Teichmüller Theory and Applications to Geometry, Topology, and Dynamics
Volume 2: Surface Homeomorphisms and Rational Functions

Errata and Notes, complete as of February 27, 2020

We thank Alastair Fletcher, Wolf Jung, Dan Margalit, and Ahmad Rafiqi
for their contributions to this list.
Mathematical errata
Page 43

Next-to-last line of caption: .9 = 1.0, not .9 = 1.0

Page 22End of first paragraph of subsection on stretch factors: “inparticular the eigenvalues of symmetric matrices with integer or rationalcoefficients are totally real”, not “in particular the eigenvalues of symmetricreal matrices (or complex Hermitian matrices) are totally real”
Page 32[added March 29, 2017] Two lines after Definition 8.6.1: “apositive integer matrix”, not “a positive matrix”.
Page 34[added March 29, 2017] Four lines before the remark: “The1-form Re !q ”, not “The 1-form !q ”.
Page 38
Proof of Proposition 9.1.4, part 2, second line: “the sequencen 7! p n of iterates on the complement of X”, not “the sequence n 7! p nof iterates”.
Page 63
Theorem 9.4.1 could be expanded by changing “a finite orbit
containing a repelling cycle” in part 2 to “a finite orbit containing a repellingor parabolic cycle”. The proof would have to be adapted.
Page 85 Second paragraph: “the norm kD f (⌧ )k of the derivative canbe bounded in terms of the projection . . . ”, not “the norm kD f (⌧ )k of
the derivative depends only on the projection . . . ”.
Page 92
Remark: “unimodal”, not “unimodular”, and footnote no. 2
should be deleted.
Page 93 The claim that part 1 of Proposition 10.4.3 0 is a special case ofTheorem 9.3.2 is not correct, since that theorem applies only to the interior.
It’s not hard to fix the proof. Either we invoke Theorem 9.4.1 to say that
Jp is locally connected, or we adapt the proof of Theorem 9.4.1 to the caseof hyperbolic polynomials.
820
Page 110
Caption to Figure 10.4.9: ✓ = .001100110100 =, not4095
594✓ = .0010011010010 =.
4095
1

Page 113
Corollary 10.4.17 is not stated correctly. The corrected algorithm is written in Bruin, Ka✏, and Schleicher (Symbolic Dynamics ofQuadratic Polynomials, monograph in preparation), who attribute it toMary Rees. We are replacing everything from “Note that since ✓ has period
k . . . ”(second sentence of the paragraph immediately before the corollary)to the end of the section by the following:Let ✓0 be the companion of ✓; set ✓i := 2i 1 ✓ and ✓i0 := 2i 1 ✓0 . Angles will
be viewed as elements of R/Z; “preimage” means preimage under t 7! 2t,i.e., angle doubling; every interval in R/Z has two disjoint preimages.
Define intervals Ik , Ik 1 , . . . , I1 , as follows: the interval Ik is the preimage of [✓/2, (✓ + 1)/2] that does not contain ✓k ; it is bounded by two angles '0k and '00k , with '0k = ✓k periodic and '00k preperiodic. Then for all
j = 1, . . . , k 1 the interval Ij is the preimage of Ij+1 bounded by preimages '0j of '0j+1 and '00j of '00j+1 , chosen so that '00j is on the side of thedividing line specified by ⌃j✓ (✓).
Corollaryin
10.4.17 The angle ✓0 is the unique angle of period k in theinterval I1 .
Proof Observe that ✓k0 2 Ik . Further '00j always lands on a point of theboundary of the component Uj of int K✓ containing xj = p✓j (0). It thenfollows that ✓j0 2 Ij , and therefore ✓0 = ✓10 2 I1 . It is the unique periodic
angle in the interval because |I1 | = 1/2k and the angles periodic of period
k are spaced 1/(2k 1) apart. ⇤Example 10.4.18 Let us find the companion of ✓ = 4/15 = .0100. Makea table✓1✓2✓3✓4.0100.1000.0001.0010

BBA?

I1 = [.001100100, .001110100]
I2 = [.01100100, .01110100]
I3 = [.1100100, .1110100]
I4 = [.100100, .110100]

We see that indeed .0011 = 3/15 is the only angle that is periodic ofperiod 4 and is contained in I1 . 4
Page 118[new Feb. 27, 2020] Equation 10.5.8: on the right side, theo2
p in the denominator should be po(k 1) .
Page 119 Figure 10.5.3, lower right: the line marked ✓ should not extend
into the pink (nor should the lines marked ✓/2 and (✓ + 1)/2, but it’s notclear they do).
Page 175
Part 3 of the remark should be: If we replace “homeomorphism” by “di↵eomorphism”, the result remains true for S 2 and S 3 , but itis false in higher dimensions. For S 3 the result is very hard [18].
2Page 227 Last paragraph: replace “if a Thurston map f is unicritical, or
if it is a topological polynomial such that Crit f ⇢ Pf ” by “if a topologicalpolynomial f is unicritical or satisfies Crit f ⇢ Pf ”.
Notes and clarifications
Page 92 Definition 10.3.2: The kneading sequence defined here is what
Dierk Schleicher calls an “itinerary”; he uses “kneading sequence” only forthe itinerary of the critical value or of the corresponding external angle.
Page 93
Theorem 10.3.3: In the case where ✓ is in Qeven /Z and there
is a Thurston obstruction, the points Xi do converge under iteration of f ,but they do not remain distinct. This was proved in greater generality by
Nikita Selinger.
Page 114
Proposition and Definition 10.5.4: “contained in the closeddisc”, not “contained in the disc”.
Page 201[added April 21, 2019] In equation C5.12, et should be et/2tand e should be e t/2 .
Non-mathematical errata
Page xNext-to-last line: “very time” should be “every time”.

Page 22We misspelled Yoccoz’s first name. It is Jean-Christophe.

Page 102
Last line of caption to Figure 10.4.3: “three red externalrays”, not “three external red rays”.
Page 148
Proof of Proposition 110.6.8: In the first line, “This is exactly”, not “This exactly”.
Page 154
Line immediately after equation 10.7.14: “felements”, not “f 1 (Z) has d|Z| elements in its inverse”.

1(Z) has d|Z|

Page 174
Second paragraph of the remark, last line: “the scope of”,
not “the the scope of”.
Page 201 [added April 21, 2019] First line after equation C5.12: “preserves the subset”, not “preserves he subset”.
Page 249
First line of subsection Similarities in the proofs: “Manyproofs in complex dynamics”, not “Many proof in complex dynamics”.
Page 251

Entry [18]: “Lecture Notes”, not “Lectures Notes”

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