b) above, we got analytical information ( is an uniform contraction) from topological information ( is an open topological disk with ) thanks to some tools from Complex Analysis (namely, Schwarz lemma). This type of argument is a recurrent theme in one-dimensional dynamics.
Note that the situation in item (b) occurs if and only if the critical orbit escapes to infinity, i.e., . Otherwise, the sequence of nested topological disks are defined for all and .
In summary, the dynamics of on falls into the classical realm of hyperbolic dynamical systems when (cf. item (b)).
From now on, we shall focus on the discussion of for . Actually, we will talk exclusively about real parameters in what follows.
3. Statement of Jakobson’s theorem
Suppose that has an attracting periodic orbit of period , i.e., and .
It is possible to show that the basin of attraction
must contain the critical point . Thus, given , there exists at most one attracting periodic orbit of .
Remark 5 If has an attracting periodic orbit , then is hyperbolic in . Indeed, this follows from the same argument in item (b) above: the basin of attraction is the interior of and the boundary (called Julia set in the literature) is disjoint from the closure of the critical orbit ; thus, is contained in the open connected set with , so that the univalent map from to a component of is a local contraction with respect to the Poincaré metric on ; since is a compact subset of , it follows that there exists such that for all and , i.e., is uniformly expanding on .
Denote by the set of parameters such that has an attracting periodic point. The hyperbolicity of on for permits to show that is contained in the interior of the Mandelbrot set .
The so-called hyperbolicity conjecture of Douady and Hubbard in 1982 asserts that coincides with the interior of . This conjecture is one of the main open problems in this subject (despite partial important progress in its direction).
Nevertheless, for real parameters , we have a better understanding of the dynamics of for most choices of .
More precisely, it was conjectured by Fatou that is dense in . This conjecture was independently established by Lyubich and Graczyk-Swiatek in 1997: they showed that hyperbolicity of is open and dense property in real parameter space . In particular, this gives an extremely satisfactory description of the dynamics of for a topologically large set of parameters .
Of course, it is natural to ask for a measure-theoretical counterpart of this result: can we describe the dynamics of for Lebesgue almost every ?
It might be tempting to conjecture that has full Lebesgue measure on . Jakobson showed already in 1981 that this naive conjecture is false: does not have full Lebesgue measure on .
Theorem 1 (Jakobson) Let be the subset of parameters such that
- there exists such that for Lebesgue almost every (where );
- preserves a probability measure which is absolutely continuous with respect to the Lebesgue measure on .
Then, the Lebesgue measure of is positive and, in fact, has density one near :
(where stands for the Lebesgue measure).
Nevertheless, Lyubich showed in 2002 that has full Lebesgue measure on , so that we get a satisfactory description of the dynamics of for Lebesgue almost every parameters .
4. Overview of Yoccoz proof of Jakobson theorem
In 1997 and 1998, Jean-Christophe Yoccoz gave a series of lectures at Collège de France where he explained a proof of Jakobson theorem based on some combinatorial objects for the study of called Yoccoz puzzles (see, e.g., the first section of this paper here by Milnor).
Our goal in this series of posts is to describe Yoccoz’s proof of Jakobson theorem by following Yoccoz’s lecture notes (distributed by him to those who asked for them) recently submitted for publication (after this announcement here).
Very roughly speaking, Yoccoz proof of Jakobson’s theorem has three steps. First, one introduces the notion of regular parameters and one shows that satisfies the conclusion of Jakobson theorem whenever is a regular parameter. Unfortunately, it is not so easy to estimate the Lebesgue measure of the set of regular parameters. For this reason, one introduces a notion of strongly regular parameters and one proves that strongly regular parameters are regular (thus justifying the terminology). Finally, one transfers the estimates on “Yoccoz puzzles” in the phase space to the parameter space (via certain analogs of Yoccoz puzzles in parameter space called “Yoccoz parapuzzles”): by studying how the real traces of Yoccoz puzzles move with the parameter , one completes the proof of Jakobson theorem by showing that
In the remainder of this post, we will implement the first step of this strategy (i.e., the introduction of regular parameters and the derivation of the conclusion of Jakobson theorem for regular parameters via arguments from Thermodynamical Formalism).
has two real fixed points and for . Note that is repulsive for all , while is attractive, resp. repulsive for , resp. . Moreover, the real filled-in Julia set equals for all .
The dynamical features of are determined by the (recurrence) properties of the critical orbit . For this reason, we consider and we will analyze the returns of the critical orbit to the central interval
and its neighborhood where , . (Observe that these definitions make sense because the fixed points and satisfy for .)
Definition 2 A point is -regular if there exists and an open interval such that and is a diffeomorphism.
Definition 3 A parameter is regular if most points in are regular, i.e., there are constants and such that
for all . The set of regular parameters is denoted by .
As we already said, our current goal is to show the following result:
for Lebesgue a.e. , and admits an invariant probability measure which is absolutely continuous with respect to the Lebesgue measure.
4.2. The “tent map” parameter
Before proving Theorem 4, let us warm-up by showing that the parameter is regular and, moreover, satisfies the conclusions of this theorem.
The polynomial is conjugated to the tent map
This conjugation can be seen by interpreting as a Chebyshev polynomial:
The hyperbolic features displayed by (described below) are related to the fact that the critical point is pre-periodic: and for all , so that the critical orbit does not enter the central interval (and, thus, it is not recurrent).
The relationship between and the tent map allows us to organize the returns of the orbits of to as follows. Let and define recursively and as
Note that the interpretation of as a Chebyshev polynomial says that
In particular, is a decreasing sequence converging to and is an increasing sequence converging to .
In terms of these sequences, the return map of to is not difficult to describe:
- recall that the critical point does not return to (because for all );
- the points at return after one iteration: ;
- for , the return map on the intervals and is precisely ; moreover, is an orientation-preserving, resp. orientation-reversing, diffeomorphism from , resp. , onto , and , ;
- for , extends into diffeomorphisms from neighborhoods of to .
The graph of the return map of to looks like two copies of the Gauss map (where denotes the fractional part): it is an instructive exercise to draw the graph of .
At this point, we are ready to establish the regularity of .
Proposition 5 The parameter is regular.
Proof: From the previous description of the return map , we see that, for , the set of -regular points is precisely
so that is a regular parameter because, for and some constant , one has
for all .
Next, we show that satisfies the conclusion of Jakobson theorem.
Proposition 6 preserves the probability measure on and
for Lebesgue a.e. .
Proof: The reader can easily check (either by direct computation or by inspection of the conjugation equation between and the tent map) that
So, our task is reduced to compute the Lyapunov exponent of . Since Lebesgue a.e. eventually enters the central interval and the normalized restriction is absolutely continuous with respect to the Lebesgue measure, it suffices to prove that
for -a.e. .
Denote by the first return time of , so that (by definition).
Note that is an ergodic -invariant probability measure such that
This asymptotic information on the return times permits to compute the Lyapunov exponent of as follows.
Since the -orbit of a point is confined to and , we see from the chain rule that
for all . By combining this estimate with (2), we obtain
for -a.e. .
In other words, it suffices to compute the Lyapunov exponent along the subsequence of return times. As it turns out, this task is not difficult. Indeed, from (1), we have
Since for , we conclude that
as . This completes the proof of the proposition.
4.3. Dynamics of when is regular
We shall use the same ideas from the previous subsection to show that satisfies the conclusion of Theorem 4 when is regular. More precisely, we will use the returns to of certain “maximal” intervals of regular points to describe the return map of as an uniformly expanding countable Markov map. As it turns out, one can easily construct a -invariant absolutely continuous probability (via the classical approach of thermodynamical formalism [i.e., analysis of Ruelle transfer operator]), and our regularity assumption on permits to convert into a -invariant absolutely continuous probability measure (by a standard summation procedure).
Definition 7 An interval is regular of order if there exists a neighborhood of such that is a diffeomorphism from to with . We denote by the inverse branch of the restriction of to .
Remark 6 By definition, all points in a regular interval of order are -regular.
As we are going to see in a moment, the definition of regular interval of order was setup in a such a way that the “margin” provided by ensures good distortion and expansion properties of .
Remark 7 In a certain sense, the previous paragraph is the real analog of Koebe distortion theorem and Schwarz lemma ensuring good properties for univalent maps in terms of the modulus of certain annuli.
4.3.1. Distortion estimates
The Schwarzian derivative measures how far is a diffeomorphism from a Möbius transformation, and, for this reason, it is extremely efficient for studying distortion effects on derivative.
In fact, the composition rule reveals that the iterates have negative Schwarzian derivative on any regular interval of order (because has negative Schwarzian derivative).
In particular, if is the inverse branch of the restriction of to the neighborhood a regular interval of order , then has positive Schwarzian derivative on .
Since a diffeomorphism with positive Schwarzian derivative satisfies the estimate (see, e.g., Exercise 6.4 at page 166 of this book of de Faria-de Melo), we conclude:
for all . In particular,
and, for all measurable subset ,
4.3.2. Expansion estimates
Let be the countable family of regular intervals of positive order which are maximal for these properties.
Given , the composition is the inverse branch associated to a regular interval . Conversely, all regular intervals contained in are obtained in this way.
The bounded distortion estimates above allow to deduce uniform expansion estimates for the first return map of on regular intervals.
Proof: The first inequality follows from Lemma 8 by setting and . The second estimate follows from the first inequality after taking into account the definition of . Finally, the third estimate follows from the second inequality, the distortion estimate in Lemma 8 and the fact that (so that , and, hence, for some ).
4.3.3. Thermodynamical formalism for regular parameters
Let be a regular parameter. By definition,
for some constants and all . In particular, if
then . Thus, most of the dynamics of the return map is described by ,
where is the order of .
For this sake, we consider the dual of the pullback operator acting on finite absolutely continuous measures on . More concretely, the (Ruelle operator) acts on densities by the formula
derived from the change of variables formula and the definition .
We will take advantage of tools from Complex Analysis in order to study .
More precisely, we want to use the standard procedure (from the usual proof of Bogolyubov-Krylov theorem) of building invariant measures as the limit of Cesaro means
where is the constant function on taking the value one. Thus, our task is to show the convergence of the sequence (and this amounts to establish adequate estimates on ).
In this direction, let us show that admits an holomorphic extension. Let be the simply connected domain . Since the critical values of are all real (for all and ), the inverse branches associated to regular intervals extend to univalent maps satisfying
Given a regular interval, let be the sign of on and define
Since for all , the series
defines a holomorphic extension to of the functions
Moreover, the distortion estimates in Lemma 8 say that
Therefore, the family , , is normal (because belongs to the closed convex hull of the normal family ) and we can extract a subsequence of converging uniformly on compact subsets of to a holomorphic function on such that
Proof: By (4), is a measure equivalent to the Lebesgue measure. Since preserves the total mass,
and is uniformly bounded (thanks to (3)), we obtain that is a -invariant measure with total mass .
Finally, the ergodicity of is proved by the following standard argument (exploiting the good distortion and expansion properties of ). Suppose that is a -invariant subset of positive , or equivalently Lebesgue, measure. Fix a Lebesgue density point and, for each , denote by the regular interval containing . By Lemma 9, we know that (exponentially fast) as . Hence, given , there exists such that
for all . By plugging this information into the distortion estimate in Lemma 8 for , , and by using the -invariance of , we get
Since is arbitrary, we conclude that as desired.
4.3.4. Lyapunov exponent of for
For any , the return time function and the logarithm of the derivative of the return function are -integrable with respect to for all . Indeed, the regularity of the parameter means that
and hence the -integrability of follows from the fact that is equivalent to the Lebesgue measure (cf. Lemma 10). Also, the expansion estimate in Lemma 9 says that is bounded from below (by ), while the fact that for all says that , so that the -integrability of follows from the corresponding fact for .
Note that (because for all ), (by Lemma 9) and
Proposition 11 Let be a regular parameter. Then, for Lebesgue almost every , one has
Proof: Recall that, for every and , we have
4.3.5. Absolutely continuous invariant measures for ,
Let us use the invariant measure of the acceleration of to build an invariant measure of on .
Remark 8 is well-defined because and for all .
Note that has total mass (by Lemma 10). Moreover, is supported on
for . Furthermore, is -invariant is a consequence of the -invariance of because differs from by a coboundary:
Also, is -ergodic: any -invariant function restricts to a -invariant function on ; hence, the ergodicity of implies that is almost everywhere constant, and so is almost everywhere constant on .
Finally, the formula (7) defining shows that it is absolutely continuous with respect to the Lebesgue measure and its density is given by
This completes the verification of the conclusion of Jakobson theorem for regular parameters .
Closing this post, let us observe that the density of is not square-integrable.
Remark 9 By inspecting the terms , and in the definition of , we see that the distortion estimates (4) imply that for almost every ,
for almost every and
for almost every . Therefore, is bounded away from zero but is not in .