doi: 10.3934/jmd.2014.8.133
J OURNAL OF M ODERN DYNAMICS
V OLUME 8, N O. 1, 2014, 133–137
ERRATUM: BILLIARDS IN NEARLY ISOSCELES TRIANGLES
W. PATRICK HOOPER AND RICHARD EVAN SCHWARTZ
In the paper “Billiards in Nearly Isosceles Triangles,” there were errors and
unclear points in the proof of the following theorem:
Theorem 1.5 For each integer n > 2, let Vn denote the obtuse isosceles triangle
π
with two acute angles of measure 2n
. Then,
1. For k = 3, 4, 5..., the triangle V2k does not lie in the interior of an orbit tile.
2. For n ≥ 3 and not a power of two, Vn does lie in the interior of an orbit tile.
The proof of this theorem lies in §9 of the paper, and the rest of the paper including the other main results of the paper are unaffected. The statement of the
theorem is correct, but there were mistakes in some of the formulas used in the
proof. This erratum corrects several results from §9 which make up the proof.
We summarize our approach below and along the way describe the associated
unclear points, errors, and corrections.
We now recall some of the context of the paper. We label the edges of our
Euclidean triangles with elements of {1, 2, 3}. The orbit type of a periodic billiard
path in a triangle is the sequence of labels of edges hit. The orbit tile of a periodic billiard path p in a triangle T is the collection of similarity-equivalence
classes of labeled Euclidean triangles which support periodic billiard paths with
the same orbit type as p. A periodic billiard path p in T which hits an even
number of edges in a period has a canonical lift p̃ to the double DT of a triangle T . We remove the vertices from DT , making it homeomorphic to a thrice
punctured sphere. We correctly prove in §9.1 that T lies in the interior of the
orbit tile if and only if p̃ is null homologous in DT . We call a billiard path p
stable if it satisfies these criteria.
Let Vn be as in the theorem. The double DVn has a (punctured) translation
surface cover S(Vn ) with Veech’s lattice property. The surface S(Vn ) is naturally
tiled by triangles isometric to Vn . We normalize the surface by applying a rotation so that at least one of the long sides of a triangle is in this tiling is horizontal.
We let φ : S(Vn ) → DVn be the covering map. The affine automorphism group
of S(Vn ) acts and has the following property:
2010 Mathematics Subject Classification: 37E15.
Key words and phrases: Triangular billiards, periodic orbits, Veech triangles, isosceles triangles,
trigonometric series, unfoldings.
IN
THE PUBLIC DOMAIN AFTER
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133
©2014 AIMS CIENCES
134
W. PATRICK H OOPER
AND
R ICHARD E VAN S CHWARTZ
Lemma 9.5a (Stability)(Restated) Suppose p is a periodic billiard path in Vn .
Then there is a lift of p to a closed geodesic p̃ in S(Vn ). Let x be the homology
class of p̃. Then x = w(γ) for some orientation-preserving affine automorphism
w of S(Vn ) and some homology class γ of a closed geodesic traveling in either the
π
horizontal direction or in the direction of angle 2n
.
This restates all but the last sentence of Lemma 9.5 of the published paper. This
portion of the Lemma is correct, and we correct the last sentence below.
Lemma 9.5 reduces the proof of Theorem 1.5 to understanding
the action of
¡
the affine automorphism group on the homology group H1 S(Vn ), Z ). We need
to do calculations in homology, so we choose a basis B. For each k ∈ {−n +
1, . . . , n − 1} we defined a homology class γk in the paper. This homology class
contains a horizontal geodesic when k is even and a geodesic in the direction
π
of angle 2n
when k is odd. We also define homology classes of curves β−1 and
β−1 around ¡the two two punctures with 2π cone angle. Together, these classes
generate H1 S(Vn ), Z ). These choices are described in the published paper and
depicted by example in Figure 1.
F IGURE 1. The surface S(V5 ) with the curves from the homology
classes γ
. . with
, γ4 , β
and β1 .from
Thethe
curves
from γclasses
Figure 1: The surface
S(V
the
homology
−4 ,.5 )
−1 curves
i with i odd
4 ,. . . , 4 ,
1 and
π
⇡
in direction
and
drawn
in dotted10lines.
Numbers
and drawn
in indicate
dotted lines. Numbers
from
odd
are
in direction
1 . The curvesare
i with i 10
identified
by translation.
indicate edges edges
identified
by translation.
For calculations, we find it convenient to work with the dual space to first
¡
¢ affine
Lemma 9.5 reduces the proof of Theorem 1.5 to understanding the action
of the
homology. We call this dual space cohomology and denote it by H 1 S(Vn ), Z .
automorphism group on the homology group H1 S(Vn ), Z). We need to do calculations in
We have prescribed
a basis for homology, and we use the dual basis as a basis
¢ a basis
homology, so1 ¡we choose
B. For each k 2 { n + 1, . . . , n 1} we defined a homology
for H S(Vn ), Z ,
class k in the paper. This homology class contains a horizontal geodesic when k is even and
B ∗of= angle
{γ∗−n+12n
,⇡γ∗−n+2
, . . k. , γis∗n−2
, γ∗n−1
, β∗−1
, β∗1 define
}.
a geodesic in the direction
when
odd.
We
also
homology classes of
curves We1 note
and that
the
two
two
punctures
with
2⇡
cone
angle.
Together,
these classes
1 around
the covering map φ : S(Vn ) → DVn induces a linear map on homol2
generate
H
S(V
),
Z).
These
choices
are
described
in
the
published
paper
and
depicted by
∼
ogy. 1SincenH1 (DVn , Z ) = Z , by choosing a basis, the covering map determines
example in Figure 1.
OURNAL OF M ODERN DYNAMICS
OLUME 8, N O. 1 (2014), 133–137
For Jcalculations,
we find it convenient to work with theVdual
space to first homology. We
1
call this dual space cohomology and denote it by H S(Vn ), Z . We have prescribed a basis
for homology, and we use the dual basis as a basis for H 1 S(Vn ), Z ,
B⇤ = {
⇤
n+1 ,
We note that the covering map
⇤
n+2 , . . . ,
⇤
n 2,
⇤
n 1,
⇤
1,
⇤
1 }.
: S(Vn ) ! DVn induces a linear map on homology.
E RRATUM : B ILLIARDS
IN NEARLY ISOSCELES TRIANGLES
135
¡
¢
two linear maps φ∗1 , φ∗−1 : H1 S(Vn ), Z → Z . (We describe this choice above
¡
¢
Equation 9.1 of the paper.) We interpret φ∗1 and φ∗−1 as elements of H 1 S(Vn ), Z
and provide correct formulas in Equation 9.1 for their representation in our
basis B ∗ .
For the calculations we also need to understand the action of the affine automorphism group on homology. By Veech’s work, the orientation-preserving
part of the affine automorphism group has the following generators:
• The map σ which acts by translation swapping two 2n-gons making up
the surface.
• The left Dehn multitwist τo in the curves γi with i odd.
• The right Dehn multitwist τe in the curves γi with i even.
The twist τo was improperly called a right Dehn multitwist in the paper. (The
calculations were done for the left Dehn multitwist.) Given this, we correctly
work out the action of these elements on homology in terms of this basis. For
clarity, we provide the derivatives of these affine automorphisms, which are
elements of SL(2, R):
·
¸
·
π
π ¸
2 + cos nπ −2 cot 2n
+ sin nπ
1 2 cot 2n
(1) D(σ) = I . D(τo ) =
.
. D(τe ) =
sin nπ
− cos nπ
0
1
We also work out the push-forward
action
on¡ cohomology
in terms of the
¡
¢
¢
dual basis. We recall that if w : H1 S(Vn ), Z → H1 S(V
¡ n ), Z is¢ an automorphism
¡
¢
of homology, then the push-forward map w ∗ : H 1 S(Vn ), Z → H 1 S(Vn ), Z is
defined to satisfy
¡
¢
¡
¢
¡
¢
¡
¢
w ∗ (η∗ ) (x) = η∗ w −1 (x) for all x ∈ H1 S(Vn ), Z and all η∗ ∈ H 1 S(Vn ), Z .
We provide correct formulas for the push-forward actions of σ, τo and τe (see
Equations (9.3), (9.5) and (9.7)), but improperly used w ∗ to denote the pushforward.
Confusion between push-forward and pull-back led to an error in the last
sentence of Lemma 9.5. It should read:
Lemma 9.5b (Stability)(Corrected) It follows that p = w(γk ) is stable if and
only if for each j ∈ {1, −1}, when w ∗−1 (φ∗j ) is written in the basis B ∗ , the coefficient of γ∗k is zero.
Statement 1 of Theorem 1.5 follows by calculation using the Lemma. This
deals with the case of Vn when n = 2m . In this case, we need to show that
for all i ∈ {1 − n, . . . , n − 1} and all w in the affine automorphism group, either
the coefficient of γ∗i of w ∗ (φ∗1 ) is nonzero or the same coefficient of w ∗ (φ∗−1 ) is
nonzero. This follows directly from Proposition 9.6 of the paper:
Proposition 9.6 (Clarified) Let n ≥ 3 be an integer and let w be an affine automorphism of S(Vn ). Then, there are odd integers r and s so that for all j ∈ {−1, 1},
(
n−1
X
r (i + n) if i odd
∗
∗
w ∗ (φ j ) ≡ j
c(i )γi (mod 2n) where c(i ) =
s(i + n) if i even.
i =1−n
J OURNAL
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V OLUME 8, N O. 1 (2014), 133–137
136
W. PATRICK H OOPER
AND
R ICHARD E VAN S CHWARTZ
The proof proceeds by induction in the affine automorphism group and is correct. This implies statement 1 of Theorem 1.5, since when n is a power of two
no such c(i ) can be zero as i + n ∈ {1, . . . , 2n − 1}.
Section 9.4 of the paper is devoted to a proof of statement 2 of Theorem
1.5. The proof proceeds by finding for each n not a power of two, a choice of
an element w of the affine automorphism group and an i ∈ {1 − n, . . . , n − 1} so
that the homology class w(γi ) projects to a null homologous homology class in
DVn . Since w acts affinely, there is always a geodesic in w(γi ) and this geodesic
descends to a stable periodic billiard path in Vn . The choices of w and i are
incorrect in the paper, and there are errors in the proofs. Correct versions of the
results are stated below.
Proposition 9.7 (Odd case) (Corrected) Suppose n ≥ 3 is odd. Let
w = (τe ◦ τ−1
o )
n−1
2
3−n
◦ τe 2
and i = n − 2.
Then, a closed geodesic in the homology class w(γi ) in S(Vn ) projects to a stable
periodic billiard path in Vn via the folding map S(Vn ) → Vn .
Proposition 9.8 (Even case) (Corrected) Suppose n > 3 is even and not a power
of two. Then n = 2a b for an odd integer b ≥ 3 and an integer a ≥ 1. Let
n
b−1
2
2
w = (τe ◦ τ−1
o ) ◦ τo
and i = n − 2a+1 .
Then, a closed geodesic in the homology class w(γi ) projects to a stable periodic
billiard path in Vn via the folding map S(Vn ) → Vn .
As in the paper, these results are best proved by calculation. It suffices to
prove:
1. The cohomology class φ∗1 + φ∗−1 = 2n(β∗1 + β∗−1 + γ∗0 ) is invariant under the
action of the affine automorphism group.
2. For each n and each w = w(n) and i = i (n), the coefficient of γ∗i in w ∗−1 (φ∗1 )
is zero.
To use Proposition 9.5 to verify stability, we also need to show that the coefficient of γ∗i in w ∗−1 (φ∗−1 ) is also zero. This follows because by statement (1) above,
the coefficients of γ∗i in w ∗−1 (φ∗1 ) and w ∗−1 (φ∗−1 ) sum to zero for any w and any
i 6= 0.
Statement (1) above can be proved easily by induction in the group. Statement (2) involves a lengthy but elementary calculation. We will omit this calculation here, but it can be found in [1].
Since we had these formulas wrong the first time around, it seems worth
discussing the extent to which we have verified them on the second pass. In
addition to the write up of the calculations posted on the arXiv, Propositions
9.7 and 9.8 have been checked for correctness in two ways both involving the
use of Mathematica. First, statement (2) above has been checked directly with
Mathematica by programming in the linear maps associated to the group elements. Second, the billiard path on triangle Vn can be determined explicitly
by finding a point on the path and the direction in which the billiard path is
J OURNAL
OF
M ODERN DYNAMICS
V OLUME 8, N O. 1 (2014), 133–137
E RRATUM : B ILLIARDS
IN NEARLY ISOSCELES TRIANGLES
137
moving in at this point. We can always choose this point to be the midpoint
M of the long side of the obtuse isosceles triangle Vn . The lifts of M to S(Vn )
are permuted by the action of the affine automorphism group. Let w and i be
as in the propositions above. The homology class γi is realized by a geodesic
passing through two lifts of M . Call one such lift M 1 . A geodesic in the class
of w(γi ) is determined by the image w(M 1 ) and the direction of the geodesic.
The point w(M 1 ) can be worked out by understanding how the affine automorphism group permutes the lifts of M . The direction of a geodesic in the class
w(γi ) is given by applying D(w) to the horizontal direction (if i is even) or the
π
direction of angle 2n
(if i is odd). This involves understanding the linear action
of derivatives of affine automorphisms, which is described by Equation 1. Once
the billiard path is found, it can be verified for stability using the standard combinatorial test for stability. This combinatorial test appears as Lemma 2.2 of the
published paper and as Lemma 3.3.1 of [2].
Acknowledgments. The authors would like to thank Yilong Yang for pointing
out the errors described above.
R EFERENCES
[1] W. P. Hooper and R. E. Schwartz, Billiards in nearly isosceles triangles, J. Mod. Dyn., 3 (2009),
159–231.
[2] S. Tabachnikov, Billiards, Panor. Synth., (1995), vi+142 pp.
W. PATRICK H OOPER <[email protected]>: Department of Mathematics, City College of
New York, 160 Convent Avenue, New York, NY 10031, USA
R ICHARD E VAN S CHWARTZ <[email protected]>: Department of Mathematics, Brown University, Providence, RI 02912, USA
J OURNAL
OF
M ODERN DYNAMICS
V OLUME 8, N O. 1 (2014), 133–137