The Shape of Thurston's Master Teapot
Harrison Bray Diana Davis Kathryn Lindsey Chenxi Wu

Linear Algebraic Techniques for Weighted Spanning Tree Enumeration
    Steven Klee and Matthew T. Stamps
    
==================================================================


We establish basic geometric and topological properties of Thurston's Master Teapot and the Thurston set for superattracting unimodal continuous self-maps of intervals.  In particular, the Master Teapot is connected, contains the unit cylinder, and its intersection with a set 𝔻×{c} grows monotonically with c.  We show that the Thurston set described above is not equal to the Thurston set for postcritically finite tent maps, and we provide an arithmetic explanation for why certain gaps appear in plots of finite approximations of the Thurston set. 




§ INTRODUCTION



In his last paper, unfinished at the time of his death, William Thurston
studied piecewise-linear maps of the unit interval <cit.>. One
concept mentioned in this paper is an object that Thurston, in his 2012 course at Cornell University, affectionately called
the Master Teapot,

 and which can be defined as follows. 
A unimodal endomorphism f of a real interval is said to be critically periodic
 if the critical point is a fixed point of some forward iterate of f, and is said to be postcritically finite if the forward orbit of the critical point is a finite set. 
If f is postcritically finite, it is easy to see that the orbit of the critical point determines a Markov partition of the interval.  
The Perron-Frobenius theorem then implies that the exponential of the topological entropy of f, e^h_top(f), is a weak Perron number - i.e. a real, positive algebraic integer that is not less than the absolute value of any of its Galois conjugates - which we call the 
the growth rate of f and denote by λ(f).
Denote by ℱ^cp the family of critically periodic unimodal continuous self-maps of compact real intervals. 
 

Then Thurston's Master Teapot is the set


    Υ_2^cp:={(z,λ) ∈ℂ×ℝ|λ = λ(f)  for some f ∈ℱ^cp, z  is a Galois conjugate of λ}.



An application of Υ_2^cp is that it can be used as a necessary condition for a weak Perron number to be the growth rate of a critically periodic unimodal map: β being such a number would imply that for each of its Galois conjugates z, (z, β)∈Υ_2^cp.  Studying the Master Teapot may also inform the open question of completely classifying the set of dilatations of pseudo-Anosov surface diffeomorphisms, which may be thought of as two-dimensional analogues of uniformly expanding interval self-maps.  We call the image of the projection of the Master Teapot to ℂ the Thurston set; this set, which we discuss later, has been the subject of several recent works (e.g. <cit.>).  Another motivation for studying these sets is that the part of the Thurston set inside the unit disk may be viewed as an analogue of the Mandelbrot set – while the Mandelbrot set may be defined as the set of parameters c ∈ℂ such that 0 belongs to the filled Julia set of the polynomial 

z ↦ z^2+c, 

the part of the Thurston set inside the unit disk coincides with the set of parameters z ∈𝔻 for which 0 belongs to the limit set of the iterated function system generated by the maps x ↦ zx+1, x ↦ zx-1. Furthermore, a forthcoming article by the last two authors will show that each horizontal slice of the Master Teapot is an analogue of the Mandelbrot set.  This topic is also connected to the theory of “core entropy,” which has been the subject of numerous recent works (see, e.g. <cit.>), as the restriction of a real quadratic polynomial to its Hubbard tree is a unimodal interval self-map.  













Thurston describes the part of the Master Teapot Υ_2^cp outside the unit cylinder as “a network of very frizzy hairs, …sometimes joining and splitting, but always transverse to the horizontal planes," <cit.> and the part inside the unit cylinder as “confined to (and dense in) closed sets that include the unit circle and increases [sic] monotonically with λ" <cit.>.
The first phenomenon is 


well-known (see e.g.  proof by Tiozzo <cit.>), but Thurston did not provide any further
explanation for the second. A main contribution in our paper is a proof of
the second phenomenon, which is that a point in the unit disc 𝔻
which is on a horizontal slice of the Master Teapot persists as the height
of the slice increases. 

[To see this phenomenon in action – that roots inside the unit cylinder persist, and also that roots outside the unit cylinder move continuously – see our video <https://vimeo.com/259921275>.]


[Persistence]
  
  For any point z∈ℂ in the open unit disk 𝔻, if
  (z,β) is in the Master Teapot, then every point above it up to
  height 2 is also in the Master Teapot. In other words,
  
    (z,λ) ∈Υ_2^cp implies {z }×
        [λ,2] ⊂Υ_2^cp.



Two corollaries of this main theorem are the following:


[Unit Cylinder] 
    The Master Teapot Υ_2^cp contains the unit cylinder S^1 × [1,2].
    


Another equivalent way to state the Unit Cylinder Theorem
<ref> is that S^1×{1} is contained in the
Master Teapot, and the Persistence Theorem <ref>
holds on the closed unit cylinder.
  
[Connectedness] 
	The Master Teapot Υ_2^cp is connected.  Furthermore,  is path-connected.  


We also proved a number of results that are not logically dependent on the main theorem above and concern other sets related to the Master Teapot.

Let the Thurston set, which we denote Ω_2^cp, be the
projection of Thurston's Master Teapot onto ℂ:

    Ω_2^cp :={z ∈ℂ|λ = λ(f)  for some f ∈ℱ^cp, z  is a Galois conjugate of λ}.



In other words, the Thurston set Ω_2^cp is the closure of the set
containing all Galois conjugates of growth rates of unimodal maps which are
critically periodic. 

A heretofore mysterious feature of plots of finite approximations of the Thurston set, formed by
bounding the length of the postcritical orbits, was the appearance of visible “gaps” or holes at
fourth roots of unity, sixth roots of unity, and certain other algebraic numbers. 




The gaps on the unit circle get filled in as the length of the
postcritical orbits approaches infinity <cit.>.  It is
known, however, that Ω_2^cp∩𝔻 does have a hole other than the large central hole around the origin <cit.>. The Gap Theorem <ref> provides an arithmetic explanation for these visible gaps in finite approximations of Ω_2^cp.  


[Gaps]
  
  For n ∈ℕ, let  ω_n denote the set of Galois conjugates of growth rates of unimodal critically periodic maps with postcritical length at most n.   Let R be one of the rings 

  ℤ[√(-D)] or ℤ[1+√(-3)/2] 
  for D=1,2, or 5, 
  and set c = inf{|z| : z ∈ R, z ≠ 0}.  
	Then for any x ∈ R, 
	
    B_r(x)(x)  ∩ω_n ⊂{x},

where 

    r(x) = 
    min{c/(2n^2 + 3n+1) |x|^n e,  1/n+1}    if  |x| ≥ 1,
    min{c/(2n^2+3n+1) e, 1/n+1}    if  |x| ≤ 1.





Let ℱ^pcf be the family of unimodal postcritically finite self-maps of real intervals.  
We define the postcritically finite Thurston set, Ω_2^pcf, as

    Ω_2^pcf :={z ∈ℂ|λ = λ(f)  for some f ∈ℱ^pcf, z  is a Galois conjugate of λ}.


In other words, the postcritically finite Thurston set Ω_2^pcf is
the closure of the set containing all Galois conjugates of growth rates of
unimodal maps which are postcritically finite. 

We proved that:

[Two Thurston Sets] 
The Thurston set Ω_2^cp and the postcritically finite Thurston set Ω_2^pcf are not equal.







The caption of Thurston's image <cit.> states that the image shows the roots of
the defining polynomials for "a sample of about 10^7 postcritically finite quadratic maps of
the interval with postcritical orbit of length ≤ 80." We suspect, based on visual comparison of plots, that Thurston's image shows
only roots of critically periodic tent maps, i.e. shows Ω_2^cp and not Ω_2^pcf (c.f. Figure <ref>).

At the moment, we do not have a good understanding of the shape of the postcritically finite Thurston set Ω_2^pcf and the 

analogously defined “teapot” Υ_2^pcf; for example, we do not know if they exhibit persistence (as in the Persistence Theorem <ref>) or connectivity (as in the Connectedness Theorem <ref>).





  

 §.§ Perspectives on the Thurston set







 Our main tool for the study of Ω_2^cp, and unimodal maps on intervals in general, is the Milnor-Thurston kneading theory. The Milnor-Thurston kneading theory <cit.> (also cf. <cit.>) provides the connection between general unimodal maps, real quadratic maps and subshifts in certain symbolic dynamical systems via entropy-preserving semi-conjugacies, and connect them to the study of infinite power series with prescribed coefficients called kneading determinants. As a result, there are numerous characterizations of Ω_2^cp from different points of view, and our results build (directly or indirectly) on a long history of research in each of these areas.
  
 1. Polynomials and power series with prescribed coefficients An alternative way to describe the kneading determinant and kneading polynomials, which predates Milnor-Thurston kneading theory, is β-expansions and Parry polynomials, which were first introduced in <cit.> for maps of the form x ↦β x 1 and later extended to a larger class of piecewise linear interval self-maps (e.g. <cit.>). Solomyak <cit.> used Parry polynomial to study the closure of the Galois conjugates of β such that x ↦β x 1 has finite critical orbit, Thompson <cit.> used it to study a set that contains the Thurston set, and the distribution of roots of Parry polynomials was studied in <cit.>.
 More generally, there is a large body of literature that investigating the roots of polynomials and power series with all coefficients in a prescribed set (see, for example, <cit.>). The polynomials most closely related to the Thurston set are perhaps Littlewood, Newman and Borwein polynomials, polynomials whose coefficients belong to the sets {± 1}, {0,1} and {-1,0,+1} respectively.

 2. Complex dynamics. Since the study of unimodal maps can be reduced to the study of real quadratic maps, the study of entropies of critically periodic unimodal maps is reduced to the study of core entropy on superattracting parameters on the real slice of the Mandelbrot set. The study of the core entropy on the Mandelbrot set is a rich subject, cf. <cit.>.

 3. Symbolic dynamics and Iterated function systems (IFS). The kind of symbolic dynamical systems semiconjugate to a real quadratic map was described in <cit.> via a combinatorial “admissibility criteria”. Using this, Tiozzo <cit.> proved that the Thurston set Ω_2^cp is connected, locally connected, and contains a uniform neighborhood of the unit circle. In particular, <cit.> shows that a point z with absolute value less than 1 is in the Thuston set Ω_2^cp if and only if 0 is in the limit set of the iterated function system generated by the two maps x ↦ zx+1 and x ↦ zx -1. This and some other related IFS are the focus of numerous works, including 
<cit.>.  In <cit.>, Calegari, Koch
and Walker used this and a related IFS to prove that the Thurston set has a hole, in addition to the obvious, large hole of radius 1/2 centered at 0. 



 §.§ Structure of the paper
  



A major consequence of the Milnor-Thurston theory is that unimodal maps on
intervals are semiconjugate to tent maps with the same entropy. This
tool is essential in our method of proof. 
 
<ref>: Preliminaries 

We define the β-itinerary of a point under a tent map, 

Parry polynomials, and give the admissibility criterion for itineraries, which are key tools in our arguments.


<ref>:
Quadractic maps, iterated function systems, and renormalization

provides background on Milnor-Thurston kneading theory and reviews the
concept of renormalization. 




<ref>: Dominant words reviews the definition and
properties of dominant words from Tiozzo's work
<cit.>. 
 



 
 <ref>: Persistence proves the main theorem, Persistence Theorem <ref>.

 <ref>: The unit cylinder and connectivity shows that
 the Master Teapot is connected inside the unit cylinder, and uses this
 structure to prove the Unit Cylinder Theorem <ref>
 and the Connectedness Theorem <ref>. 



 <ref>: Gaps in the Thurston set explains why there appear to be “holes” near primitive roots of unity in the finite approximations of the Thurston set. We show that these holes are associated to discrete subgroups, proving the Gap Theorem <ref>. 

<ref>: Ω_2^cp and Ω_2^pcf are not
equal shows that the periodic and preperiodic Thurston sets are not equal,
proving the Two Thurston Sets Theorem <ref>.




 §.§ Acknowledgements

The authors gratefully acknowledge Giulio Tiozzo, Daniel Thompson, Sarah Koch, and Dylan Thurston for helpful conversations.  This work began at the AMS Mathematics Research Communities program in June 2017. The authors are immensely grateful to the MRC program for stimulating this collaboration, and to Daniel Thompson introducing us to this subject while at the MRC.  This material is based upon work supported by the National Science Foundation under Grant Number DMS 1641020. The first author was supported in part by NSF RTG grant 1045119.  The third author was supported in part by NSF DMS grants 1901247 and 1401133.



§ PRELIMINARIES







 <cit.>
  
  Every continuous self-map g of an interval with finitely many turning
  points and with h_top(g) > 0 is semi-conjugate to a uniform
  λ-expander PL(g) with the same topological entropy h_top(g) =
  logλ. If g is postcritically finite, so is PL(g).
 

 Thus,
to understand Thurston's Master Teapot, it will suffice to study these more
rigid dynamical system. 






 §.§ Tent maps






Denote the unit interval by I=[0, 1]. For fixed β∈ (1, 2], the tent map of slope β is the continuous, piecewise linear map
f_β of the unit interval I defined by:
 

    f_β= β x     x≤1β
     2-β x    
        x>1β.





For a continuous self-map f of an interval with finitely many turning
points, the topological entropy h(f) is equal to the following
limit:

    h(f) = lim_n →∞1/nlog(Var(f^n)),
 
where Var(f) denotes the total variation of f <cit.>. Then
a straightfoward calculation confirms that for a tent map f_β, the
growth rate, which is the exponential of the topological entropy, is
equal to β. In other words, e^h(f_β)=β. 








Via Milnor-Thurston's entropy-preserving semi-conjugacy
<cit.>, the critical point of a unimodal map is sent to the
unique preimage of 1 under the associated tent map. Then a tent map
f_β is said to be postcritically finite if the f_β-orbit
of 1 is finite, and f_β is said to be critically periodic if 1 is a
periodic point for f_β. Now we have alternative interpretations of 

Thurston's Master Teapot, the Thurston set, and the postcritically finite
Thurston set: 

  
  * The Master Teapot Υ_2^cp is the closure of the
    set of all pairs (z,β) in ℂ×ℝ for which
    z is a Galois conjugate of β, and β is 
    the growth rate of a critically periodic tent map;
  
  * the Thurston set Ω_2^cp is the closure of the set of all
    Galois conjugates of growth rates of critically periodic tent maps;
  
  * the postcritically finite Thurston set Ω_2^pcf is the
    closure of the set of all Galois conjugates of growth rates of postcritically
    finite tent maps. 


























 §.§ Combinatorial itineraries


The dynamics of the tent map f_β can be represented by a relatively
simple Markov coding: there is a Markov partition of the unit interval into
two subintervals labeled 0 and 1, and we represent the f_β-orbit of
any point x with an itinerary sequence whose n-th term is 0 or 1
depending on which subinterval contains the nth iterate of x.  We will
make this representation more precise later in this section, since we will
extensively use the Markov coding of a tent map in this work.  First, we
will define essential abstract data of sequences and words in the alphabet
{0,1}. 


   
  We will use the term string to refer to an ordered list of letters
  in some alphabet, and this list may be either finite or infinite. We
  adopt the convention that a word is always a finite string, and a
  sequence is always an infinite string. An itinerary is also
  assumed to be an infinite string. We often concatenate a word w with
  the notation w^n, which is the word created from repeating w exactly
  n times. Similarly, w^∞ is the sequence created by repeating
  w infinitely many times. 




  The sequence of signs associated to a sequence w=(w_1 w_2 …)
  ∈{0, 1}^ℕ is the sequence e_w:ℕ→{-1,+1}
  defined by 

  
    e_w(j) = {[         +1  if w_j=0,;         -1 if w_j =1. ].

    The sequence of cumulative signs associated to a sequence w=(w_1
    w_2 …)∈{0, 1}^ℕ is the sequence s_w ℕ→{+1,-1} defined by
    s_w(1)=1 and
    
    s_w(j+1)=∏_k=1^j e_w(k)

    for j≥1. In other words, the (k+1)^st sign s_w(k+1) is equal to 1 if
    and only if the sum of the first k entries of the sequence w is even. 
    If w is a finite string, the cumulative sign of w is defined
    as ∏_k e_w(k).




    The sequence of digits associated to a sequence w=(w_1 w_2
    …)∈{0, 1}^ℕ is the sequence d_w:ℕ→{0,2} defined by d_w(i) = 2w_i.  




 §.§ Ordering on the set of strings
 

 
 
 for formatting only


 
  * Define the ordering ≤_E on the set of sequences in  {0,1}^ℕ as follows.  




Given two distinct sequences
w=(w_1w_2…) and v=(v_1v_2…) in {0,1}^ℕ, define
w <_E v if and only if at the first integer n such that w_n≠ v_n, 

    {[        w_n<v_n if  s_w(n)=+1,;        w_n>v_n if  s_w(n)=-1. ].

Note that s_w(n)=s_v(n) by definition since n is the first index at
which the sequences w and v differ. 


  * Define the ordering ≤_E on the set of words in the alphabet {0,1} as follows.  Given two words w and v, write 
w <_E v if and only if w^∞ <_E v^∞.  






  It is straightforward to check from the definition of twisted
  lexicographic ordering that if a word
  a has positive cumulative sign, then for any strings v,w, we have
  w<_E v if and only if aw<_E av. Similarly, if a has negative
  cumulative sign, then w<_E v if and only if aw>_Eav. 
  


Now we can define the concept of β-itinerary as below:


  Let I_0^β=[0, 1/β], I_1^β=[1/β, 1]. The
  β-itinerary of the tent map f_β is the sequence
  w=(w_1w_2…) satisfying the following two conditions:
  
  
  * f^n(1)∈ I_w_n+1^β.
  
  * Among all the sequences satisfying the preceding condition
    (1), w is
    the minimal such sequence under the twisted lexicographical ordering. 
  


It is obvious that if f_β is not critically periodic, there is a unique sequence satisfying condition (1) which has to be the β-itinerary. If f_β is critically periodic, one can easily check that the β-itinerary can be equivalently defined explicitly as follows: if f_β^k(1)=1
  and k is minimal, then this is the itinerary w^∞ where w has
  length k and the last digit of w is chosen such that w has positive
  cumulative sign. From this observation one can see that this definition is consistent with the standard kneading theory definition of the itinerary of 1 under f_β, which is the limit of the itineraries of x_i under f_β, where x_i∈ [0, 1], lim_ix_i=1, and the forward orbit of x_i never hits any critical point. 



 §.§ Parry polynomials


The definition below for a Parry polynomial is motivated by the concept of β-expansion.

   

  Let w be a word in the alphabet {0,1}. Set f_0^z(x)=zx and
  f_1^z(x)=2-zx.  Then the Parry polynomial for w is
  
    P_w (z)     := s_w(p+1)(f_w_p^z∘ f_w_p-1^z∘…∘
        f_w_1^z(1)-1) 
        = z^p-s_w(1)d_w(1)z^p-1-⋯-s_w(p)d_w(p) - s_w(p+1) 
        = (z-1)(z^p-1+s_w(2)z^p-2+…+s_w(p)).




When f_β is critically periodic, the first line of equation
(<ref>) confirms that for any word w for which
w^∞ is a β-itinerary, we have that β is a root of the
Parry polynomial P_w.  
Thus, the minimal polynomial for β is a factor of P_w for any word
w such that w^∞ is a β-itinerary.  
As a final observation, P_w is also never irreducible over the integers, 
as it always has a
factor of (z-1). At times it will be important for our arguments to
ensure that the Parry polynomial has only this one extra factor of (z-1),
i.e. has exactly two irreducible factors. 





 §.§ Admissible itineraries

















  





 








Let σ{0,1}^ℕ→{0,1}^ℕ be the
standard shift map, defined by σ(w_1w_2w_3…)= (w_2w_3…).  

Milnor-Thurston developed a combinatorial criterion for a sequence in
{0,1}^ℕ to be realized as an itinerary of the critical
value under a 
quadratic map from the family g_c x↦ x^2+c, where  
c is real. A quadratic map is given an itinerary in the same procedure as
for a tent map; partition the domain of the map into two intervals whose
intersection is the critical point, and the left interval receives a coding
value of 1 while the right interval receives a coding value of 0.




 <cit.>




A sequence a=(a_n) in {0,1}^ℕ is an itinerary of the
critical value of a quadratic map if and only if 

σ^j(a) ≤_E a for all j ∈ℕ. 





As a corollary of Milnor-Thurston's semi-conjugacy from Theorem
<ref>, a sequence in
{0,1}^ℕ which is realizable as an itinerary of 
of 1 under a tent map is also realizable as an itinerary of the critical
value of a quadratic map, as 1 is the image of the critical value under
semi-conjugacy. 
Thus, Theorem <ref> introduces a necessary combinatorial
condition on 

β-itineraries which we call admissibility:

 
  A sequence a=(a_1a_2…) in the alphabet {0,1} is admissible
  in the Milnor-Thurston sense if for all positive integers j, the
  shifted sequence satisfies the inequality
  
    σ^j(a)≤_E a.

  Then a word w is admissible if and only if the sequence w^∞ is
  admissible. 








On the other hand, the converse is more subtle because the Milnor-Thurston
semi-conjugacy is not a true conjugacy, the reason being that a critically
periodic tent map is semi-conjugate to infinitely many quadratic maps with
different post-critical itineraries. However we do have the partial
converse which will be sufficient for our purposes:


  Let w be a word in the alphabet {0,1} with positive cumulative
  sign.  If w is admissible 
  and the Parry polynomial associated
  with w can be factored into z-1 and another irreducible factor, then

  w^∞ is the β-itinerary
  for some β∈(1,2]. 



Theorem 12.1 of <cit.> tells us that if w^∞ is
admissible, it must be the itinerary of c under some quadratic map g_c:
x↦ x^2+c (here we let I_1=(-∞, 0], I_0=[0, ∞)).
  Because a quadratic map is unimodal, we can find some tent map f_β
  semi-conjugate to the quadratic map g_c. Suppose (w')^∞ is the
  β-itinerary, and w' has minimal length. 



  The proof of Lemma 12.2 in <cit.> implies (which can also
  be checked by bookkeeping) that the 

  itinerary of the critical value c under any quadratic map g_c 
  which is semi-conjugate to f_β must lie between (w')^∞ and
  (w”)^∞, where w” has the same length as w' and
  agrees with w' except for the last letter. Hence, (w')^∞≤_E
  w^∞≤_E (w”)^∞, which implies that the length of w must be
  a multiple of the length of w'.

  Because β is a root of the Parry
  polynomial associated with w, w' and w”, the fact that the Parry
  polynomial of w has only two irreducible factors implies that w and
  w' have the same length. Hence w=w' or w=w”. The condition that w
  has positive cumulative sign precludes w=w”, so w=w'. 





  
Note that the converse of Proposition <ref> is false; if we write 
the β-itinerary as w^∞, this string 
is admissible in the Milnor-Thurston sense, 
but the Parry polynomial P_w may have more than two irreducible factors,
even if w is minimal length and has positive cumulative sign. 

For a critically periodic tent map f_β, we call the Parry polynomial
associated with the β-itinerary the Parry
polynomial of f_β and denote it by P_β.  In the case that
f_β is postcritically finite, 


a similar procedure using the sum of a power series produces a polynomial
associated to a preperiodic word, and hence to the preperiodic 
β-itinerary.




 §.§ Irreducibility
 





To check that a Parry polynomial has only two irreducible factors, we will
use two lemmas from  <cit.>, which are derived from
Eisenstein's criterion.



  <cit.> 
Let d=2^n - 1 with n ≥ 1, and choose a sequence ϵ_0,ϵ_1,…,ϵ_n with each ϵ_k ∈{± 1} such that ∑_k=0^d ϵ_k ≡ 2 4.  Then the polynomial 

    f(x):= ϵ_0 + ϵ_1 x + … + ϵ_d x^d

is irreducible in ℤ[x]. 



<cit.> 
Let f(x) = 1 + ∑_k=1^d ϵ_k x^k be a polynomial with ϵ_k ∈{± 1} for all 1 ≤ k ≤ d and ϵ_k = -1 for some k.  If f(x) is irreducible in ℤ[x], then for all n ≥ 1, the polynomial f(x^2^n) is irreducible in ℤ[x].  






§ QUADRACTIC MAPS, ITERATED FUNCTION SYSTEMS, AND RENORMALIZATION






In this section we elaborate on the connections to quadratic maps and
iterated function systems, including consequences of kneading theory for
the combinatorial itineraries we will study in this work. We close the
section with Proposition <ref>, which is a sufficient criterion for
the Persistence Theorem <ref>.



 §.§ Iterated function system description of the Thurston set
 

Here we associate a limit set Λ_z to a nonzero complex parameter
z in the open unit disk 𝔻. 
A point z ∈𝔻∖{0} defines a contracting iterated function system (IFS) generated by the two maps

    f_0^z:x ↦ zx+1,    f_1^z:x ↦ zx-1.
 
The attractor or limit set Λ_z of this IFS is defined
to be the unique fixed, nonempty, compact set S ⊂ℂ such
that S = f_0^z(S) ∪ g_1^z(S).  The existence and uniqueness of this
attractor and statements of some of its fundamental properties are due to Hutchinson <cit.>.  It is straightforward to see that:

 <cit.>  
  The limit set Λ_z associated to 
  z∈𝔻∖{0} is contained in the open ball of radius
  1/1-|z| around the origin in the compex plane.



Our work is motivated by Tiozzo's description of Ω_2^cp∩𝔻 in <cit.>, which is as follows:
<cit.>


  The Thurston set Ω_2^cp intersected with 𝔻 is equal to
  the set of all compex numbers z whose associated limit set Λ_z
  contains the origin. 
 


Milnor and Thurston showed that any tent map f_β is semiconjugate to
a quadratic map g_c: x↦ x^2+c for c∈ [-2, 1/4], and this
semiconjugacy preserves the data of the Markov coding and hence the
entropy <cit.>.
The kneading series of a quadratic map g_c and a number x is a power series

    K(x, t) = 1 + ∑_n=1^∞η_nt^n,


where η_n(1) is the cumulative sign η_n(x) = ∏_i=0^n-1 sign(g_c^i(x)).

The kneading determinant is

    K_c(t) =  
    K(c,t)  if the critical point is not periodic under f_c, 
    lim_C → c^+K(C,t)  if the critical point is periodic under f_c
    .


When g_c has periodic critical orbit, K_c(t)=P_c, knead(t)/1-t^n, where P_c, knead is called the kneading polynomial.

 The semiconjugacy between f_β and g_c sends the critical value 1 to c, and intervals I_o^β and I_1^β to [0, ∞) and (-∞, 0] respectively. Hence, η_n(c)=s_w(n), which implies that:

    (t-1)t^n-1P_c, knead(t^-1)=P_β(t).

  

The following are classical from kneading theory; see <cit.>:

  Fix a parameter c in [-2, 1/4] with associated quadratic map g_c:
  x↦ x^2+c with entropy h, and let s=e^h be the growth rate of
  this map. 
  
  
  * <cit.> 
    The growth rate 
    s is a continuous function of c.
  
  * <cit.> s>√(2) if
    and only if c<-1.
  
  * <cit.> If s>1, 1/s is the smallest positive
    root of the kneading polynomial P_c, knead.
  

Furthermore, 
Milnor and Thurston introduce an ordering on the additive group
ℤ[[t]] of formal power series with integer coefficients is
defined by setting α = a_0 + a_1t + … > 0 whenever a_0 = …
= a_n-1 = 0 but a_n > 0 for some n ≥ 0 <cit.>.
This induces an ordering on formal power series as follows: if a,b are
distinct formal power series with integer coefficients, then 
a>b if and only if a-b>0. 



The following is an immediate consequence of <cit.>. We include proofs for completeness. 


 
For tent maps, the kneading determinant is a monotonically decreasing function of the growth rate.  




For the real one-parameter family of maps f_a(x) = (x^2-a)/2,
<cit.> asserts that the kneading determinant
D(f_a) ∈ℤ[[t]] is monotonically decreasing as a function of the
parameter a; and Corollary 13.2 asserts the growth rate is monotonically
increasing as a function of a.  The family of maps  {f_a} takes on
all possible growth rates; this can be seen from the fact that  f_a is
conjugate to the map g_(-a/4)(z) = z^2+(-a/4) via the conjugation map
h(z)=z/2, growth rate is a continuous function of c (Proposition <ref>(1)), and the Intermediate Value Theorem. 


 
Let f_β be a tent map with kneading determinant a and let w be
the β-itinerary; let f_β' be a tent map with kneading
determinant b and let w' be the
β'-itinerary. If a > b, then w <_E w'. 



  From Remark <ref>, a= 1+∑_i=1^∞ s_w(i)
  t^i, b=1+∑_i=1^∞ s_w'(i) t^i. Let n be the smallest
  natural number such that s_w(n) ≠ s_w'(n).  
  We must have s_w(1)=s_w'(1), so we may assume n ≥ 2. 
  
  By definition, a>b implies s_w(k)=s_w'(k) for all k=1,…,n-1,

  and 
  s_w(k)=(-1)^w_k-1s_w(k-1), so we must have w_j =w'_j for 1 ≤ j
  ≤ n-2 and w_n-1≠ w'_n-1.


  Since s_w(n)>s_w'(n), the two possibilities are:


    s_w(n-1)=s_w'(n-1) = +1,     w_n-1 = 0,    w'_n-1 = 1,  or
 

    s_w(1,n-1)=s_w'(n-1) = -1,   w_n-1 = 1,    w'_n-1 = 0.

In both cases, we have w <_E w'.



Combining Lemma <ref> and Lemma <ref>, we have:

 
If 1<β<β'≤ 2, the β-itinerary w and 
β'-itinerary w' satisfy the inequality w<_Ew'.




 §.§ Renormalization
 

We will develop the proof of the Persistence Theorem <ref> using a
combinatorial approach of Tiozzo <cit.>. Certain
tent maps f_β admit itineraries with strong combinatorial properties.
Due to the renormalization phenomenon, if the slope β is at most 
√(2) then it is impossible for any associated itinerary to satisfy this
strong combinatorial property. Renormalization is how we and Tiozzo
compensate for this obstruction. 

One of the renormalization or “tuning” procedures on the Mandelbrot set (see e.g. <cit.>) implies the following:



  If β∈ (1, √(2)], then f_β 
  has periodic critical orbit of length 2k 
  if and only if f_β^2 has periodic critical orbit of length
  k. 

This is a well-known fact and we give a short proof below for completeness.


Consider the intervals J_1^β=[2-β, 2 1+β],
J_2^β=[2 1+β, 1]. Then it is straightforward to
check that f(J_1)⊂ J_2, f(J_2)⊂ J_1, and f^2
J_2→ J_2 is a unimodal piecewise linear map with constant slope
β^2 and critical value equal to 1, and is clearly conjugate to the
tent map of slope f_β^2. Hence, f_β^  n, the
nth iterate of f_β, fixes
1 
if and only if (f_β^2|_J_2)^  n/2, the n/2-iterate of
the restriction, also fixes 1.





The Renormalization Lemma <ref> motivates the following
definition:

  
A tent map f_β with growth rate 
β in the interval (1,2] is defined to be
renormalizable if β≤√(2), and is otherwise nonrenormalizable. 





With the Renormalization Lemma <ref>, 


we reduce the Persistence Theorem <ref> to the
following proposition, which is essentially persistence restricted to
growth rates of nonrenormalizable tent maps:




  Let β∈ (1,2] be the growth rate of a critically periodic tent
  map and let z∈𝔻 be a root of the Parry polynomial of
  β. Then for any real number y satisfying 2 >y>max{β,
    √(2)} and any real number ϵ>0, there exist a real number
    β' within ϵ of y such that
  
  
  * one of the Galois conjugates of β' is within distance ϵ
   of z, and 
  
  * a Parry polynomial of β' is of the form (x-1)f(x), where
    f(x^2^n) is irreducible in ℤ[x] for all natural numbers n.
  




The proof of Proposition <ref> will appear at the end of Section
<ref>. 
We confirm here that this proposition is indeed sufficient to prove 
the Persistence Theorem <ref>.




  Suppose (z, β) is a pair in the Master Teapot Υ_2^cp.
  We want to show that if y∈ [β, 2], then (z,
  y)∈Υ_2^cp.
    Note that it suffices to consider
  y>β. 

  Let ϵ>0. By definition of the Master Teapot, there exists a 
  β_0 that is the growth rate of some tent map
  f_β_0 with periodic critical orbit, z_0 a Galois conjugate of
  β, such that |z_0-z|<ϵ and |β_0-β|<ϵ.
  Then we
  may choose ϵ small enough that y>β_0 as well. 
  
  If
  y>√(2) then Proposition <ref> along with the triangle inequality
  directly implies existence of 
 
  β' within ϵ of y, such that one of the Galois
  conjugates of β' is within 2ϵ distance from z, as
  desired. 

  It remains to consider y<√(2). Fix an integer n 
  such that y∈(2^1
  2^n+1, 2^1 2^n].  








  By the Renormalization Lemma <ref>, β_0^2^n is the
  growth rate of some tent map with periodic critical orbit, is clearly
  less than y^2^n, and
  z_0^2^n is a Galois conjugate of β_0^2^n because the Galois group
  consists of field automorphisms.  

  Since √(2)<y^2^n≤ 2,  
  again by Proposition <ref>, 
  we can find some β' within
  ϵ of y^2^n, such that one of the Galois conjugates z' of
  β' is within 2ϵ of z^2^n. The second condition in
  Proposition <ref> implies that all the 2^n-th roots of z' are
  Galois conjugates of (β')^1 2^n. The conclusion follows.









§ DOMINANT WORDS
 


In this section we will review Tiozzo's definition of dominant words in <cit.> and the properties of dominant words proved in <cit.>.

In <cit.>, Tiozzo considers the tent map with slope 2 f_2, then any points x in unit interval has a correspondence with a infinite sequence w_x in {0, 1}^ℕ, which is its Markov coding, where the two subintervals are I_0=[0, 1/2] and I_1=[1/2, 0]. It is obvious that f_2 is semiconjugate to the shift map σ under this correspondence, and x<y implies that w_x<_Ew_y where <_E is the twisted lexicographical order defined earlier.

For any finite word w of length n, let x be the point on the 
unit interval labeled by w^∞, then the Cylinder set C_x is
defined as the  subinterval which contains x and f_2^n+1 sends it
homeomorphically to the unit interval I; in other words, C_x is the
closure of points with a coding starting with w· w_1. Here w_1 is the first letter of w.


<cit.> A finite
  word w in the alphabet {0, 1} is called
  dominant if and only if it satisfies the following two conditions:
  
  
  * w has positive cumulative sign.
  
  * For any 1≤ k≤ n-1, f_2^k(C_x) lies to the left of C_x and their interiors are disjoint.
  


Now, because x<y implies w_x<_Ew_y, and f_2^k(C_x) is the closure of points whose itineraries start with a suffix of w followed by the first letter of w,

we can make the definition of dominance more explicit as below:

 
  
	Let w be a word in the alphabet {0,1} that starts with 10
	and has positive cumulative sign.  Then w is dominant if and only
	if for any proper suffix b of w, the word (b1) is (strictly)
	smaller than the prefix of w of length |b|+1 in the twisted
	lexicographical ordering <_E. 


A word w in the alphabet {0,1} is irreducible if there exists no shorter word w_0 in the alphabet {0,1} and integer n ≥ 2 such that w=(w_0)^n. 


The main result we will cite from Tiozzo's work is the following, which can
be read from the proof of <cit.> in
<cit.>. One can read it from the first paragraph of the proof of Lemma 10.6 on page 689, and the third paragraph of the proof of Proposition 10.5 on page 692.


  
  If β∈(√(2),2] and w is a word in the alphabet {0,1}
  such that
  w^∞ is a β-itinerary, then for any positive integer 
  n there exists a word w' in the alphabet {0,1} that is a power of
  some dominant word such that w^nw' is also a dominant word. 


Tiozzo uses this observation to prove that the growth rates associated to
dominant words are dense in the interval [√(2),2].

The following lemma is straightforward to show from calculation:

Any β-itineraries for β∈ (1, 2] start with 10.  


Because of Lemma <ref>,  the dominant itineraries obtained in Proposition <ref> must satisfies the assumptions of Lemma <ref>.





§ PERSISTENCE


The goal of this section is to prove Proposition <ref>, which is a
version of persistence restricted to growth rates of nonrenormalizable tent
maps; that
is, growth rates 
that are larger than √(2). As discussed in Section
<ref> following the statement of the proposition, the
Persistence Theorem <ref> follows by
renormalization. 








 §.§ Constructing dominant extensions


The development of persistence for growth rates of nonrenormalizable tent
maps 

hinges on a series of technical combinatorial lemmas. The goal of this
subsection is the proof of the Extension Lemma
<ref>.



Assume w_1 is dominant, w_2 is admissible and irreducible, 
n is a positive integer such that 

    2n|w_2| > |w_1| > n|w_2|,


 w_1^∞>_E w_2^∞, and w_2^n has positive cumulative sign. 

Then (w_1w_2^n)^∞ is admissible.




It suffices to show that 
    σ^k(w_1w_2^n)^∞≤_E (w_1w_2^n)^∞
 for all k<|w_1|+n|w_2|.
  If 1<k<|w_1|, denote by b the proper suffix of w_1 of length |w_1|-k. Then (b1) is a
  prefix of σ^k (w_1w_2^n)
  because the first
  letter of w_2 is 1 by admissibility and Lemma <ref>.
  By dominance of w_1 and Lemma <ref>, (b1) is
  smaller than the prefix of w_1 of length |b|+1 in the twisted lexicographical ordering, 
  which proves
  
    σ^k(w_1w_2^n)=bw_2^n<_E w_1
 and provides the desired inequality. 

  If k=|w_1|, for contradiction, see that existence of n such that w_2^n≥_E w_1 implies
  
    w_2^∞ <_Ew_1^∞≤_E (w_2^n)^∞=w_2^∞,

  which is impossible given the assumption that w_2^∞ is smaller than
  w_1^∞ in the twisted lexicographical ordering. Thus, 
    σ^|w_1|(w_1w_2^n)^∞
      =w_2^n (w_1w_2^n)^∞ <_E
      (w_1w_2^n)^∞.
 

  Lastly, we consider the shift by k where |w_1|<k<|w_1|+n|w_2|. Let r=k-|w_1|, so that
  1<r<n|w_2|. 
  See that σ^rw_2^n>_Ew_1 is impossible, because σ^rw_2^n>_Ew_1 
  and admissibility of w_2 implies
  
    w_1^∞<_Eσ^r(w_2)^∞≤_E w_2^∞,
 a contradiction. We conclude that σ^rw_2^n≤_E w_1.
  If this inequality is strict, we are done: we would have 
  
    σ^k(w_1w_2^n)
      =σ^|w_1|+r(w_1w_2^n)=σ^rw_2^n<_Ew_1
 as desired.

  We must now consider when this inequality is not strict; in other words,
  σ^rw_2^n is a prefix of w_1. We will need to prove that such
  a string must always have cumulative negative sign. If it does, then |w_1|-r<|w_1|
  implies 
  
    σ^|w_1|-r(w_1w_2^n)^∞≤_E (w_1w_2^n)^∞

  by dominance of w_1 discussed above. Then by Remark <ref> and the fact that
  σ^rw_2^n has negative cumulative sign,

  
    (w_1w_2)^∞ =σ^rw_2^nσ^|w_1|-rw_1 w_2^n (w_1w_2^n)^∞   ≥_E
        σ^rw_2^n(w_1w_2^n)^∞
       = σ^k-|w_1|w_2^n(w_1w_2^n)^∞
       = σ^k(w_1w_2^n)^∞.


  It remains to prove that if σ^rw_2^n is a prefix of w_1, then it cannot have cumulative
  positive sign. Consider the suffix b=σ^rw_2^n of w_2^n. Since w_2^n is admissible,
  b≤_E a, where a is the prefix of w_2^n of the same length. 

  
 

  Since w_2^∞ <_E
  w_1^∞, moreover a is smaller than or equal to the prefix of w_1 of the same length,
  which is assumed to be equal to b. Then b≤_E a≤_E b implies equality, and we conclude
  w_2^n=ac=db=da. 

  Now 
  
    w_2^∞=(ac)^∞=(da)^∞≥_E a (da)^∞,

  implying
  
    (ca)^∞≥_E(da)^∞=w_2^∞≥_E (ca)^∞

  because we assumed that a has positive cumulative sign (see Remark <ref>) and w_2 is admissible, hence 
  (ca)^∞=(ac)^∞. Then
  
    w_2^∞= (ac)^∞ =a(ca)^∞=a(ac)^∞=a^2(ca)^∞ =
    	  ⋯=a^∞

  implies a=w_2^m for some m because w_2 is irreducible. 
  Then
  w_1=af = w_2^mf for some suffix f, and again by dominance of w_1 and
  Lemma <ref>,
  
    w_1^∞=(w_2^mf)^∞=w_2^m(fw_2^m)^∞≤_E
    	  (w_2^mw_1)^∞=w_2^2m(fw_2^m)^∞≤_E…≤_E w_2^∞,

  which contradicts the assumption that w_1^∞>_Ew_2^∞. 


Now we want to further make sure that (w_1w_2^n)^∞ is a
β-itinerary and has a Parry polynomial with a large irreducible factor.
However, this would require some slight modification of the construction as
below:



  We say that a string 
  v is an extension of a 
  word w if w is a proper prefix of
  v. If v is finite then such a v is a finite extension of w. 




  
Let w_1 be dominant such that w_1>_E10· 1^|w_1|-2 and w_2 be 
admissible and irreducible, w_1^∞>_E w_2^∞, and assume
there exists an m such that 

    2m|w_2|>|w_1|>m|w_2|.



Then there exists 
a finite extension w_1' of w_1 and an integer m'≥ m such that (w_1'w_2^m')^∞ is
admissible, |w_1'|>m'|w_2|, and
P(z^2^k)/(z^2^k-1) is an irreducible polynomial for any k≥ 0, where P is the Parry polynomial of (w_1'
    w_2^m'). 



The following Lemma will give us a recipe for extending w_1, as needed
for the Extension Lemma. 



  Let w be a dominant word, such that w_1>_E10· 1^|w_1|-2.
  Then the words
  
    w 10 1^κ 10 1^|w|01 1^|w| and w 10 1^κ 10 1^|w|10 1^|w|

  for any odd natural number κ>|w|, and
  
    w 1^κ 10 1^|w|01
        1^|w| and w 1^κ 10 1^|w|10 1^|w|

  for any even natural number κ>|w|, are all dominant extensions of w.

  Moreover, for each κ, the sums of the coefficients of the kneading
  polynomials for the two extensions differ by 2.
  





  The parity condition on κ is to guarantee that the new word has an
  even number of 1s, which is part of the definition of dominance.

  We apply the alternate definition of dominance from Lemma
  <ref>. Let w' be one of the possible
  extensions in the statement of the Lemma. Let b be any suffix of w'.
  If a prefix of b is a suffix of w, then (b1) is smaller than the
  prefix of w' of the same length in the twisted lexicographical ordering
  by dominance of w and the construction of w'. If not, then if b
  starts with 0 or 11, and the desired inequality is immediate, so the
  interesting case is if b starts with 10 and no prefix of b is a
  suffix of w. By construction, including our choice of κ>|w| in
  the κ odd case, we are comparing 10· 1^|w|-1 with w·
  1, and the former must be smaller by assumption.


  For any natural number κ, odd or even, there are now two choices
  to extend w to a dominant word. The two choices only differ by an
  exchange of 01 with 10 in one position.  This exchange will change
  the sum of the coefficients of the kneading polynomials by a factor of 2.






  §.§.§ Proof of extension lemma





  We need to choose for w_1' one of the extensions of w_1 from Lemma
  <ref>, and select n, κ, and
  m' so that |w_1'| has length 2^n-1-m'|w_2| and
  
    2m'|w_2|>|w_1'|>m'|w_2|.

  To do so, first define constants C_1=1+|w_1|+m|w_2| and C_2=|w_2|. 
  Then choose n for which 
  
    2^n>max{C_2(10m+3)+C_1,18C_2+C_1}
 and define 
  
    k_n=⌈2^n-C_1/2C_2⌉-2,     
        k_n'=⌈2^n-C_1/2C_2⌉-3.

  The two options k_n and k_n' are needed for parity reasons. 
  Choosing 2^n>C_2(10m+3)+C_1 ensures that 
  
    k_n>k_n'>10m,

  which becomes useful later in the proof when we define the length of the
  extension. The choice of 2^n>18C_2+C_1 and the definition of k_n,k_n'
  ensures (respectively) that 
  
    3k_n>3k_n'>2^n-C_1/C_2>2k_n>2k_n'.

  Let m'=k_n+m if this is even, and else, replace k_n with k_n'. We
  will proceed with the notational choice m'=k_n+m and assume m' is
  even, but note that the needed inequalities hold for both k_n and
  k_n'. 

  Now, replacing C_1,C_2 with their definitions, applying Equation
  (<ref>), and invoking the assumed
  relationship between |w_1| and |w_2|, we see that  
  
    3m'|w_2|>3k_n|w_2|+m|w_2|+|w_1|>2^n-1>2k_n|w_2|+m|w_2|+|w_1|>2m'|w_2|,

  which implies 
  
    2m'|w_2|>2^n-1-m'|w_2|>m'|w_2|.

  We now adjust the extension w_1' of w_1 to have length
  |w_1'|=2^n-1-m'|w_2|, so that (w_1'w_2^m') has total length
  2^n-1. 

  If |w_1| is odd, then 
  κ=(2^n-1-m'|w_2|)-6-3|w_1| is even, as needed for 
  
    w 1^κ 10 1^|w|01
        1^|w|   and    w 1^κ 10 1^|w|10 1^|w|

  to both be dominant extensions of w_1 by Lemma
  <ref>, each of length 2^n-1-m'|w_2|.

  If |w_1| is even, then 
  κ=(2^n-1-m'|w_2|)-4-3|w_1| is odd, as needed for 
  
    w_1 1^κ 10 1^|w_1|01
        1^|w_1|   and    w_1 1^κ 10 1^|w_1|10 1^|w_1|

  to both be dominant extensions of w_1 by Lemma <ref>, each of length
  2^n-1-m'|w_2|. In all the above cases, κ>|w_1| follows from Equation
  (<ref>). 

  For each choice, w_1^∞>_Ew_2^∞ implies
  w_1'^∞>_Ew_2^∞, and w_2^m' has positive cumulative sign
  because we ensured that m' is even. Combined with Equation
  (<ref>), we have all the necessary hypotheses to
  apply Proposition <ref> and conclude that
  (w_1'w_2^m')^∞ is admissible. We also designed w_1' so that
  |w_1'|>m'|w_2|.

  The sum of the coefficients of the kneading polynomial of w_1'w_2^m'
  is even, because it has 2^n coefficients, each of which is either -1
  or +1.  By the final observation in Lemma <ref>,
  we can choose the extension so that the sum of the 
  coefficients of the kneading polynomial for w_1'w_2^m' is equivalent to
  2  4.  Since the kneading polynomial has degree 2^n-1, we apply
  Lemma <ref> to conclude irreducibility. 




 §.§ Controlling Galois conjugates and entropies of concatenations



 
Let w_2 be a 

word whose Parry polynomial has a root at z_0 ∈𝔻.  Then for any ϵ > 0, there exists an integer N = N(ϵ,w_2) ∈ℕ such that n > N implies that for every 

word w_1 
for which w_1w_2^n is admissible,
the Parry polynomial associated to (w_1w_2^n) has a root within distance ϵ of z_0. 



First, for any word w, denote the Parry polynomial associated to w by P_w.  
Let D be the closed disk of radius ϵ centered at z_0, and let C be the boundary of D.  Without loss of generality, assume that ϵ is small enough that D ⊂𝔻, and that D contains no root of P_w_2 except z_0.  




For any n ∈ℕ, it is straightforward to see that 


    P_w_1w_2^n(z) = z^n|w_2|P_w_1(z) + (z^(n-1)|w_2| + z^(n-2)|w_2|+ … + 1 )P_w_2(z).



Set α = min_z ∈ C |P_w_2(z)|, which exists and is positive by compactness
and the assumption that D contains no root of P_w_2 except z_0. 
Set 

    0<β  
    
     := min_z ∈ C(1-|z|^|w_2|)/(1+|z|^|w_2|).




Then for all z ∈ C, we have 


    |(z^(n-1)|w_2| + z^(n-2)|w_2|+ … + 1)P_w_2(z)| ≥|
    	1-(z^|w_2|)^n/1-z^|w_2|| α≥1-|z^|w_2||/1+|z^|w_2||α≥βα>0,

	where the middle nonstrict inequality follows the triangle inequality and that
	|z^|w_2||<1. 


Set 1> m: = max_z ∈ D |z|.  Also for all z ∈ C, since all coefficients of P_w_1 have absolute value at most 3,


    | z^n|w_2|P_w_1(z) |  ≤ | z|^n|w_2|(1+3 ∑_i=0^∞ |z|^i ) ≤
    	m^n|w_2|(1+3 ∑_i=0^∞ m^i ).



Therefore, for sufficiently large n ∈ℕ depending only on w_2, we have 


    | z^(n-1) |w_2|P_w_1(z) |  < βα/2.

Consequently, the winding number around 0 of the image of C under P_w_1w_2^n equals the
winding number around 0 of the image of C under the map


    z ↦(z^(n-1)|w_2| + z^(n-2)|w_2|+ … + 1)P_w_2(z).
  
The winding number of the image around 0 is related to number of zeros via the Argument Principle; for a holomorphic function f and a simple closed contour Γ, the number N of zeros of f inside Γ is given by

    N = 1/2π i∫_Γf^'(z)/f(z)dz = 1/2π i∫_f(Γ)dw/w,
 where w=f(z).  
Since P_w_2 has a root in D, this implies P_w_1w_2^n also has a root in D for
sufficiently large n.  
 

 


 §.§ Dominant approximations of growth rates


Finally, we prove that we can approximate y∈(√(2),2) with growth rates
corresponding to extensions of dominant strings. 




  Let w_1 be an admissible word such that w_1^∞ is a
  β-itinerary for some β>1.  For any ϵ>0, there
  exists an integer N = N(ϵ,w_1) such n>N implies that for
  every word w_2 for which (w_1^n w_2)^∞ is a
  β'-itinerary, β' is within distance ϵ of β.
  


  Let w' be the (β-ϵ)-itinerary, and let w” be the
  (β+ϵ)-itinerary. Let N be large enough that the
  N|w_1|-prefix of w', w_1^∞ and w” are all distinct. Then we
  must have w'<_E(w_1^nw_2)^∞<_Ew” so β'∈ (β-ϵ,
  β+ϵ). 


Combining Lemma <ref>,
Proposition <ref>, and the fact that slopes of tent maps
with periodic critical orbits are dense, the following result is evident:
 

  For all y∈(√(2),2) and all ϵ>0, there exists a sequence of
  dominant words (w_n)_i=1^∞ such that for any admissible
  extension w_n' of w_n, including the empty extension, if
  (w_n')^∞ is a β-itinerary, then β is within
  ϵ of y. 
  


 



 §.§ Proof of Proposition <ref>


Now we prove Proposition <ref>, which will finish the proof of the Persistence Theorem <ref>.


  Let w be an irreducible word in the alphabet {0,1} such that 
  w^∞ is the β-itinerary. 

  Then z is a root of the
  Parry polynomial of w, since this is equal to the Parry polynomial of
  β.  
  If y = β the statement is trivial, so
  assume y > β.  Fix 
    0<ϵ<y-β/2.
 Construct the
  sequence of dominant words (w_n) as in 
  the Dominant Approximations Lemma <ref>; 
  the words w_n satisfy that for any admissible extension w_n' of
  w_n, if (w_n')^∞ is some β'-itinerary, then
  β' is within ϵ of y. 

  We will show there is a subsequence
  of (w_n) with corresponding extensions (w_n') which meet this
  criteria and whose corresponding growth rates have controlled Galois
  conjugates.

  Passing to subsequences as needed, we may assume that |w_n| →∞
  as n →∞, since there are only finitely many words of bounded
  length. 
	
  For each n, let M_n=⌈|w_n|/|w|⌉ - 2.
  Then 
  
    2M_n|w|≥ 2(|w_n|/|w|-2)|w| = 2|w_n|-4|w|.


  Since 2|w_n|-4|w| > |w_n| if and only if |w_n|>4|w|, we have from
  equation (<ref>) that 
  
    |w_n|>4|w|  ⟹    2M_n |w| > |w_n|.



  Observe that	
  
    |w_n|
        =|w_n|/|w||w| > ( ⌈|w_n|/|w|⌉-2 )|w| 
        =M_n|w|    for all n

  and |w_n| →∞.  Therefore, for all n large enough that
  |w_n|>4|w|, there exists a positive integer M_n such that 
  
    2M_n|w| > |w_n| > M_n |w|.

  Note also that M_n →∞ as n →∞.  	
  Thus, for sufficiently large n, the hypotheses of 
  the Extension Lemma <ref> 
  hold, using w_n in place of w_1 and w in place of w_2.  
  Then 
  we conclude there exists an integer m_n'>M_n and a dominant extension
  w_n' of w_n so that (w_n'w^m_n')^∞ is admissible and the
  polynomial 

  
    P_w_n'w^m_n'(x^2^k)/1-x^2^k

  is irreducible for all k≥ 0, where P_w_n'w^m_n' is the Parry
  polynomial of the admissible word w_n'w^m_n'. Hence, by
  Proposition <ref>, (w_n'w^m_n')^∞ is a
  β'-itinerary, and we have the criteria needed to apply the final
  conclusion of 
  the Dominant Approximations Lemma <ref>); 
  that is, β'∈ [y-ϵ,
  y+ϵ].  Since M_n →∞ as n →∞ and
  m_n^' > M_n, we have m_n' →∞ as n→∞.  
  

  We know z is a root of P_w so

  by the Nearby Roots Lemma <ref>),
  for sufficiently large n ∈ℕ, we conclude P_w_n'
  w^M_n' has a root within ϵ of z. 









§ THE UNIT CYLINDER AND CONNECTIVITY





 
The bottom level of the Master Teapot is the unit circle, i.e. 
    Υ_2^cp∩ (ℂ×{1}) = S^1 ×{1}.
  



  We will first show S^1 ×{1}⊂Υ_2^cp.  By
  Proposition <ref>, there exists some growth rate β∈ (1,2)
  of a critically periodic tent map such that β has a Galois
  conjugate z ∈𝔻 satisfying the condition that, for all k
  ∈ℕ, every 2^k-th root of z is a Galois conjugate of
  β^1 2^k.   Repeatedly applying renormalization to (z,
  β)∈Υ_2^cp - by which we mean considering the set of
  2^k points of the form (z^1/2^k,β^1/2^k), all of which are in
  Υ_2^cp, as k →∞ - and then taking the set of limit
  points, we get that S^1 ×{1}⊂Υ_2^cp.




To show  Υ_2^cp∩ (ℂ×{1}) ⊂ S^1 ×{1}, suppose there exists a point (y,1)∈ such that |y|≠ 1. 
Since 1 has no nontrivial Galois conjugates, (y,1) ∈ℂ×ℝ must be the the limit of a sequence of points (α_n,β_n) ∈ℂ×ℝ such that β_n is the growth rate of a superattracting tent map and α_n is a Galois conjugate of β.  
Thus, reindexing the sequence as necessary, we have that for any k>0,
there exists β_k with 1<β_k<1+1/k with Galois
conjugate α_k, so that |α_k-y|<ϵ. Now by
renormalization, β_k^2^n_k≤ 2 is the slope of a critically
periodic tent map, where n_k is the maximal value of N for which
β_k^2^N≤ 2.  The fact that α_k^2^n_k is a Galois conjugate of β_k^2^n_k follows immediately from the definition of a Galois automorphism.  Thus (α_k^2^n_k,β_k^2^n_k) ⊂Υ_2^cp.  

Now, |α_k| is bounded away from 1 for k sufficiently large (because α_k → y), and n_k→∞ as k →∞, since β_k → 1 as k →∞.  Consequently, either α_k^2^n_k→ 0  or α_k^2^n_k→∞ as k→∞. This is a contradiction because

    Ω_2^cp⊂{z ∈ℂ: 1/2 ≤ z ≤ 2}

 by  <cit.> 

and that the projection of Υ_2^cp onto the first coordinate is Ω_2^cp.   






As in the proof of Proposition <ref>, let β be the slope of a critically periodic tent map such that β has a Galois conjugate  z ∈𝔻 and for all k ∈ℕ, and every 2^k-th root of z is a Galois conjugate of β^1 2^k. Then we have

    {(z', β') ∈ℂ×ℝ^+ :  (z ')^2^k=z  and β'=β^1 2^k for some  k ∈ℕ}⊂Υ_2^cp.
 The Persistence Theorem <ref> then implies that 

    {(z', y) ∈ℂ×ℝ^+ : (z')^2^k=z, y≥β^1 2^k for some  k ∈ℕ}⊂Υ_2^cp,
 
Taking the closure of the above set, we obtain that the unit cylinder
𝕊^1 × [1,2] is a subset of Υ_2^cp.




 
Connectivity of the part of the Master Teapot outside of the unit cylinder
is due to Tiozzo <cit.>.  More specifically, by
<cit.>, for any point (z,β) ∈ℂ×ℝ such that β is the growth rate of a
critically periodic tent map, z is a Galois of β, and |z| > 1,
there exists a continuous path (γ(x),x) in Υ_2^cp
connecting (z,β) to a point (w,1) with w ∈ S^1.   Consequently,
since the unit cylinder is a subset of Υ_2^cp by the Unit
Cylinder Theorem <ref>, and since Υ_2^cp is
closed, this implies  Υ_2^cp∩ ({z:|z| ≥ 1}×ℝ) is connected.  By <cit.>, 

the Thurston Set is connected and contains an open annulus containing S^1.  By the Persistence Theorem <ref>, the projection to ℂ of part of the top level of the Master Teapot that is inside the unit cylinder agrees with the Thurston Set, i.e. Υ_2^cp∩ (𝔻×{2}) = (Ω_2  ∩𝔻) ×{2}.  Also by the Persistence Theorem <ref>, the part of the Master Teapot inside the unit cylinder is connected.  Thus, the entire Master Teapot, Υ_2^cp, is connected.  
 






§ GAPS IN THE THURSTON SET
 

Plots of finite approximations of the Thurston set consisting of the roots of all defining polynomials associated to superattracting tent maps of critical orbit length at most n, for fixed n ∈ℕ, have  “gaps" at certain algebraic integers, some of which are on the unit circle.  The Thurston set contains a neighborhood of the unit circle <cit.>, but these gaps get filled in more slowly with n than some other regions.  
See Figure <ref> for a picture of a finite approximation of the Thurston set, and Figure <ref> for a closeup of one such gap. In this section, we prove an arithmetic justification for gaps:



 
Let α be an algebraic integer such that ℤ[α] is a discrete subgroup of ℂ and let x ∈ℤ[α]. Set 
c = min{|z| : z ∈ℤ[α], z ≠ 0}.
Suppose there exists a superattracting tent map with postcritical length n whose growth rate has a Galois conjugate of the form x+ϵ for some ϵ∈ℂ with |ϵ| ≤1/n+1.  
Then 



  * if |x| ≥ 1, then 
c/(2n^2 + 3n+1) |x|^n e≤ϵ.

  * if |x| ≤ 1, then 
c/(2n^2+3n+1) e≤ϵ.




Fix x ∈ℤ[α] and suppose there exists a real number β associated to a generalized PCF β-map with m intervals and postcritical length n that has a Galois conjugate of the form x+ϵ for some ϵ∈ℂ with |ϵ| ≤ 1.   

Then β is the root of the associated Parry polynomial P_β,E;

    0 = z^n+1-(a_0z^n + a_1 z^n-1+⋯ + a_n) - 1,
 
where a_i ∈{-2,0,2}. Hence (x+ϵ) is also a root of P_β,E:

    0 = (x+ϵ)^n+1 - (a_0(x+ϵ)^n + a_1(x+ϵ)^n-1+ ⋯ + a_n)-1.
 
Therefore

    1-x^n+1+a_0x^n + … +a_n    = (x+ϵ)^n+1-x^n+1 -( a_0((x+ϵ)^n-x^n) 
            + a_1((x+ϵ)^n-1-x^n-1) +… + a_n-1((x+ϵ)-x)).

We have 1-x^n+1+a_0x^n + … +a_n ∈ℤ[α], so c ≤ |1-x^n+1+a_0x^n + … +a_n|.  Then by the triangle inequality, 


    c    ≤ |1-x^n+1+a_0x^n + … +a_n|  
       ≤ |(x+ϵ)^n+1-x^n+1| + |a_0||(x+ϵ)^n-x^n| + |a_1||(x+ϵ)^n-1-x^n-1| … |a_n-1||(x+ϵ)-x|.


We now restrict to the case |x| ≥ 1.  
For any k ≤ n+1, by the binomial theorem, the triangle inequality, and |ϵ| ≤1/n+1, 


    | (x+ϵ)^k - x^k |    =   | ∑_i=1^k [ k; i ] x^k-iϵ^i |   ≤  ∑_i=1^k | [ k; i ] x^k-iϵ^i | 
       ≤  ∑_i=1^k | k^i/(k-i)! x^k-i1/(n+1)^i-1ϵ|   =  ∑_i=1^k | ( k/n+1)^i-1k/(k-i)! ϵ x^k-i| 
       ≤  ϵ k |x|^k-1∑_i=1^k 1/(k-i)!  =  ϵ k |x|^k-1∑_i=0^k-11/i!
       ≤  ϵ k |x|^k-1∑_i=0^∞1/i!  =  ϵ k |x|^k-1 e.


Combining equations (<ref>) and (<ref>) yields


    c    ≤ϵ (n+1) e |x|^n + 
    |a_0| ϵ n e |x|^n-1
    + … + 
    |a_n-1| ϵ 1 |x|^0 e| 
       ≤ϵ (n+1) e |x|^n (1 + |a_0| + … + |a_n-1| ) 
       ≤ϵ (n+1) e |x|^n (1 +2n ).


Thus for |x|≥ 1,

    c/e (1+2n) (n+1) |x|^n≤ϵ.



We now restrict to the case |x| ≤ 1.  In this case, the estimate (<ref>) becomes 

    |(x+ϵ)^k - x^k| ≤ϵ ke.


Combining equations (<ref>) and (<ref>) yields

    c ≤ϵ (n+1) e(1+|a_0| + |a_1| + … +|a_n-1|) ≤ϵ (n+1) e(1+2n) .
 
Hence, for |x| ≤ 1, 

    c/(n+1)(1+2n) e≤ϵ.





In view of Theorem <ref>, it suffices to classify the discrete subgroups of ℂ.
The classification of discrete subrings of ℂ is well-known, and
we include it for completeness: firstly, because it is a discrete additive
subgroup, it is either ℤ or a lattice of rank 2. If it is the
latter case, let {1, a} be a basis of the lattice, then a must be an
algebraic integer of degree 2, in other words, the discrete subring must be of the form ℤ[a] where a is an
algebraic integer of degree 2, hence it must be contained in the ring of integers of an algebraic field of degree 2. There are only 4 such rings of integers that contains some element not on the real line and has absolute value less than 2, which are
 ℤ[√(-1)], ℤ[√(-2)], ℤ[√(-5)], or
ℤ[1+√(-3)/2]. 










§ Ω_2^CP AND Ω_2^PCF ARE NOT EQUAL


In this section we prove the Two Thurston Sets Theorem <ref>, that Ω_2^cp and Ω_2^pcf are not equal. A finite approximation of Ω_2^cp is shown in Figure <ref>, and a finite approximation of Ω_2^pcf is shown in Figure <ref>.




As outlined in section <ref>, a point z ∈𝔻 is in Ω_2^cp if and only if 0 is in the limit set of the iterated function system generated by f_z,g_z, where
 
    f_z:x ↦ zx+1,    g_z:x ↦ zx-1.
 
Denote the alphabet {f_z,g_z} by ℱ_z and denote the alphabet of inverses {f^-1_z,g^-1_z} by ℱ^-1_z.
 For a word w=w_1,…,w_n in the alphabet ℱ_z or in the alphabet ℱ^-1_z, define the action of w on ℂ by 

    w(x) = w_n ∘…∘ w_1 (x).


 
Fix z ∈𝔻∖{0}.  If there exists n ∈ℕ such that 


    min{ |v(0)| : v ∈ (ℱ^-1_z)^n } > 1/1-|z|,

then z ∉Ω_2^cp.



Suppose z ∈𝔻∩Ω_2^cp.  Then 0 is in the limit set Λ_z.  
Since Λ_z = f_z(Λ_z) ∪ g_z(Λ_z), it follows that Λ_z is fixed by taking the union of the images of Λ_z under all words of length n, for any n ∈ℕ:

    Λ_z = ⋃_w ∈ (ℱ_z)^n w(Λ_z).
 Hence, for
any n ∈ℕ, each point in Λ_z is the image of a point
Λ_z under some word in ℱ_z of length n.  In
particular, 0 is the the image of a point in Λ_z under some word
in ℱ_z 

of length n.  Since Λ_z ⊂ B_1/1-|z|(0) by Lemma <ref>, this implies that for any n ∈ℕ,

    ( ⋃_v ∈ (ℱ^-1_z)^n v(0) ) ∩ B_1/1-|z|(0) ≠∅.









  Let β be the leading root of the polynomial
  
    P(x)=x^12 - 2x^11 + x^10 - 2x^9 + x^8 - 2x^7 + 2x^6 - 2x^5 + 4x^4 - 2x^3 + 4x^2 - 4x + 2.

  (The value of β is approximately 1.94848.) 
  By computation, the minimal β-itinerary is
  
    w=1000011100(101000)^∞.

 









  Because P is irreducible, any roots of P must be in Ω^pcf. 
Let p be the root of P with approximate value  

    p ≈ 0.5393738531461442 + 0.4050155839374199i.

Since |p| is approximately 0.674509, p ∈𝔻∩Ω_2^pcf.  

 Let ℱ^-1_p be the alphabet consisting of the two maps f_p^-1 and g_p^-1, where

    f_p^-1:x ↦x-1/p,    g_p^-1:x ↦x+1/p.

Computation shows that

    min{ |v(0)| : v ∈ (ℱ^-1_p)^5 }≈ 4.3792,

which is much bigger than 1/1-|p|≈ 3.07228. Consequently, Lemma <ref>  implies that p ∉Ω_2^cp.








alpha




Harrison Bray, University of Michigan, Department of Mathematics, 530 Church Street, Ann Arbor MI, 

Diana Davis, Swarthmore College, Department of Mathematics and Statistics, 500 College Avenue, Swarthmore PA, 

Kathryn Lindsey, Boston College,  Department of Mathematics, Maloney Hall, Fifth Floor, Chestnut Hill, MA, 

Chenxi Wu, Rutgers University, Department of Mathematics, 110 Frelinghuysen Road, Piscataway, NJ, 

