Core entropy and biaccessibility of quadratic polynomials

0.59 MB·

by  Wolf Jung

Core entropy and biaccessibility
of quadratic polynomials
Wolf Jung
Gesamtschule Aachen-Brand
Rombachstrasse 99,52078Aachen,Germany.
4
E-mail: [email protected]
1
0
2

n Abstract
a
J For complex quadratic polynomials, the topology of the Julia set and the dynamics

0 areunderstood fromanother perspective byconsidering theHausdorffdimension of
2
biaccessinganglesandthecoreentropy: thetopologicalentropyontheHubbardtree.

] These quantities are related according toThurston. Tiozzo [60] has shown continu-
S
ity on principal veins of the Mandelbrot set M. This result is extended to all veins
D
here, and it is shown that continuity with respect to the external angle q will imply
.
h
continuity in the parameter c. Level sets of the biaccessibility dimension are de-
t
a scribed, which are related to renormalization. Hölder asymptotics atrational angles
m
are found, confirming the Hölder exponent given by Bruin–Schleicher [11]. Partial
[
results towards local maxima at dyadic angles are obtained as well, and a possible

1 self-similarity ofthedimensionasafunctionoftheexternalangleissuggested.
v
2
9
7 1 Introduction
4
.
1
0 For a real unimodal map f(x), the topological entropy h=logl is quantifying the com-
4 plexity of iteration: e.g., the number of monotonic branches of fn(x) grows like l n.
1
: Moreover, f(x) is semi-conjugate to a tent map of slope l ; so l is an averaged rate
v ±
of expansion, which is a topological invariant [38, 42, 2]. Consider a complex quadratic
i
X polynomial f (z)=z2+cwithitsfilled Juliaset K ,which isdefined in Section 2:
c c
r
a
At least in the postcritically finite case, the interesting dynamics happens on the
•
Hubbard tree T K : other arcs are iterated homeomorphically to T , which is
c c c
⊂
folded over itself, producing chaotic behavior. The core entropy h(c) is the topo-
logicalentropyof f (z)on T [1, 31, 57].
c c
On the other hand, for c = 2 the external angles of these arcs have measure 0,
• 6 −
and the endpoints of K correspond to angles of full measure in the circle. This
c
phenomenonisquantifiedbytheHausdorffdimensionB (c)ofbiaccessingangles:
top
thebiaccessibilitydimension[63, 67, 51, 10, 36, 11].
Accordingto Thurston,thesequantitiesarerelated by h(c)=log2 B (c), whichallows
top
·
tocombinetoolsfromdifferentapproaches: e.g.,B (c)iseasilydefinedforeveryparam-
top
eter c in the Mandelbrot set M, but hard to compute explicitly. h(c) is easy to compute
1
andtoanalyzewhen f (z)ispostcriticallyfinite. Butforsomeparametersc,thecoremay
c
becometoo largebytakingtheclosureoftheconnected hullofthecritical orbitin K .
c
The postcritically finite case is discussed in Section 3. A Markov matrix A is associated
to the Hubbard tree, describing transitions between the edges. Its largest eigenvalue l
gives the core entropy h(c) = logl . An alternative matrix is due to Thurston [21, 22].
Or l is obtained from matching conditions for a piecewise-linear model with constant
expansion rate. This approach is more convenient for computing specific examples, but
the matrix A may be easier to analyze. Known results are extended to all renormalizable
maps f (z). HereAisreducibleorimprimitive,anditsblocksarecompared: puresatellite
c
renormalization gives a rescaling h(c c) = 1h(c). Lower bounds of h(c) for b -type
p∗ p
Misiurewicz points and for primitive centers show that in the primitive renormalizable
b b
case, the dynamics on the small Julia sets is negligible in terms of entropy, so h(c) is
constant on maximal-primitivesmall Mandelbrot sets M M. Moreover, it is strictly
p
⊂
monotonic between them. An alternative proof of h(c) = log2 B (c) shows that the
top
·
externalanglesofT havefinitepositiveHausdorffmeasure.
c
In Section 4, the biaccessibility dimension is defined combinatorially for every angle
q S1andtopologicallyforeveryparameterc M. Whencbelongstotheimpressionof
∈ ∈
theparameter ray R (q ), we haveB (c)=B (q ) [10]. This relationmeans that non-
M top comb
landing dynamic rays have angles of negligible Hausdorff dimension, but a discussion
of non-local connectivity is avoided here by generalizing results from the postcritically
finite case: the biaccessibility dimension is constant on maximal-primitive Mandelbrot
sets, and strictly monotonic between them. Components of a level set of positive B (c)
top
are maximal-primitive Mandelbrot sets or points. Examples of accumulation of point
components are discussed. In [11, 60] the Thurston relation h(c) = log2 B (c) was
top
·
obtained for all parameters c, such that the core T is topologically finite. The proof
c
extendsto compacttrees withinfinitelymanyendpoints.
In an email of March 2012 quoted in [21], Thurston announced proofs of continuity for
B (q )byHubbard,Bruin–Schleicher,andhimself. InMay2012,aproofwithsymbolic
comb
dynamics was given in [11], but it is currently under revision. In the present paper, it is
shown that continuity of B (q ) on S1 will imply continuity of B (c) on M. Again, a
comb top
discussionofnon-localconnectivitycanbeavoided,sincethebiaccessibilitydimensionis
constant on primitiveMandelbrot sets. See version 2 of [11] for an alternativeargument.
Tiozzo [60] has shown that h(c) and B (c) are continuouson principal veins of M; this
top
resultisextendedto allveinshere.
In Section 5, statements of Bruin–Schleicher, Zakeri, and Tiozzo [10, 11, 65, 66, 60]
on the biaccessibility dimension of M are generalized to arbitrary pieces. In Section 6,
Markovmatricesareusedagaintoshowageometricscalingbehaviorofthecoreentropy
for specific sequences of angles, which converge to rational angles; for these examples
the Hölder exponent of B (q ) given in [11] is optimal. The asymptotics of sequences
comb
suggests the question, whether the graph of B (q ) is self-similar; cf. the example in
comb
Figure1. PartialresultstowardstheTiozzoConjecture[60]areobtainedaswell,whichis
concerning local maxima of B (q ) at dyadic angles. Some computations of character-
comb
isticpolynomialsaresketchedinAppendixA.Fortherealcase,statementsonpiecewise-
linear models [38] and on the distribution of Galois conjugates [58, 59, 61] are reported
in Appendix B to round off the discussion. See [10, 11] for the approach with symbolic
dynamicsand [21, 22] forthestructureofcriticalportraits.
2
The present paper aims at a systematicexpositionof algebraic and analytic aspects; so it
contains a mixture of well-known, extended, and new results. I have tried to give proper
creditsandreferencestopreviousorindependentwork,andIapologizeforpossibleomis-
sions. SeveralpeopleareworkingonreconstructingandextendingBillThurston’sresults.
IhavebeeninspiredbyhintsfromordiscussionswithHenkBruin,GaoYan,SarahKoch,
MichaelMertens,Dierk Schleicher, Tan Lei, and GiulioTiozzo.
n=0 n=1
→
ւ
n=2 n=3
→
ւ
n=4 n=5
→
Figure 1: The biaccessibility dimension is related to the growth factor l by B (q ) =
comb
log(l (q ))/log2. Consider zooms of l (q ) centered at q = 1/4 with l = 1.69562077. The
0 0
widthis0.201 2 n andtheheightis1.258 l n. Thereseemstobealocalmaximumatq and
× − × 0− 0
akindofself-similaritywithrespecttothecombinedscalingby2andbyl . SeeExample6.1for
0
theasymptotics ofB (q )onspecificsequences ofanglesq q .
comb n→ 0
3
2 Background
A short introduction to the complex dynamics of quadratic polynomials is given to fix
some notations. The definitions of topological entropy and of Hausdorff dimension are
recalled, and conceptsfornon-negativematrices arediscussed.
2.1 Quadratic dynamics
Quadratic polynomials are parametrized conveniently as f (z)= z2+c. The filled Julia
c
set K contains all points z with a bounded orbit under the iteration, and the Mandelbrot
c
set M contains those parameters c, such that K is connected. Dynamic rays R (j )
c c
are curves approaching ¶ K from the exterior, having the angle 2pj at ¥ , such that
c
f (R (j ))=R (2j ). They are defined as preimages of straight rays under the Böttcher
c c c
mapF : C K C D. ParameterraysR (q )approach¶ M [40,49];theyaredefined
c c M
\ → \
intermsoftheDouadymapF : C M C DwithF (c):=F (c). Thelandingpoint
b b M \ → \ M c
isdenotedbyz=g (j )orc=g (q ),respectively,buttheraysneednotlandforirrational
c M b b
angles,seeFigure3. Thereare twocases ofpostcriticallyfinitedynamics:
When the parameter c is a Misiurewicz point, the critical value z = c is strictly
• preperiodic. Both c ¶ M and c ¶ K have the same external angles, which are
c
∈ ∈
preperiodicunderdoubling.
When c is the center of a hyperbolic component, the critical orbit is periodic and
•
contained in superattracting basins. The external angles of the root of the compo-
nent coincide with the characteristic angles of the Fatou basin around z = c; the
characteristicpointmay havemoreperiodicangles inthesatellitecase.
In both cases, the Hubbard tree [16, 26] is obtained by connecting the critical orbit with
regulated arcs, which are traveling through Fatou basins along internal rays. Fixing a
characteristicangleq ofc,thecircleS1=R/Zispartitionedbythediameterjoiningq /2
and (q +1)/2 and the orbit of an angle j S1 under doublingis encoded by a sequence
∈
of symbols A,B, or 1,0, . There is a corresponding partition of the filled Julia set, so
∗ ∗
pointsz K aredescribedbysymbolicdynamicsaswell. Thekneadingsequenceisthe
c
itinerary∈ofq orofc, see[26, 49, 10].
M consists of the closed main cardioid and its limbs, which are labeled by the rotation
number at the fixed point a ; the other fixed point b is an endpoint of K . A partial
c c c
order on M is defined such that c c when c is disconnected from 0 in M c . See
′ ′
≺ \{ }
Sections3.3and4.1forthenotionofrenormalization[14,15,25,39,48,28],whichisex-
plainingsmallJuliasetswithinJuliasetsandsmallMandelbrotsetswithintheMandelbrot
set. Primitive and satellite renormalization may be nested; a primitivesmall Mandelbrot
set will be called maximal-primitive, if it is not contained in another primitive one. A
pure satellite is attached to the main cardioid by a series of satellite bifurcations, so it is
notcontainedin aprimitiveMandelbrotset.
2.2 Topological entropy
Suppose X is a compact metric space and f : X X is continuous. The topological
→
entropy is measuring the complexity of iteration from the growth rate of the number of
4
distinguishable orbits. The first definition assumes an open cover U and considers the
minimalcardinalityV (n)ofasubcover,suchthatallpointsinasetofthesubcoverhave
U
thesameitinerarywithrespect toU fornsteps. See, e.g., [17, 31]. Thesecond definition
is using the minimal numberVe (n) of points, such that every orbit is e -shadowed by one
ofthesepointsfornsteps [8,37]. Wehave
1 1
htop(f,X):=sUupnlim¥ n logVU(n)=elim0limn s¥up n logVe (n). (1)
→ → →
For a continuous, piecewise-monotonic interval map, the growth rate of monotonic
branches (laps) may be used instead, or the maximal growth rate of preimages [42, 17].
The same result applies to endomorphisms of a finite tree [2, 31]. Moreover, f is semi-
conjugate to a piecewise-linear model of constant expansion rate l when h (f,X) =
top
logl > 0; this is shown in [38, 17, 2] for interval maps and in [4] for tree maps. See
Section B for the relation to the kneading determinant and Section 4.4 for continuity re-
sults [38, 42, 43, 17, 2]. If p : X Y is a surjective semi-conjugation from f : X X
→ →
to g:Y Y, then h (g,Y) h (f,X). Equality follows when every fiber is finite, but
top top
→ ≤
fibercardinalityneed notbeboundedglobally[8, 37].
2.3 Hausdorff dimension
The d-dimensional Hausdorff measure is a Borel outer measure. It is defined as follows
foraboundedsubsetX R orasubsetX S1 =R/Z:
⊂ ⊂
m (X):= lim inf(cid:229) U d (2)
d i
e 0 U | |
→ i
Here the coverU of X is a countable family of intervalsU of length U e . They may
i i
| |≤
beassumedtobeopenorclosed,alignedtonestedgridsornot,buttheimportantpointis
thattheymaybeofdifferentsize. Whenanintervalisreplaced withtwosubintervals,the
sum may grow in fact when d <1. In general, the Hausdorff measure of X may be easy
toboundfromabovebyusingintervalsofthesamesize,butitwillbehardtoboundfrom
below,sincethisrequires to find an optimalcoverwithintervalsofdifferentsizes.
The Hausdorff dimension dim(X) is the unique number in [0,1], such that m (X) = ¥
d
for 0 d < dim(X) and m (X) = 0 for dim(X) < d 1. For d = dim(X), the Haus-
d
dorff m≤easure m (X) may be 0, positive and finite, or≤¥ . The Hausdorff dimension of a
d
countableset is 0 and the dimensionof a countableunion is thesupremum ofthe dimen-
sions. Again, dim(X) may be easy to bound from above by the box dimension, which
correspondstoequidistantcovers,butitishardertoboundfrombelow. Sometimesthisis
achieved by constructing a suitable mass distribution according to the Frostman Lemma
[19, 65]. When X S1 isclosed and invariantunderdoublingF(j )=2j , theHausdorff
⊂
dimensiondim(X)is equalto thebox dimensionaccordingto Furstenberg [20].
2.4 Perron–Frobenius theory
ThePerron theory of matrices with positiveelements was extended by Frobenius to non-
negativematrices,see[24]. WeshallneedthefollowingfeaturesofasquarematrixA 0:
≥
5
There is a non-negative eigenvalue l with non-negative eigenvector, such that all
•
eigenvalues of A are l in modulus. l is bounded above by the maximal sum of
≤
rowsorcolumns,and boundedbelowbytheminimalsum.
A 0 is called reducible, if it is conjugate to a block-triangular matrix by a per-
• ≥
mutation. It is irreducible (ergodic) if the corresponding directed graph is strongly
connected. Then l is positive and it is an algebraically simple eigenvalue. The
eigenvectorofl is positive,and othereigenvectorsarenotnon-negative.
An irreducible A 0 is called primitive (mixing), if the other eigenvalues have
• ≥
modulus<l . Equivalently,An >0element-wiseforsomen.
IfAisirreduciblebutimprimitivewith p>1eigenvaluesofmodulusl ,itscharac-
•
teristic polynomial is of the form xkP(xp). The Frobenius normal form shows that
thereare p subspaces mappedcyclicallybyA.
IfB Aelement-wisewithB=A andAisprimitive,thenl >l . Thisisproved
B A
• ≥ 6
by choosing n with An >0, noting Bn An, so Bn+1 is strictly larger than An+1 in
at least one row and one column, and≥finally B2n+1 > A2n+1. Now fix e > 0 with
B2n+1 (1+e )A2n+1 andconsiderhigherpowersto showl 2n+√1 1+el .
B A
≥ ≥
To obtain l numerically from A 0, we do not need to determine the characteristic
≥
polynomialanditsroots: forsomepositivevectorv computev =Anv recursively,then
0 n 0
l =lim n v convergesslowly. IfAisirreducibleandprimitive,l =lim v / v
n n+1 n
k k k k k k
willconvpergeexponentiallyfast.
3 Postcritically finite polynomials and core entropy
Suppose f (z) is postcritically finite and consider the Hubbard tree T . Since the col-
c c
lection of vertices is forward invariant, each edge is mapped to one edge or to several
adjacentedges. ThustheedgesformaMarkovpartition(strictlyspeaking,atessellation).
By numbering the edges, the map is described by a non-negativematrix A with entries 0
and1,suchthatthe j-thcolumnisshowingwherethe j-thedgeofT ismappedby f (z).
c c
The Markov matrix A is the transition matrix of the Markov partition and the adjacency
matrixoftheMarkovdiagram. Often thetransposed matrixisused instead. In theprepe-
riodic case, no postcritical point is mapped to the critical point z=0. So we still have a
Markov partition when the two edges at 0 are considered as one edge, but mapping this
edgewillcovertheedgebefore z=ctwice, resultinginan entry of2 in A.
Definition3.1 (Markovmatrix and coreentropy)
For a postcritically finite quadratic polynomial f (z), the Markov matrix A is the transi-
c
tion matrix for the edges of the Hubbard tree T . Its highest eigenvalue l gives the core
c
entropyh(c):=logl . Equivalently,h(c):=h (f ,T ) isthetopologicalentropyonT .
top c c c
The i-th row of A says which edges are mapped to an arc covering the i-th edge. Since
f : T T issurjectiveandatmost2:1,thesumofeachrowofAis1or2,sothehighest
c c c
→
eigenvalue of A satisfies 1 l 2. The largest entries of An are growing as l n when
l >1. (Notasnkl n: accor≤ding≤toSection3.3,Amaybereduciblebutl >1c≍orresponds
to a unique irreducible block, which need not be primitive.) Since the entries of An give
6
the number of preimages of edges, the same estimate applies to the maximal cardinality
of preimages f n(z ) in T . There are various ways to show that logl is the topological
c− 0 c
entropyof f (z)on theHubbard treeT :
c c
By expansivity,every edgeis iteratedto an edgeatz=0,so thepreimages of0are
•
growing by l n even if the edges at 0 correspond to an irreducible block of lower
eigenvalues. Sothenumberofmonotonicbranchesof fn(z)isgrowingbythesame
c
rate, whichdeterminesthetopologicalentropyaccording to[2].
This criterion is obtained from the definition according to Section 2.2 by showing
•
that the open cover may be replaced with closed arcs having common endpoints,
andthat themaximalgrowthrateisattainedalready forthegivenedges[17, 31].
According to [4], f (z) on T is semi-conjugate to a piecewise-linear model with
c c
•
constant expansion rate l when the topological entropy is logl > 0. See also
[38, 17] for real parameters and [60] for parameters on veins. Now the highest
eigenvector of the transposed matrix A is assigning a Markov length to the edges,
′
such that f (z) corresponds to multiplying the length with l . Note that according
c
tothedecomposition(3)inSection3.3,theedgesinaprimitivesmallJuliasethave
Markovlength0 and willbesqueezed topointsbythesemi-conjugation.
3.1 Computing the core entropy
Letusstartwithfourexamplesofpostcriticallyfiniteparameterscinthe1/3-limbofM.
Theexternalangleq =3/15givesaprimitivecenterofperiod4. Theotherexamplesare
preperiodicandtheedgesatz=0areunited. q =1/4definesab -typeMisiurewiczpoint
of preperiod 2 and q = 9/56 gives an a -type Misiurewicz point of preperiod 3 and ray
period3;itissatelliterenormalizableofperiod3andtheMarkovmatrixAisimprimitive
ofindex3. Andq =1/6 givesc=i,a Misiurewiczpointwithpreperiod1 and period2.
q = 3 q = 1 q = 9 q = 1
15 4 56 6
0=c4 7→c1 b c =c3 7→c3 a c =c4 7→c4 c3 7→c2
c c c c
s 1 s 1 s 1 s 1
❚ ❚ ❚ ❚
❚ ❚ ❚ ❚
c c
❚ s4 sc ❚ sc ❚s4 sc ❚ sc
3 3 3 3
✔ ✔ ✔ ✔
✔ ✔ ✔ ✔
✔s ✔s s✔ ✔s
c c c c
2 2 2 2
0 0 1 1
 1 0 0 0  0 0 2 0 0 2 0 0 2
0 1 0 0  1 0 0   1 0 0   1 0 1 
 
 0 1 1 0  0 1 1 0 1 0 0 1 0
       
l 4 2l 1=0 l 3 l 2 2=0 l 3 2=0 l 3 l 2=0
− − − − − − −
l =1.395337 l =1.695621 l =1.259921 l =1.521380
Figure 2: Examples of Hubbard trees and Markov matrices defined by an angle q . Here f (z)
c
mapsc=c c c c andtheedgesarenumberedsuchthatthefirstedgeisbeforec =c.
1 2 3 4 1
7→ 7→ 7→
7
Instead of computing the characteristic polynomial of the Markov matrix A, we may use
a piecewise-linear model with expansion constant l >1 to be matched: in Figure 2, the
edgesorarcs froma to c , c , c havelengthl , l 2, l 3 when[0, a ] haslength1.
c 1 2 3 c
Forq =3/15,wehave[ a ,c ] [a ,c ],so l (l 3 2)=1. ±
c 3 c 4
− 7→ −
Forq =1/4,wehave[ a ,c ] [a ,c ],so l (l 3 2)=l 3.
c 3 c 3
Forq =9/56,wehave−c = a 7→, sol 3 =2. −
3 c
−
Forq =1/6,wehave[ a ,c ] [a ,c ],so l (l 3 2)=l 2.
c 3 c 2
− 7→ −
Example3.2(Lowest periods and preperiods inlimbs)
The p/q-limb of M contains those parameters c, such that the fixed point a has the
c
rotationnumber p/q. Theprincipalveinisthearcfrom0totheb -typeMisiurewiczpoint
ofpreperiodq 1. TheexamplesinFigure2canbegeneralizedbyconsideringsequences
−
of Markov matrices or by replacing l 3 2 with l q 2 for the piecewise-linear models.
Thisgivesthefollowingpolynomialsfo−rl : −
b -typeMisiurewiczpointofpreperiod q 1: xq xq 1 2=0 [1]
−
− − −
Primitivecenter ofperiodq+1: xq+1 2x 1=0
a -typeMisiurewiczpointofpreperiod−q: x−q =2
Theseequationsshowthath(c)isnotHöldercontinuouswithrespecttotheexternalangle
q as q 0, seeSection 4.5.
→
Example3.3(Sequences onprincipal veins)
On the principal vein of the p/q-limb, the b -type Misiurewicz point is approached by
a sequence of centers c and a -type Misiurewicz points a of increasing periods and
n n
preperiods. The corresponding polynomials for l are obtained from piecewise-linear
n
modelsagain, orby consideringa sequence ofmatrices. The polynomialis simplifiedby
summinga geometricseries andmultiplyingwithl 1:
−
Centerc ofperiodn q+1: xn+1 xn 2xn+1 q+x+1=0
n −
≥ − −
a -typeMisiurewiczpointa ofpreperiodn q: xn+1 xn 2xn+1 q+2=0
n −
These polynomialsimply monotonicityof l≥and give−geom−etric asymptotics by writing
n
xn+1 xn 2xn+1 q =xn+1 q (xq xq 1 2); note that the largest root l of the latter
− − − 0
− − · − −
polynomialcorresponds totheendpointoftheveinaccording toExample3.2:
Forc wehavel l K l n withK = l 0+1 >0.
n n ∼ 0− c· 0− c q (q 1)/l 0
Fora wehavel l K l n withK = − −2 >0.
n n ∼ 0− a· 0− a q (q 1)/l 0
See Proposition 6.2 and Appendix A for a deta−iled−computation and Remark 6.4 for the
relationtoHöldercontinuity.
Example3.4(Sequences onthe real axis)
Thereal axis istheprincipalveinofthe1/2-limb;settingq=2 inExample3.3,dividing
byl +1 and notingl =2 at theendpointc= 2 gives:
0
−
c ofperiod n 3: xn 2xn 1+1=0,l 2 2 2 n
n − n −
a ofpreperiod≥n 2:−xn+1 xn 2xn 1+∼2=−0, l· 2 4 2 n
n ≥ − − − n ∼ −3· −
Nowconsiderthea -typeMisiurewiczpointa withtheexternalangleq =5/12andwith
2
l = √2, which is the tip of the satellite Mandelbrot set of period 2. It is approached
a
from above(with respect to , i.e., from the left) by centers c of periods n=3,5,7,...
′n
≺
related to the Šharkovski˘ı ordering. The entropy was computed by Štefan [54], and the
centers ofeven periods before a (to theright)are treated analogously. Again, geometric
2
asymptoticsareobtainedfrom thesequenceofpolynomialsforl :
n′
c ofperiod n=3,5,7,...: xn 2xn 2 1=0, l l + 1 l n
′n − − − n′ ∼ a √2· a−
8
c ofperiod n=4,6,8,...: xn 2xn 2+1=0, l l 1 l n
′n − − n′ ∼ a−√2· a−
Note that for even periods n, the polynomial for l is imprimitive of index 2, which is
n′
related tothesatelliterenormalizationaccording toSection 3.3.
Constructing the Hubbard tree T from an external angle q is quite involved [10]. In
c
[21, 22] an alternative matrix F by Thurston is described, which is obtained from an
externalanglewithoutemployingtheHubbardtree. Actually,onlythekneadingsequence
oftheangleisrequired to determineF from theparts ofT 0 :
c
\{ }
Proposition3.5 (Alternativematrixby Thurston and Gao)
From a rational angle or from a -periodic or preperiodic kneading sequence, construct
∗
atransitionmatrixF. Thebasicvectorsrepresent non-orientedarcsbetween postcritical
j
points c = f (0), j 1, and [c ,c ] is mapped to [c ,c ] by F unless its endpoints
j c j k j+1 k+1
≥
areindifferentpartsofK 0 : then itismapped to[c ,c ]+[c ,c ].
c 1 j+1 1 k+1
\{ }
Nowthelargesteigenvaluesof theThurstonmatrixF and theMarkovmatrixA coincide.
Thiscombinatorialdefinitioncorrespondstothefactthatanarccoveringz=0ismapped
2:1toanarcatz=cby f (z). Arcsatc areomittedinthepreperiodiccase,becausethey
c 0
would generate a diagonal 0-block anyway. Note that in general F is considerably larger
than A; it will contain large nilpotent blocks and it may contain additional blocks, which
seem to be cyclic. In the case of b -type Misiurewiczpoints, a small irreducible block of
F isobtainedin Proposition3.8.3.
Proof: Gao [22] is using a non-square incidence matrix C, which is mapping each arc
to a sum of edges, so AC =CF. Consider the non-negative Frobenius eigenvectors to
obtain equality of the highest eigenvalues: if Fy = l y then Cy is an eigenvector of A
F
with eigenvalue l . The transposed matrices satisfy F C =C A and if A x=l x, then
F ′ ′ ′ ′ ′ A
C x is an eigenvector of F with eigenvalue l . (Note thatCy andC x are not 0, because
′ ′ A ′
C has anon-zero entry ineach rowand each column.)
Asanalternativeargument,defineatopologicalspaceX asaunionofarcs[c ,c ] K ,
c j k c
⊂
which are considered to be disjoint except for common endpoints. There is a natural
projectionp : X T andaliftF : X X of f (z),suchthatp isasemi-conjugation
c c c c c c c c
and F is the transi→tion matrix of F . No→w any z T has a finite fiber p 1(z) = x
c c c− i
∈ { } ⊂
X and we have the disjoint union F n(x) = p 1(f n(z)). Choosing z such that the
c c− i c− c−
cardinality of f n(z) is growing bySl n shows l l . And choosing z such that the
cardinalityofFc−n(x )is growingby l An givesl A ≤l .F
c− 1 F F ≤ A
Note that F is determined from the kneading sequence of c =c, which can be obtained
1
fromtheexternalangleq asan itinerary. Alternatively,consideramatrixwherethebasic
vectors represent pairs of angles 2j 1q ,2k 1q ; it will be the same as F except in the
− −
{ }
preperiodic satellite case, where c is entering a repelling p-cycle of ray period rp > p.
1
(This cycle contains the characteristic point of a satellite component before the Misi-
urewiczpointc.) The matrixof transitionsbetween pairs ofangles will bedifferent from
F, but the largest eigenvalue will be the same: pairs of equivalent angles are permuted
cyclically and correspond to eigenvalues of modulus 1 [22]. Arcs of X with at least one
c
p-periodic endpoint are represented by several pairs of angles, but when a topological
space is built from multiple copies of arcs, it comes with a semi-conjugation to X or to
c
T again.
c
9
3.2 Estimates of the core entropy
The edges of theHubbard tree are connecting the marked points: the critical orbit fn(0),
c
n 0, whichincludesall endpoints,and additionalbranch points.
≥
Lemma 3.6(Modified Markovmatrix)
For a postcritically finite f (z), the Hubbard tree T and the associated Markov matrix
c c
A may be changed as follows, without changing the highest eigenvalue l , and without
changingirreducibilityorprimitivity(except for item4 andforc= 1 in item2):
−
1. If c is preperiodic, the two edges at z=0 may be considered as one edge by removing
z=0 fromthemarked points;thusan eigenvalue0 hasbeen removed fromA.
2. For c in the 1/2-limb, the unmarked fixed point a may be marked, splitting one edge
c
intwo. Thisgives an additionaleigenvalueof 1.
−
3. An unmarkedpreimageofa marked periodicor preperiodicpointmaybemarked.
4. Extend theHubbard treeT byattachingedges towards thefixed point b and/orsome
c c
ofitspreimages. This givesadditionaleigenvaluesof 1and 0,and itmakes A reducible.
Thesemodificationsmay be combined, and item 3 can be applied recursively. Theproof
is deferred to Appendix A. Figure 2 gives examples of item 1. Items 2 and 3 are applied
inSection 3.3. Item4 showsthatforabiaccessibleparameterc, theHubbardtreeT may
c
beextendedinauniformwayforallparametersonthevein. E.g.,intherealcasewemay
replace T = [c, f (c)] with [ b ,b ]. Item 4 is used as well to prove Proposition 6.6.
c c c c
−
Note that a matrix with an eigenvalue 0, -1, or 1 will be reduced by a conjugation in
GL(QN),but onlypermutationsareconsidered foraFrobeniusirreduciblematrix.
Proposition3.7 (Monotonicity ofcore entropy, Penrose andTao Li)
Coreentropyismonotonic: forpostcriticallyfinitec c wehaveh(c) h(c).
′ ′
≺ ≤
Not all parameters are comparable with the partial order . In particular, many param-
≺
eters c are approached by branch points c before them, and parameters c in different
′ ′′
branches are not comparable to c. — The proof below employs Hubbard trees. Tao Li
′
[31]usedthesemi-conjugationfromtheangledoublingmap,seeSection4.2. Penrosehad
obtaineda moregeneral statementforkneading sequences [44]. See also Proposition4.6
formonotonicitywithrespect to externalangles.
Proof: The periodicand preperiodic points marked in the Hubbard tree T of f (z) move
c c
holomorphicallyfor parameters inthewakeofc. TheHubbard treeT contains thechar-
c
′
acteristicpointz correspondingtocandtheconnectedhullT T ofitsorbitishomeo-
c c
′ ⊂ ′
morphic to T , but the dynamics of f (z) is different in a neighborhood of z=0, which
c c
′
is mapped behind the characteristic point. There is a forward-invariant Cantor setC T
⊂
defined by removing preimages of that neighborhood. To obtain the lower estimate of
h(c), either note that the preimages of a suitable point in C under f (z) correspond to
′ c
′
those in T under f (z), or consider a semi-conjugation p from C T to T . If c is a
c c c c
⊂ ′
center, hyperbolicarcs inT mustbecollapsedfirst.
c
Aparameterc M isab -typeMisiurewiczpoint,if fk(c)=b ;theminimalk 1isthe
c c
∈ ≥
preperiod. ThefollowingresultswillbeneededforProposition3.11. Theweakerestimate
log2
h(c) isobtainedfrom CorollaryE in [1];it doesnotgiveProposition3.11.2.
≥ k+1
10