On Hausdorff dimension of polynomial not totally disconnected Julia sets
Feliks Przytycki Anna Zdunik
math.DS Primary: 37F35, Secondary: 37F10


[
    S. Riedel
    
=============


 We prove that for every polynomial of one complex variable
of degree at least 2 and Julia set not being totally disconnected
nor a circle, nor interval, Hausdorff dimension of this Julia set is larger than 1.
Till now this was known only in the connected Julia set case.

We give also an example of a polynomial with non-connected but not totally disconnected 
Julia set and such that all its 
 components comprising of more than single points are analytic arcs, thus resolving a question by
  Christopher Bishop, who asked whether every
such component must have Hausdorff dimension larger than 1.
 




§ INTRODUCTION


In the sequel we refer to connected components of subsets of the complex plane ℂ
that are not singletons (sets comprising of single points) as non-trivial components.



Christopher  Bishop in <cit.>, commenting on his result on the existence of an entire transcendental function with
one-dimensional Julia set, asked the following question:


The connected components of the Julia set constructed in this paper are all
either points or continua of Hausdorff dimension one.(...)
However, the situation for polynomials is
open. If a polynomial Julia set is connected, then it is either a generalized
circle/segment or has Hausdorff dimension strictly greater than 1 (this
follows from work of Zdunik [51] and Przytycki [36]). Is this also true of
the non-trivial connected components when the Julia set is disconnected?
In other words, if J(p) is disconnected, its every connected component is
either a point or a set of Hausdorff dimension strictly greater than 1?


We recall that for every polynomial p of degree at least 2, the Julia set J(p)
is the boundary of the basin of attraction to ∞ under the action by p.



In this article, we  answer the above   questions and related ones.
We start with key


Let f:ℂ→ℂ be a polynomial of degree d≥ 2. If  the Julia set J(f) is disconnected then
every non-trivial connected component J' of J(f) is eventually periodic, i.e. f^ℓ(J') is periodic with period k for some integers ℓ, k. Furthermore, either
f^ℓ(J')  is an analytically
embedded interval (i.e. analytic arc),
 and f^k on it is analytically conjugate to    ± Chebyshev polynomial, or J' has Hausdorff dimension,
 and even hyperbolic dimension, greater than 1.


This theorem is proved in Section <ref>. An example is presented at Figure <ref>.





In Section  <ref> we complete the picture, including possible analytic components and proving the following theorem, which is the main result of this note.

 Let f:ℂ→ℂ be a polynomial of degree d≥ 2,  such that J(f) is  not  totally disconnected. Assume also that f
is not complex affine conjugate to a map z↦ z^d or a ± Chebyshev polynomial.
Then (J(f))>1. Even more, the hyperbolic dimension, _ (J(f)) is greater than 1.




Recall that  hyperbolic dimension of the Julia set J(f), analogously of any other f-invariant subset of J(f),
_ hyp(J(f)) is  defined as supremum of Hausdorff dimensions of isolated hyperbolic forward invariant subsets of J(f). A compact set L⊂ℂ is hyperbolic (or expanding) forward invariant for f if f(L)⊂ L and there exists n∈ℕ such that |(f^n)'(z)|>1 for all z∈ L. The set L is said to be isolated (or repelling) if there exists its neighbourhood U in J(f) such that if a trajectory (f^j(z))_j≥ 0 is in U, then it is in L.

Thus, the inequality


    (J(f))≥_ hyp(J(f))

holds trivially.
See <cit.> for the discussion on equivalent definitions of the hyperbolic dimension,
in particular the one we shall use here, introduced in Proposition <ref>.




So, in view of Theorem <ref>, if J(f) is disconnected but it has a non-trivial connected component, then _ (J(f))>1, even
if the dimension of every non-trivial component is equal to one.
In the latter case the dimension larger than 1 is achieved
due to a Cantor set of other components.
To prove it, we follow a strategy by Irene
Inoquio-Renteria and Juan Rivera-Letelier in <cit.>, where they proved
that geometric pressure P(t)  is larger than -t times Lyapunov exponent of any probability
invariant measure on the Julia set J(f) of a rational function on the Riemann sphere
f:ℂ→ℂ (for the definition see Section <ref>, provided the exponent is positive. See Proposition <ref> for the definition of the geometric pressure and
Remark 12.5 in <cit.>.

Theorem <ref> strengthens the result of <cit.>, where the analogous theorem was proved for polynomials with connected Julia set.



Finally, we show in Section  <ref> that the situation we deal with in the proof of Theorem <ref> really may happen:
in Proposition <ref> we provide an example of a family of polynomials of degree 3 with
non-trivial  components of the  Julia set  J(f), each of which is an analytically embedded interval.

This resolves Christopher Bishop's question quoted above in the negative.



Remark.  One can also consider radial (or conical) Julia set J_ r(f)⊂ J(f) which consists of all those points z∈ J(f)  for which there exists δ>0 such that for infinitely many n∈ℕ the map f^n admits a holomorphic inverse branch f^-n:B=B(f^n(z),δ)→ℂ sending f^n(z) to z, for the Euclidean balls B in the case  f is a polynomial and balls in the spherical metric for f a rational function.

Then
_ (J(f))=(J_ r(f)), see e.g. <cit.> for this, in the setting including also rational and meromorphic functions, and see references therein; other versions of conical Julia sets for rational functions were introduced in <cit.>. Therefore in Theorem <ref> we can write

    (J_ r(f))>1.







§ BASIC NOTIONS, REPELLING BOUNDARY DOMAINS AND PROOF OF THEOREM <REF>


For a polynomial f:ℂ→ℂ of degree d≥ 2 one considers the set of points escaping to infinity

    A_∞(f):={z∈ℂ: f^n(z)→∞  as n→∞}.

Here f^n=f^∘ n=f∘ f ...∘ f n-times the composition of f. It is clear that this is exactly the set of points with the forward orbits (f^n(z))_n=0,1,... unbounded. The set A_∞(f) is called the basin of attraction to infinity. It is open and connected (since otherwise its bounded component would contain a pole for an iterate of f by Maximum Principle). Define the filled-in Julia set: K(f):=ℂ∖ A_∞(f) and Julia set J(f): =∂ K(f)=∂ A_∞(f) where ∂ means the boundary,  as in the Introduction. Clearly both K(f) and J(f) are completely invariant, namely f^-1(K(f))=K(f) and f^-1(J(f))=J(f).

A point z∈ℂ is said to be critical (or f-critical) if
the derivative f'(z) is equal to 0. The basin A_∞(f)∪∞ is simply-connected, that is J(f) be connected, if and only if the forward trajectories of all critical points are bounded.
At the other extreme if all critical points escape to the infinity then  J(f) is totally disconnected. However the opposite implication is false!. For more details see e.g. <cit.>, and <cit.> in particular Theorem 3.8 and an example following it.

Another definition of Julia set is that it is the complement of Fatou set
F(f) on which the sequence
f^n is normal, that is for every its subsequence and a compact set K⊂ F(f)  a   subsubsequence of it is uniformly convergent on K. This definition is valid also for
a rational function f:ℂ→ℂ 
acting on the Riemann sphere ℂ=ℂ∪∞.
The Julia set J(f) is compact, completely f-invariant, namely f^-1(J(f))=J(f), and a closure of repelling periodic orbits contained in it. For entire transcendental functions f the definition is the same.

Intuitively, the action of f on J(f)
is `chaotic'  and at points in J_ r(f) yields a similarity of infinitesimal parts of J(f) with its big pieces, thus exhibiting a fractal local geometry/nature of J(f).









We go back to a polynomial f.
Choose R so large that K(f) is a subset of 𝔻(0,R)),
the Euclidean disc with radius R and origin at 0,
and moreover

    f^-1(𝔻(0,R))⊂𝔻(0,R).

Then, by its complete invariance, K(f)⊂⋂_n=0^∞ f^-n(𝔻(0,R)).
Directly by the definition of K(f) the opposite inclusion holds, hence the equality holds.

Let C be a connected component of  the filled-in Julia set K(f).
Denote by U_n(C) the unique connected component of f^-n(𝔻(0,R)) containing C.
The boundary ∂ U_n(C) is disjoint from K(f) since ∂𝔻(0,R) is, and f(K(f))=K(f).
Notice also that by (<ref>)

    U_n+1(C)⊂ U_n(C).

Finally notice that all U_n are simply-connected (topological discs) by Maximum Principle.



Define
C':=⋂_n=0^∞ U_n(C).
By U_n(C)⊃ C we get C'⊃ C.
On the other hand note that C' is connected as the intersection
of the decreasing sequence of compact connected sets U_n(C).


It is also clear from the definition of C' that it consists of non-escaping points, hence it is contained in K(f). Therefore C'⊂ C.
We conclude with

    C=⋂_n=0^∞ U_n(C).



One fact pivotal for our paper is

[Qiu & Yin <cit.>, Kozlovski & van Strien <cit.>]
For a polynomial f of degree at least 2
if a component
of filled-in Julia set K(f) is not a point, then its forward orbit contains a periodic component containing a critical point.





We refer to periodic components that contain a critical point as critical periodic components.



In the proof of Theorem <ref> we shall use in fact a weaker version of Theorem A, that if the Julia set of a polynomial is not totally disconnected then there exists a non-trivial critical periodic component of K(f).



Theorem A
was proved
for degree 3 polynomials by Branner and Hubbard <cit.> and independently by Yoccoz, but for higher degree polynomials stayed not proven
for a long time.



Consider any non-trivial  component of K(f). We can replace it by a periodic component C which is its image under an iterate of f,  i.e., f^k(C)=C for some k≥ 1. Since replacing f by f^k does not change the Julia set, we may assume and we do assume from now on that f(C)=C.

[polynomial-like maps]
A polynomial-like map is a triple (U,U',Φ), where U and U' are open subsets of ℂ
homeomorphic  to a disc, with U' relatively compact in U and Φ:U'→ U holomorphic, proper,
of degree Φ≥ 2.

Proper


means that preimage of every compact set is compact. 

Degree is the number of preimages

# (Φ^-1(z)) for an arbitrary z∈ U
 counted with multiplicities.

It does not depend on z.



The notion polynomial-like was introduced by Adrien Douady and John Hubbard in <cit.>, Section I.1.
See also <cit.> or e.g. <cit.>, where
a smoothness or analyticity of ∂ U were additionally assumed. This allowed to extend Φ
continuously to ∂ U' and define proper by Φ(∂ U')=∂ U. One can always achieve this analyticity by a slight decreasing of the Douady and Hubbard U and U', see <cit.>. In our case we adjust R, see below.

Note that degree above is well defined for U' and U being open connected domains not necessarily simply connected (topological discs). This will be used in the proof of Lemma <ref>).



For a polynomial-like Φ the filled-in Julia set and the Julia set
can be defined in the same way as for a polynomial, namely

    K(Φ)={z∈ U': Φ^n(z)∈ U'  for all  n∈ℕ}   and   J(Φ)=∂ K(Φ).






Back to our component C, for  an arbitrary n∈ℕ,  F:=f|_U_n+1(C) maps U_n+1(C) onto U_n(C) (the same C). So since F is proper and due to (<ref>),  the triple
(U_n(C), U_n+1(C), F) is
a polynomial-like map.
 
 
 The boundaries of the domain and range are analytic if ∂𝔻(0,R) is taken disjoint from the forward orbits of critical points. We shall use the latter property to guarantee that for every
 n∈ℕ,  each two distinct components of f^-n(𝔻(0,R)) have disjoint closures.


If C is a fixed by f non-trivial critical component of K(f), then for each n large enough
the map F defined above is polynomial-like, with
C being its  filled-in Julia set.




Though F is polynomial-like for each n, there is no reason to expect that K(F)=C; only K(F)⊃ C must hold, since forward trajectories of points in C do not leave U_n+1(C) by the forward invariance of
C. So Lemma <ref> is not void.

 Let us write here U_n for U_n(C) for all n.
We claim that for n large enough
U_n+1 is  the only connected component of f^-1(U_n) contained in U_n. No other component
intersects U_n.



Indeed, if a component V of f^-1(U_n) intersects U_n, it must have its closure entirely in U_n.  Otherwise V
 would intersect ∂ U_n. Taking the images under f^n+1  we would conclude that
 𝔻(0,R)∩ f(∂𝔻(0,R))≠∅ contradicting (<ref>).

 Suppose now, there is a component V of f^-1(U_n) with closure entirely in U_n, different from U_n+1. Then there is a critical point of f in
 W':=U_n∖ (V∪U_n+1).
Otherwise f:W'→ U_n-1∖U_n=W  would be a covering map from W' to the annulus W.
This contradicts the equality χ(W')=(f|_W') χ(W) for Euler's characteristics since χ(W') is negative and
χ(W)=0.  See Figure <ref>.





Since the number of critical points of f is finite, they cannot happen in W' for n large enough.
The claim follows.





We conclude that  F^-k(U_n)=U_n+k for all k>1, so
⋂_k=0^∞ F^-k(U_n(C))=

⋂_k=0^∞ U_n+k(C)=C.




Notice finally that degree of F is larger than 1, since C hence U_n+1 is assumed to contain a critical point.





Remarks 1. Notice that the assumption that C is critical need not be specified. It follows automatically from the assumption that C is non-trivial. Indeed, if there were no critical point in U_n+1 then
the moduli of all annuli U_n∖U_n+1 for n large enough would be positive equal to each other so C would be a point, contradicting the non-triviality of C.

2. Notice that for F as in Lemma <ref>,
(F)<(f). Indeed, C=K(F) is by definition completely invariant for F, namely F^-1(C)=C.

At the same time there exists a component
C' of f^-1(C), in K(f) by its complete invariance for f, different from C,
since otherwise C would be completely invariant also for f so K(f)=K(F).
Then f(U_n+1(C'))=U_n(C) for all n∈ℕ, hence
f maps C' onto C thus adding to (F) an additional contribution.[A more general approach to the proof, more standard, is via the topological exactness of f on J(f), compare
a footnote to Comment 1 in Section <ref>, yielding the density of
⋃_n=0^∞ f^-n({z}) in J(f) for each z∈ J(f). This density will be used also in Proof of Proposition <ref>.]





Below we recall the definition of  RB-domain, which was introduced in <cit.>, with the name introduced in <cit.> together with crucial applications of this notion.


[RB-domain]
Let Ω be a simply connected open domain in the Riemann sphere ℂ with #(ℂ∖Ω)>2.
Assume there exists a holomorphic map defined on a neighborhood U of  ∂Ω such that

    f(U∩Ω)⊂Ω,    f(∂Ω)=∂Ω  and  ⋂_k=0^∞ f^-k(U∩Ω)=∂Ω.


Then Ω  is called a repelling boundary domain (RB-domain).


 The domain Ω:=ℂ∖ C   together with f restricted to U:=U_n+1(C) with
n sufficiently large, as in Lemma <ref>,
is a repelling boundary domain (RB–domain). Indeed, by this Lemma, f|_U_n^-1(C)=C which yields f(U∩Ω)⊂Ω.



Now we rely on:

[see <cit.>, Theorem A']
 If Ω is an RB-domain then either _ hyp(∂Ω)>1  or  ∂Ω
 is an analytic Jordan curve or an analytically embedded interval, where f is topologically conjugate to z^d
 or to a ± Chebyshev polynomial, respectively.



Theorem B was stated as a conjecture in <cit.>.
A comparison of harmonic measure ω on ∂Ω to ℋ^1 which is the Hausdorff measure  in dimension 1, is the key. For ω absolutely continuous it was proved in <cit.> that ∂Ω was analytic.
 It was also claimed in <cit.> that for ω singular (∂Ω)>1 could be proved by adapting the methods of
<cit.>.
A detailed proof of this appeared in <cit.>. Analogously to <cit.>, the proof not
only showed that (∂Ω)>1, but, actually, the
hyperbolic dimension of  ∂Ω for f restricted to it, was larger than 1. For a survey see <cit.>.


Combining the above Observation and  Theorem B, concludes the proof of Theorem <ref>, except it does not
exclude
C             
such that ∂ C=∂Ω is an analytic Jordan curve.

So, now we consider the latter case.

Denote ∂ C by γ.
Denote by Ω' the bounded (internal) component of ℂ∖γ, and consider now Ω'  instead of the external one  ∖ C, which was considered above.


The domain Ω'  is forward invariant for f since f(C)=C (as we replaced f by its adequate iterate) and due to Maximum Principle.

We already know that Ω is an RB–domain. Let U be a neighbourhood coming from the definition of RB-domain (see(<ref>)). We  know (see Observation <ref>)
that any set U_n, with n sufficiently large can be taken as U in this definition.

Since γ =∂ C is an analytic curve, there exists a neighbourhood W of γ , for which the Schwarz Reflection Principle applies.

Define U_n:=U_n∩Ω and choose n, additionally, so large that U_n⊂ W.
Take U:=U_n for this n.
We have by (<ref>):

    f(U∩Ω)⊂Ω,    f(∂Ω)=∂Ω  and  ⋂_k=0^∞ f^-k(U∩Ω)=∂Ω=γ .


Denoting by U' the reflection of U with respect to γ, we see that the same holds true with U replaced by the set U^ r, being the union of U', U and the arc γ along which their closures intersect,  and Ω replaced by Ω', i.e.

    f(U^ r∩Ω')⊂Ω',    f(∂Ω')=∂Ω'
      and  ⋂_k=0^∞ f^-k(U^ r∩Ω')=∂Ω'=γ .



In other words,  Ω' is also a repelling boundary domain.  Since  f(U) contains
the closure of U in Ω, by symmetry f(U') contains the closure of U' in  Ω'.
In other words
the compact set
Ω'∖ U' is mapped by holomorphic f into its interior.
It follows that there is an attracting fixed point ( a sink)  in Ω', see e.g. <cit.>,  and
Ω' is

the immediate basin of attraction to this sink[“immediate” stands for a connected component of the basin, which contains the sink]. One can also refer to the observation that (due to RB) f is a contraction on Ω'∖ U' in the hyperbolic metric on Ω'[One can also argue that, as Ω' is connected bounded and invariant, it is a component of the Fatou set for f. As being an RB-domain it cannot be neither parabolic, nor Siegel disc, nor Herman ring, so it must be an immediate basin of attraction to a sink. See e.g. <cit.>.].

Using <cit.> we now conclude that γ=∂ C is a circle and f(z)=z^d in some complex affine coordinates.  (See also  <cit.> or    <cit.>). But this contradicts the assumption that J(f) is disconnected.[We are grateful to  Fei Yang for bringing our attention to this absence of analytic Jordan curves.]



Remark. A key feature on the domain Ω' is that it is f-invariant, unlike
Ω containing components of f^-1(C). So having the nonempty interior of C we can apply
<cit.>, what does not work for Ω. The RB-property is weaker than
being an immediate basin of attraction to a sink.

The common feature is the repulsion
of the boundary to the side of a domain Ω, which for a pullback g=R^-1∘ f∘ R of f
 for a Riemann mapping R:𝔻(0,1) →Ω allows to extend g holomorphically beyond ∂𝔻(0,1), expanding,  see <cit.>, thus allowing easily to invoke Gibbs measures, see e.g. <cit.>.





§ PROOF OF THEOREM <REF>


We shall use a version of Bowen's formula, which can be found in <cit.>,
see also <cit.>,  for a strengthened version.

Let f be a rational map of degree d≥ 2. There exists an exceptional set E⊂ℂ of Hausdorff dimension 0, such that for every t≥ 0 (even every real t)
and for every z∉ E the limit

    P(t,f;z)=lim_n→∞1/nlog∑_v∈ f^-n(z)1/|(f^n)'(v)|^t

exists, and is independent of z∈ℂ∖ E. Denote the common value as P(t,f). It is called the geometric pressure.  The function t↦ P(t,f) is continuous and non-increasing. Moreover P(0,f) is positive
(equal to log d>0, the topological entropy).
The following formula holds:


    _ hyp(J(f))=inf{t>0: P(t,f)≤ 0}.


In words, _ hyp(J(f)) is the first zero of P(t,f).

There is an abundance of points not belonging to E (i.e. non-exceptional). Those are for example all points which are not post-critical (not in the forward f-trajectory of an f-critical point)
and which belong to  basins of attraction to periodic orbits.






In view of Theorem <ref>, to prove Theorem <ref> it remains only to prove the following 


Let f:→ be a polynomial of degree d≥ 2 with disconnected Julia set. Suppose there exists a periodic
connected component C of K(f)
being
an analytic arc.
Then _ (J(f))>1.











We shall prove that
 P(1,f)>0, which implies, by Proposition <ref>, that
 _ (J(f))>1.
Consider
 F=f:U_m+1(C)→ U_m(C), with some fixed m  large enough, so that F is a polynomial-like map and C its
 Julia set, here equal to the arc K(F),
 see Lemma <ref> (there m was denoted n).




For x∈ U_m∖ C denote

    L_n(F,x):=
    log∑_F^n(y)=xexp(-log |(F^n)'(y)| )=
    log∑_F^n(y)=x |(F^n)'(y)|^-1.


To continue the proof of Proposition <ref> we need two lemmas (Lemma <ref> and Lemma <ref>) , which we formulate and prove below. 


There exists C_0 > 0 such that for all x∈ U_m∖ C and n>0 

    L_n(F,x) ≥ -C_0.




Let Ω= ∖ C.
We know from the proof of Theorem <ref> that Ω is an RB-domain.
Recall that C=∂Ω is an analytically embedded interval,
as assumed in Proposition <ref>.





We recall the final step of the proof of the analyticity of C provided in <cit.>), which will allow us to deduce also the inequality (<ref>).

So assume
that C is an analytically embedded interval, with endpoints, say -1, 1.
Now we use Zhukovsky's function, namely the
ramified (branched) covering map Π:ℂ→ℂ, ramified over -1 and 1:  Π(z)=1/2(z+1/z)
and the preimage γ=Π^-1(C), see Figure <ref>. Then γ is an analytic Jordan curve, as proved
in <cit.>.[The property of a closed continuous  arc joining -1 and 1, saying that the Jordan curve being its preimage under Π is analytic, can be assumed to be a definition of the analyticity of the arc
(called therefore an analytic arc or an analytically embedded interval).]
 


 
 



The curve γ divides the
sphere into two disc 𝒟_i, i=1,2 and Π(𝒟_1)=Π(𝒟_2)=ℂ∖ C. The map F:U_m+1∖ C → U_m∖ C can be lifted to holomorphic maps G_i defined on the respected components W_i:=𝒟_i∩Π^-1(U_m+1∖ C)),
so that
G_i(W_i)⊂𝒟_i. Next consider just one of these components,
say W_1[A priori we do not know that both G_1 and G_2 are mutual holomorphic extensions through γ, but we need only one of them.].








By Carathéodory's Theorem (local version), see e.g. <cit.>, and Schwarz reflection principle, we can extend G_1 holomorphically from 𝒟_1 to a map G acting on a neighbourhood of γ. Then the map G is expanding on γ
because of uniform convergence  of G^-n on a neighbourhood of γ to γ which is a nowhere dense set. This implies  that the limit functions are constant, hence uniform
 convergence of derivatives (G^-n)' to 0, by normality, see e.g. <cit.> or <cit.>.
 Due to the just mentioned expanding property, the domain 𝒟_1 bounded by γ is an
RB-domain for the action of G. (It is seen also directly, by
  taking as a neighbourhood of γ=∂𝒟 in the definition of RB, the set U=Π^-1(U_m).








Remark.
Here is a different argument for the existence of the extension of G from W_1 beyond γ not using Schwarz reflection principle: Consider the lift of F on say U_m+1 to G on the set W:=Π^-1(U_m+1), which is a neighbourhood of γ (the both sides and γ itself), for the branched covering Π, directly by the formula

    G:= Π^-1∘ F ∘Π.

Because of Π^-1 it is not a priori clear that G in this formula is well defined.
 Here is an explanation. First define G:W_1→𝒟_1 as G_1 above. Fix an arbitrary z∈ W_1 and denote w:=G(z). Next extend  G using (<ref>) along curves in W starting from z mapped to w, using (<ref>), the curves omitting

    A:=Π^-1( (F) ∪{-1,1}),

where (F) is the set of F-critical points.
Notice that A is precisely the set mapped by F ∘Π to 1 or -1, the critical values of Π, namely where
Π^-1 has singularities. This is so because F is topologically conjugate to ± Chebyshev polynomial. So all the singularities of G defined by (<ref>) are holomorphically removable.
Notice finally that W  is a topological annulus, with the fundamental group generated by the homotopy class of any curve γ ' in W_1 starting and ending at z running once along W_1.
 The growth of the prolongation G along γ' is 0, because it starts and ends at w.
 It is so because γ'⊂ W_1, where G has been well-defined. 
So G being an analytic continuation along curves is a well-defined (single-valued) holomorphic function on W.
In fact this G coincides with the extension G found before, since these maps are holomorphic and coincide in 𝒟_1.



Let us return to the proof of Lemma <ref>.
Let x∈ U_m∖ C. Our aim is  to find a lower bound for L_n(F,x).
Denote by w the unique preimage of x under Π in 𝒟_1.
The map Π gives a bijection between the sets {v∈ G^-n(w)} and {y∈ F^-n(x)}.
Putting y=Π(v) we obtain:

    (F^n)'(y)=(G^n)'(v)·Π'(w)/Π'(v)

and, consequently,

    ∑_y∈ F^-n(x)1/|(F^n)'(y)=1/|Π'(w)|·∑_v∈ G^-n(w)1/|(G^n)'(v)|· |Π'(v)|.




Let us  estimate  the expression above, but without  Π',
that is let us estimate L_n(G, w) (defined for G as we did for F in
(<ref>)).
First assume w∈γ.
Denote by Le the Euclidean length of curves contained in our analytic curve γ.
There exists a constant c>0 such that for each n and every w∈γ,
choosing an arbitrary w'∈γ∖{w}, denoting by γ(w)
the open arc  γ∖{w'} we have

    ∑_G^n(v)=w |(G^n)'(v)|^-1≥ c ·∑_G^n(v)=w Le(G_v^-n(γ(w)))/ Le(γ(w))≥


    c · Le (γ)/ Le(γ)=c>0.

This is due to bounded distortion for iterates (or by Koebe distortion lemma) by a constant c,
see <cit.>. The subscript v at G^-n means the component containing v.
Due to bounded distortion w∈γ can be replaced by an arbitrary point w in a neighbourhood of γ and the same estimate as in (<ref>) is achieved for for such points.




We could define   so-called transfer operator (Perron-Frobenius-Ruelle) for potential ψ:=-log |G'|
acting on a continuous function φ

    ℒ_ψ (φ)(w):=∑_G(v)=w (expψ(w)))φ(w).

Then the expression estimated in (<ref>) could be written as ℒ^n_ψ().
Compare (<ref>).

To continue (<ref>) we need to estimate from below the value
ℒ_ψ^n(φ)(w) for potential -log |G'|, where φ=|Π'|.
The function φ has value zero at two points, but ℒ_ψ^2(φ) is  positive, thus bounded from below  by some constant c_1>0.
So,

    ℒ_ψ^n(φ)(w)=ℒ_ψ^n-2(ℒ_ψ^2(φ))(w)≥ℒ_ψ^n-2(c_1·)(w)=c_1·ℒ_ψ^n-2()(w),

and the last term is bounded below by c>0 as in (<ref>).

Since 1/Π'(w) is bounded away from 0 in Π^-1( U_m), since Π' is upper bounded, we are done.



For all n=0,1,... write F_n:=f|_U_n+1(C) the polynomial-like mappings considered in
Section <ref>.
Consider now an arbitrary m∈ℕ such that
C is the filled-in Julia set for F_m, existing by Lemma <ref>.


There exists an integer N_1>0 and a
connected component V of f^-N_1(U_m), such that V⊂ U_m∖ U_m+1.



Notice that there exists k: 0≤ k ≤ m for which

there exists a component
U' of f^-1 (U_m-k) in U_m-k, different from U_m+1-k.
Otherwise C=K(f) which is not the case as C is a connected component and K(f) is disconnected.
Compare Proof of Lemma <ref>.


Consider
V', an arbitrary component of (f^k|_U_m)^-1(U'). It is a subset of U_m and
a component of f^-(m+1)(𝔻(0,R)).
It is disjoint
from U_m+1, hence bounded away from C.

Note that f^m+1 maps V' onto the disc 𝔻(0,R) and
U_m ⊂𝔻(0,R)).  Select
V, an arbitrary component of f^-(m+1)(U_m) in V'.  It satisfies the assertions of our Lemma
with N_1:=m+1.















Notice that V intersects J(f) because U_0 does and J(f) is completely invariant for f.
More precisely V contains branched holomorphic images of C, namely components
of f^-N_1(J(f)). On Figure <ref> these are small 'basilicas' surrounding the big critical fixed one.





Having Lemmas  <ref> and <ref> at our disposal, we continue the proof of Proposition  <ref>.
In order to simplify   the notation, we pass again to the iterate of f, replacing now f by f^N_1 and F by F^N_1.
Recall again that this modification does not change the Julia set.
To simplify further the  notation, denote
  U_0:=U_m(C), U_1:=U_m+N_1(C).
So, from now on, after this modification, the component V becomes just a component of f^-1(U_0) in U_0 different from U_1=U_1(C). We denote f|_V by F_V. This pairs up with F on U_1
denoted also as F_U_1.



Now we build an infinite collection of multivalued maps ϕ_ℓ, ℓ∈ℕ,  as follows:
Let us note that F_V:V→ U_0 is a holomorphic proper map onto U_0. Since the degree  of
this map may be larger than one, the inverse may be not well defined. However,
we  shall  use the notation h: U_0→ V to denote the multivalued
inverse of F_V.
Thus,  h  assigns to a point x∈ U_0 a collection 
of its
preimages for F_V.




Consider a family of multivalued  maps ϕ_ℓ:U_0→ U_0, ℓ=0,1,…,
defined as  multivalued (branched) inverses

    ϕ_ℓ:=F^-ℓ∘ h

where F^-ℓ:U_0→ U_ℓ
are multivalued holomorphic maps given by  multivalued (branched) inverses of F^ℓ.
Since F^-ℓ is pre-composed by h it is meaningful even when restricted to V.

So, each multivalued map ϕ_ℓ
assigns to a point x∈ U_0 some  collection of its preimages under f^n, with n=n_ℓ=ℓ +1, all of them being in U_0, even in U_ℓ.




For x∈ U_0 and
ℓ∈ℕ∪{0} we write
∑ |ϕ_ℓ'(x)|
to denote the summation of derivatives which runs over all branches of the multivalued map ϕ_ℓ.
If a critical point c and its f-image are met then we can put in this sum ∞, as inverse
of forward derivative 0.  In fact it does not matter since we can restrict to x not post-critical.

For every  ℓ∈ℕ∪{0} and x∈ U_0 denote 

    L^*_ℓ (f,x)=log∑ |ϕ_ℓ'(x)|,

where the summation runs over all branches of the multivalued map ϕ_ℓ.
Denote also by ℓ a sequence ℓ=(ℓ_1,…ℓ_k), and put

    n_ℓ=n_ℓ_1+… +n_ℓ_k.



Finally, for a sequence  ℓ=(ℓ_1,…ℓ_k), denote


    L_ℓ(f,x):= L^*_ℓ_k(f, y_k-1)+...+L^*_ℓ_2(f,y_1)+ L^*_ℓ_1(f,x),

 where y_1=ϕ_1(x), ..., y_k-1=ϕ_k(y_k-2) (remember that the maps ϕ_j are multivalued, so we consider sums over their values).
The star ^* means we consider only preimages along first h and next F^-ℓ_i.
In other words, more formally,

    L_ℓ(f,x)=log∑|ϕ'_ℓ(x)|,

where the summation runs over all branches of the multivalued function

    ϕ_ℓ=ϕ_ℓ_k∘ϕ_l_k-1∘…∘ϕ_ℓ_1.




It is important to note that all the multivalued functions ϕ_ℓ are mutually distinct, compare  free Iterated Function System in
<cit.>.
Indeed, let ℓ≠ℓ',
but n_ℓ=n_ℓ'.
Let i be the first integer such that ℓ_i≠ℓ'_i. Then, supposing that n_ℓ_i>n_ℓ'_i
we compose in ℓ' after ℓ'_i with h with range in V, whereas in ℓ still within ℓ_i  with F^-1
having range
U_1. Further compositions by branches of f^-1 preserve distinction.



Denote by Σ^* the set of all finite sequences ℓ=(ℓ_1,…ℓ_k), k≥ 1.
For an arbitrary x∈ U_0
denote

    Λ_N(x)=∑_ℓ∈Σ^*: n_ℓ=Nexp L_ℓ(f,x).


 The star ^* again means we consider only preimages along first h next F^-1.

 For every N≥ 1 and non-exceptional, in particular not post-critical, x∈ U_0,

    P(1,f; x)≥lim inf_N→∞1/NlogΛ_N(x).




In view of Proposition <ref>  it is sufficient to prove

    ∑_y∈ f^-N(x)1/|(f^N)'(y)|≥Λ_N(x)

This is however obvious, because on the left hand side all y∈ f^-N(x) appear, whereas on the right hand side
only selected ones.



Let Λ_N:=inf_x∈ U_0Λ_N(x). Then

    lim inf_N→∞1/NlogΛ_N>0.




Put a:=inf_x∈ U_0 (∑ |h'(x)| ), and b:=exp(-C_0) where C_0 comes from Lemma  <ref>.

Consider all  sequences ℓ=(ℓ_1,…ℓ_k) such that n_ℓ=N, for each  k≤ N. The number of such sequences
can be calculated
in the following way:

For each k≤ N, there are N-1k-1 ways of choosing k-1 positions m_1,… m_k-1 from the sequence {1,… N-1}.[The ℓ has length N, but its first place is occupied by the beginning of the first block. So the only choices for the beginnings of remaining k-1 blocks are the remaining N-1 places.]
Having these positions chosen, we assign to them the values ℓ_1=m_1-1, ℓ_2=(m_2-m_1)-1, ℓ_k=(N-m_k-1)-1 and  the (multivalued) map
ϕ_ℓ with  ℓ=(ℓ_1,…ℓ_k) and n_ℓ=(ℓ_1+1)+(ℓ_2+1)+… +(ℓ_k+1)=N.



 Then we have the estimate

 
    exp L_ℓ(f,x)≥ a^kb^k

 and, consequently,

 
    ∑_ℓ∈Σ^*: n_ℓ=Nexp L_ℓ(f,x)≥∑_k=1^NN-1k-1(ab)^k =ab(1+ab)^N-1.





Remark.Let us note that a calculation in a similar spirit appeared in <cit.> and in <cit.>.
The authors used there a method of generating functions (some power series with coefficients related to a value similar to Λ_N).
Our case is simpler, in a sense that the set of admissible values n_ℓ forms an arithmetic sequence. So, the straightforward calculation provided above is sufficient.[However in Section <ref>, Comment 1, we use the general, power series, version.]





Obviously, combining Proposition <ref> and Proposition <ref>, together with Proposition  <ref>, we conclude the proof of Proposition <ref>.

Indeed,  Proposition <ref> and Proposition <ref> imply immediately that for every non-exceptional point x∈ U_0 we have that

    P(1,f;x)>0,

which implies (see Proposition <ref>) that P(1,f)>0.
Moreover,  Proposition <ref> asserts that

    HD_(J(f))=inf{t>0: P(t,f)≤ 0}.

Since the function  t↦ P(t, f) is continuous and non–increasing for t≥ 0,
we conclude immediately that

    HD_(J(f))>1.






§ EXAMPLE



To complete the answer to the question of Christopher Bishop, it remains to ask whether there exists a polynomial with disconnected Julia set, and a connected component of J(f) being an analytically embedded interval.

In the following proposition we provide a family of maps with this property.


Consider a family of   cubic polynomials

    f_ε,β(z):=
    ε z^3+z^2-β,

with ε  and β real, except ε=0 for which the polynomials are quadratic.
Then there exists an analytic curve Γ of parameters (ε,β), which is the graph of an analytic function ε↦β(ε) for ε≈ 0,
passing through the  point (0,2), and such that for every (ε, β)∈Γ with ε> 0 the Julia set of f_ε,β is disconnected,
and contains infinitely many components being analytic arcs.



We start with the quadratic Chebyshev polynomial f_0,2(z)=z^2-2. Its Julia set is just the interval I:=[-2,2].


Consider now the  polynomials f_ε,β
with ε real and close to 0, and β real and close to 2.
The parameters (ε,β)=(0,2) will be called initial parameters.

The map f_ε,β has a real repelling fixed point, denoted by p_ε, β, close to p_0,2=2, and 0 is a (not moving) critical point of f_ε,β.


There exists an analytic curve Γ  of parameters (ε, β), passing through the initial parameters (0,2) for which


    f^2_ε,β(0)=p_ε,β.

Moreover ∂Γ/∂β≠0 at the initial parameters and in consequence
Γ is the graph of an analytic function ε↦β(ε).


This is a straightforward calculation.
Denoting
γ(ε, β)=f^2_ε,β(0),
we have γ(ε, β)=-εβ^3+β^2-β,
so

    grad(γ)(0,2)=(-8,3).


Denoting by p(ε,β) the fixed point of f_ε,β close to 2,
we calculate (differentiating  the implicit function)

    grad(p)(0,2)=(-8/3, 1/3).



Thus, 
    grad(γ-p)(0,2)≠(0,0)

and therefore, there exists a smooth curve of parameters (ε, β), passing through the initial parameters (0,2) for which

    f^2_ε,β(0)=p_ε,β.

In particular (∂(γ-p)/∂β) (0,2) = 3-1/3 ≠0
hence by implicit function theorem
Γ is indeed the graph of an analytic function β(ε).

Notice that f_ε,β(0)=-β. The interval I_ε,β=[-β, p_ε, β] is thus invariant under
f_ε,β and the map
f_ε,β: I_ε,β→ I_ε,β is
two-to-one, with critical point at 0, hence topologically conjugate to the quadratic Chebyshev polynomial z^2-2 on I as above.



Suppose from now on that ε>0, ε≈ 0 and β=β(ε). Let us note that the Julia set J(f_ε,β) is not connected, as the trajectory of the second critical point c=-2/3ε tends to infinity.
The latter holds since f_ε,β(-2/3ε)>p_ε,β

 is repelled to infinity under further action of f_ε,β.

Denote by C the connected component
of J(f_ε, β) containing I_ε,β, so in particular it is fixed under f_ε,β and is non-trivial.


By  Lemma <ref>, there exist connected  neighbourhoods of C,

    C⊂ U_1⊂U_1⊂ U_0,
 such that the map  F:=(f_ε,β)_|U_1:U_1→ U_0 is a polynomial-like map,
and C is its filled-in Julia set. Since the Julia set of f_ε,β is disconnected, then by  Remark 2. after
Proof of Lemma <ref>, the degree of
F
is not maximal, so it is equal to 2.
By <cit.>, see also more recent <cit.>,  F
is quasiconformally conjugate to a true quadratic
polynomial.  Therefore,  preimages of every point in C are dense in C. But the preimages under  F of points from
I_ε,β remain in I_ε,β, which implies that

    C=I_ε,β.


So, each map f_ε,β with (ε, β)∈Γ is  a polynomial of degree 3 with disconnected Julia set,  for which the Julia set has an invariant component  being a true interval on which the degree of the map is equal to 2.
So, for each such map the filled-in Julia set, here equal to the Julia set,  has a collection of countably many non-trivial components, each of them being an analytic arc; this collection is formed by the invariant analytic arc and all its preimages under the  iterates of f_ε,β.

Note also that these are the only non-trivial components of the filled-in Julia set K(f_ε,β).
Indeed, by Theorem A all non-trivial components are eventually periodic and by Remark 1 after Lemma <ref>
every non-trivial periodic component of the filled-in Julia set has to contain a critical point in its orbit.
In our situation,  there are two critical points; one of them is escaping, and the other one is already contained in the invariant interval I_ε,β.




 







§ FINAL REMARKS AND QUESTIONS



COMMENT 1.  For f as in Proposition <ref> and ∂ C the boundary of a connected components C of filled-in Julia set
K(f), one can ask under what assumptions it holds




    (∂ C)< (J(f)).




It does not hold in general. It may happen that the component C has empty interior (so is the Julia set of the map F introduced in Lemma <ref>), but it satisfies
 (J(F))=HD(J(f))=2 and even _ hyp(J(F))=2.
Indeed,
just in the example f_ε,β(z)=ε z^3 + z^2-β with ε>0 (but β≠β(ε)), we find ε≈ 0 and complex β≈ 2
such that _ hyp(J(F))=2, by finding[See <cit.> or <cit.>.
The family ℱ of quadratic-like maps f_ε,β for an arbitrary small ε for |β|<3 and |z|<3 is Mandelbrot-like, as a small perturbation of the family
z^2-β. So there is 1-to-1 (homeomorphic) correspondence by hybrid (in particular quasiconformal) equivalences of its elements with connected filled-in Julia sets, to polynomials z^2+c with c in the Mandelbrot set. This is a
parameter version of Douady-Hubbard's Straightening Theorem. ]
parameters so that the quadratic polynomial
hybrid equivalent to F satisfies this, <cit.>.
Note that a quasiconformal conjugacy cannot drop neither Hausdorff nor hyperbolic dimension down from 2, see  <cit.>.



Proving Proposition <ref> we proved and used the fact
that f  is “almost hyperbolic”, here: topologically (analytically) conjugate to a ± Chebyshev polynomial
on a neighbourhood of C.
An even easier case is where f is hyperbolic (expanding) on C. Then Lemma <ref>, leading to (<ref>),
holds easily.
In fact (<ref>) holds 
under more general assumptions, namely if F is non-uniformly hyperbolic here in the version that it satisfies Topological Collet-Eckmann, TCE, see <cit.>. One out of several definitions equivalent to TCE,
called Backward Exponential Shrinking,
says:

There exist λ>1 and r>0  such that for every x ∈∂ C=J(F),
every n > 0 and every connected component W_n of F^-n(𝔻(x, r)) for the Euclidean disc
𝔻(x,r) with radius r and origin at x,

   it holds     (W_n) ≤λ^-n.


  Assume for simplification

 that ∂ C contains just one f-critical point. We shall apply an analogon of
 Lemma <ref>,
 where F is  replaced by a
Canonical induced map G,
 which is a  map of return (not first return !) to a nice set 𝒲 for iteration of F,  G(x):=F^m(x)(x). We refer here to  a theory developed by the first author and J. Rivera-Letelier in <cit.> and
 <cit.>. In particular see Section 3 of the latter paper. G^-1 forms an Iterated Function System with infinitely many branches from 𝒲 into itself, with bounded distortion.
 Consider the pressure
 
    𝒫(t,p,G):=
     lim_n→∞1/nlog∑_v∈ G^-n(w)exp S_n,G(ψ_G)(v)

 for
 ψ_G:=-tlog (|G'|-pm)
 where S_n,G(ψ_G):=∑_j=0^n-1ψ_G ∘ G^j,
 not depending on w due to bounded distortion.
 Compare Proposition <ref> and (<ref>).
 Now we apply <cit.> which says that 𝒫(t,P(t,F), G)=0
 
 for t=t_0 being zero of P(t,F)
 Next denote ψ:= -t_0 log|F'|.
 Let h
be an additional branch of f^-m_0 from 𝒲 into itself for some m_0∈, compare
Lemma <ref>.[The nice set 𝒲 is a small neighbourhood of the set
(F,J) of all
F-critical points in J(F). If #(F,J)>1, it has more than one component.  So this needs some care. In place of the IFS one applies a Graph Directed Markov System  and defines 𝒫
appropriately, see <cit.>. The branch of h above means a branch for each component of 𝒲,
where it is comfortable to choose  m_0 common.
Such a choice is possible due to the topological exactness of f on J(f), what means that for every non-empty set A⊂ J(f) open in
J(f) there exists n∈ such that f^n(A)=J(f). Compare <cit.>.]
Now,
for each ℓ∈ℕ consider the multivalued function ϕ_ℓ:=G^-ℓ∘ h.
We obtain the key estimate
 similar to (<ref>) in Proof of Proposition <ref>
 
    log∑_y∈ϕ_ℓ (x)exp S_m_ℓ(ψ)(y) >-C_0

for  m_ℓ:=m(y)+m(G(y))+... m(G^ℓ-1(y))+m_0.
Here S_n means S_n,F:=∑_j=0^n-1ψ∘ F^j.
The estimate (<ref>) follows from <cit.> and <cit.>; G is hyperbolic, but with infinitely many branches.
For each ℓ define the series



Φ_ℓ (s):=∑_N=1^∞ A_ℓ, N s^N for complex s,
with coefficients

    A_ℓ,N=∑_y∈ϕ_ℓ(x), m_ℓ(y)=Nexp S_N ψ(y).

Notice that (<ref>) yields

    Φ_ℓ (1)=∑_N  A_ℓ,N> exp -C_0.

So, for every L there exists N,
such that for every ℓ<L,

    ∑_n≤ N A_n,ℓ≥1/2exp -C_0.

So, for Φ(s):=∑_ℓΦ_ℓ(s) we get

    Φ(1)≥∑_n≤ N, ℓ<L A_n,ℓ≥1/2 L exp -C_0>2

for L large enough. Hence also Φ(s)>2 for some s<1.

So, Φ(s)+Φ(s)^2+ ... =∞, hence the radius of convergence of  Φ is at most s.
So P(t_0, f )>P(t_0, F)=0. 
Hence the (unique) zero of the function P(t, f), equal to _ hyp(J(f))  (=(J(f)) due to TCE),
is strictly larger than t_0=_ hyp(J(F))=(J(F)).


Notice that in the situation of Propositon 5 , given ℓ,  Φ_ℓ=A_m_ℓ,ℓ s^m_ℓ, where the time of each backward branch for its iterate of f is fixed, equal to m_ℓ,  whereas here it is not.



 









Question. How TCE can be weakened so that (<ref>) still holds?



COMMENT 2.  Theorems
<ref>
and
<ref>
hold for f being polynomial-like, so in the situation more general than for f being a polynomial. Proofs are the same.[We do not know whether we can refer to Theorems <ref>
and
<ref> for polynomials using straightening, since
quasi-conformal homeomorphisms may in general change Hausdorff dimension larger than 1 to equal to 1.]



COMMENT 3.  A positive answer to the following conjecture would generalize Theorem <ref> to rational maps:

 
Let f:ℂ→ℂ be a rational function  of degree at least 2. Assume that  the Julia set J(f) is connected. Then f is either a finite Blaschke product
in some holomorphic coordinates or a quotient of a Blaschke product by a rational function of degree 2,
or (J(f))>1.


If f has an attracting periodic orbit, then the positive answer is given by Theorem B applied to
Ω being the immediate basin of attraction to a point of the orbit for an iterate of f. Indeed Ω is simply connected by the connectivity of J(f). More precisely, either _(J(f))>1 or due to
<cit.> it is like in the assertion of the conjecture.

We believe that the same holds if f has a parabolic periodic orbit.



Another possibility is that the Fatou set consists only of Siegel discs and their preimages
for iterates of f.

Before discussing this, consider  f=P_α, a quadratic polynomial e^2π iαz +z^2
such that
α is real irrational of sufficiently high type, namely all the coefficients
in its continued fraction
expansion are larger than certain constant.  Then, if α∈ℬ∖ℋ, i.e. satisfying Brjuno
condition, (equivalent to the linearizability at 0, yielding a Siegel disc S), but not satisfying Herman condition (in particular not
Diophantine), the Siegel disc is hairy outside. Namely there is a Cantor bouquet of lines
in J(P_α) having together Hausdorff dimension 2. See <cit.> and references therein.

If α is Diophantine, of constant type, namely all the coefficients in its continued fraction expansion are smaller than a constant, then  (∂ S)>1, see <cit.>.



In those situations however, P_α are polynomials so they have basins of attraction to infinity in their Fatou sets and Theorem 2

proves the Conjecture.
But maybe for a rational map f the above theory can also be applied, so
in the “Siegel” cases above, Conjecture <ref> holds true.



In the remaining case of Herman rings, the situation is expected to be similar to Siegel's.



Notice that a positive answer to Conjecture <ref> follows immediately from the Comment 2,
for f being renormalizable.
Renormalizable means here, that there exist an integer r≥ 2, a compact connected

set C having more than one point, intersecting J(f), and its neighbourhoods U and U',  such that F=f^r|_U':U'→ U is polynomial-like with the filled-in Julia set equal to C.

Unlike before, we do not assume here that ∂ C is a connectivity component of J(f).

See examples of such renormalizations for rational Newton maps in <cit.>.




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