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Comptes Rendus Mathématique
Volume 355, n° 2
pages 222-225 (février 2017)
Doi : 10.1016/j.crma.2016.09.004
Received : 2 April 2014 ;  accepted : 13 September 2016
Periodic points in the intersection of attracting immediate basins boundaries
Points périodiques à l'intersection entre les frontières de bassins immédiats attractifs
 

Bastien Rossetti
 Laboratoire Émile-Picard, Université Paul-Sabatier, 31062 Toulouse, France 

Abstract

We give conditions under which the intersection between two attracting immediate basins boundaries of a rational map contains at least one periodic point.

The full text of this article is available in PDF format.
Résumé

Nous donnons des conditions suffisantes pour que l'intersection entre les frontières de deux bassins immédiats attractifs d'une fraction rationnelle contienne au moins un point périodique.

The full text of this article is available in PDF format.

For a rational map  ,   denotes the Julia set of R ,   the set   and   the set of   that are not in the closure of a connected component of  . A point   is said to be eventually periodic if there exists a   such that   is periodic. By sink we mean a connected component of an attracting immediate basin.

Theorem 1

Let f be a rational map with two distinct sinks   and   (not necessarily in the same cycle) such that  . Assume that   and   are simply connected, and   and   are locally connected.

1.
If the intersection   contains no critical point with infinite orbit and is disjoint from the ω-limit set of every recurrent critical point, then   contains a periodic point.
2.
Assume furthermore that each component of   that is eventually mapped to   or to   is simply connected. If   contains no accumulation point of   nor  , then the subset of eventually periodic points in   is non-empty and dense in  .

As a particular case of part 2 of Theorem 1, if   then the set of eventually periodic points in   is non-empty and dense in  . Nevertheless, the theorem does not require   to be finite.

Here is an example of a non-empty intersection between two sink boundaries with no periodic point in the intersection. Let us consider  , where   and   (  denotes the unit circle in  ) is such that   has rotation number θ . The map   has been studied in [[3]]. The map   has two attracting fixed points 0 and ∞. The intersection between the boundaries of the corresponding sinks is non-empty and included in  . This intersection contains no periodic point since   is topologically conjugate to  . One notes that in this example the intersection contains the point 1, which is a critical point with an infinite orbit.

To prove the theorem, we assume that   and   are fixed, for otherwise we work with an iterate of f . Since there will not be confusion, we will note  ,   and  .

Proof of part 1

We assume that   does not contain a critical point with finite orbit nor a parabolic point, for otherwise   would contain a periodic point.

A point   is said to be multiple if it belongs to the impression of at least two prime ends in  . Using the expansion of f on  , it is easy to show that a multiple point of   in   is eventually periodic. Thus we assume that   contains no multiple point of   nor  .

In this context we show, using Theorem 3, that   is distance-expanding with respect to the spherical metric, that is there exist  ,   and   such that for any  , if   then  . Then we find a periodic point in   using the Theorem 4 dealing with periodic points for distance-expanding maps.

Lemma 2

The restriction   is distance-expanding with respect to the spherical metric.

Proof

By Theorem 3 below, there exists an integer   such that  . By continuity of the map  , there exist   and a neighborhood U of   such that  . By compactness of  , there exists   such that if  , then the geodesic Γ between   and   lifts to a path γ from x to y with  . Thus we get  .  □

Theorem 3

([[2]]) Let g be a rational map of degree at least 2, and   be a compact forward invariant set containing no critical point nor parabolic point. If Λ is disjoint from the ω-limit set of every recurrent critical point, then there exists   such that   for every  .

Theorem 4

([[5]], chapter 4) Let   be a compact metric space. If   is continuous, open and distance-expanding, then there exists   such that the following holds: if there exist   and   such that  , then X contains a periodic point.

Lemma 5

The restriction   is open.

Proof

Let   and assume   is not open.

There exists a sequence   converging to some  . Let   be such that  . Since   contains no critical point, there exist a neighborhood U of x and a neighborhood V of y such that   is a homeomorphism. Thus for n large enough  , the point   is well defined and  .

We show now that   so that O is not open. It is clear that   since  . For any n , there exists a Fatou component   such that   and  . The following assertion finishes the proof of the lemma.

Assertion 6

For n large enough  ,  .

Proof

Otherwise, for some   there exists a Fatou component B such that  ,   and  . The boundary ∂B has finitely many connected components, thus each one of them is locally connected. Let   be either B or  . There exists a connected component   of   such that  . Since   is simple in  , there exists a unique connected component   of   such that  . Hence  . Since  , we have   and  , which contradicts the injectivity of  .  □

Now we apply Theorem 4 to finish the proof of part 1. Let w be an accumulation point of the orbit of some  . There exist   such that  , where α is the constant in Theorem 4. Hence  , and we get a periodic point in  .

The proof of part 2 uses ideas and techniques developed by K. Pilgrim in his thesis ([[4]], chapter 5). In case where f is hyperbolic and  , part 2 is a corollary of his work.

We assume that  , for otherwise f is conjugate to   for some  , and the conclusion follows. Up to make a quasi-conformal deformation, we also assume that all the critical points in   have a finite orbit (see [[1]], theorem VI 5.1; this is why we assume that each component of   that is eventually mapped to   or to   is simply connected).

Let   be the degree of   and let   be an isomorphism conjugating f with  . For  , set  . Since   is locally connected,   extends continuously to  .

Denote χ the set of chords , that is the set of   such that  . If   is periodic, then the point   is periodic. For any chord α and any set  ,   will denote the isotopy class of α rel X . For any distinct  , the complement   has at least two connected components and at most three, with points of   in each of them. For any  ,   if and only if one connected component of   contains all but two points of   (these two points being the extremities of the chords).

Set  . From the hypothesis of part 2, the set   is finite. Thus if   is such that  , then the point   is eventually periodic. We denote   the set  . For any   we denote   the set  .

The proof of part 2 is as follows. We equip χ with the Hausdorff distance  , so that it is a compact metric space. Pick  . If the sequence   is eventually cyclical then α is eventually periodic (Lemma 10). Otherwise, noting that   contains twice the same element (Lemma 11), we build a sequence   by a series of adjustments (Lemma 7) such that   converges (Lemma 8) to a chord β with the following property: either  , or   is eventually cyclical. This proves the existence of a periodic point in  . The density part will follow from the fact that we can build β as close as we want to α .

Let   (resp.  ). A lift of α is the closure of a connected component of  . If a lift of α is a chord, then it belongs to   (resp.  ).

Lemma 7

Let  ,   and   be such that  . There exists a unique chord   isotopic to α rel  , such that   and   for every  . Furthermore,  .

Proof

Since   is a lift of  , there exists a unique lift   of   such that  . In particular,  . For each  , we construct inductively a unique   such that   and   for any  . Note that  .  □

Lemma 8

For any  , there exists   such that:  , if   then  . As a consequence, if   for every   then  .

It follows from the following assertion:

Assertion 9

For every   there exists   such that : for any  , if   then in at least two connected components of   lie an open ball centered at a point of   and with radius η.

Proof

By contradiction, assume that there exists  , a sequence   tending to 0, and a sequence   such that, for any  :   and there does not exist two connected components of   in which lies an open ball centered at a point of   and with radius  . By compactness of  , we choose an accumulation point   of   and up to extraction  . We have  . If   has three connected components, then we note   one of the two connected components that are Jordan domains and we note   the connected component that is not a Jordan domain. If   has two connected components, then we note them   and  . In any case, there exist   and   such that  ,  . For n large enough,   and   are included in two distinct connected components of  . This is a contradiction as soon as  .  □

Proof of Lemma 8

Let   and η as in the assertion. Since  , there exists   such that each one of the two balls of the assertion contains a point of  . Hence there is a point of   in at least two connected components of  , thus  .  □

Lemma 10

For any  , if the sequence   is cyclical, then α is periodic.

Proof

Assume that there exists   such that   for any  . In particular,  . By Lemma 7, there exists a unique chord   such that   and for all  ,  . This chord is  . Thanks to Lemma 7, we also have  . Since this is true for any  , we conclude by Lemma 8 that  .  □

Lemma 11

For any  , there exist   distinct such that  .

Proof

Assume that for any   distinct, we have  . Let us show that the set   or the set   accumulate on  , which contradicts the hypothesis of part 2 of Theorem 1.

Since   is compact, up to extraction the sequence   accumulates on a chord β . Since for any   distinct at least two connected components of   contain a point of  , one can construct a non-stationary sequence  , which accumulates on a point  .

Assume that   is infinite and up to extraction that  . There exist finitely many distinct connected components   of  , which are distinct from   and   and such that  . Each   is included in  , but by construction there is an infinite subset of   whose elements are pairwise separated by chords, thus there is an infinite subset of   included in  . Since we assume that the extremities of the chords are the only points of   in  , we conclude that there is an infinite subset of   included in  .

Hence, up to extraction, we have   or  . In particular,  , and  .  □

Proof of part 2

Let α be a chord. We have three cases.

Case 1:  . Thus   is eventually periodic, as explained before.

Case 2:   and   is eventually cyclical. Then α is eventually periodic by Lemma 10, and the point   is eventually periodic.

Case 3:   and   is not eventually cyclical. Let us build from α a chord β fitting case 1 or 2.

By Lemma 11 there exist   and   such that  . Set  . By Lemma 7, there exists a chord   such that   for any  , and  . Thus   for any  , and  . We build inductively a sequence of chords   such that:

(i)
  for any  , and
(ii)
  for any   and  .

Assertion 12

The sequence   converges to a chord β whose point   is eventually periodic.

Proof

The convergence follows from (i) and Lemma 8. The limit β is a chord since χ is compact. If   then β fits case 1. If  , then we get for the limit   for every   and  . Hence   is cyclical, and β fits case 2.  □

Thus there exists an eventually periodic point in  .

To finish the proof, let us explain now the density. The chord β is in  . Applying Lemma 7 to  , we obtain a chord   such that   is eventually periodic. Using Lemma 11, we can have N as large as we want. Thus we can build a sequence   converging to α , such that every   is eventually periodic. Since   and   are locally connected, the sequence   converge to  .  □

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References

Carleson L., Gamelin T.W. Complex Dynamics  : Springer (1995). 
Mañé R. On a theorem of Fatou Bol. Soc. Bras. Mat. 1993 ;  24 : 1-11
Petersen C.L. Local connectivity of some Julia sets containing a circle with an irrational rotation Acta Math. 1996 ;  177 : 163-224
Pilgrim K.M. Cylinders for iterated rational mapsPhD thesis.   CA, USA: University of California at Berkeley (1994). 
Przytycki F., Urbański M. Conformal Fractals: Ergodic Theory Methods  :  (2010). 


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