CUBO A Mathematical Journal
Vol.14, No¯ 03, (59–61). October 2012

Erratum to “on the group of strong symplectic
homeomorphisms”

Augustin Banyaga

Department of Mathematics,

The Pennsylvania State University,

University Park, PA 16802.

email: banyaga@math.psu.edu, augustinbanyaga@gmail.com

ABSTRACT

We give a proof of the estimate (1.1) which is the main ingredient in the proof that the

set SSympeo(M, ω) of strong symplectic homeomorphisms of a compact symplectic

manifold (M, ω) forms a group [1].

RESUMEN

Probamos la estimación (1.1) que es el principal elemento en la demostración que el

conjuntos SSympeo(M, ω) de homeomorfismos simplécticos fuertes de una variedad

simpléctica compacta (M, ω) genera un grupo [1].

Keywords and Phrases: C0-symplectic topology; Strong symplectic homeomorphism.

2010 AMS Mathematics Subject Classification: 53D05; 53D35.



60 Augustin Banyaga CUBO
14, 3 (2012)

1 Erratum

In the paper [1] mentioned in the title, the “constant E” at page 60 may be infinite ( so proposition

2 is meaningless). Therefore, some of the estimates on pages 63 to 65 based on E, needed to show

that ∫1

0

osc(vnt − v
m
t ) → 0 (1.1)

as n, m → ∞ may not hold true. Here is a direct proof of (1.1).

First simplify the notations by writing kHm for (µkt )
∗
H

m, H for H and omitting t. The

function vn := vnt satisfies nH
n − Hn = dvn. Fix a point ∗ in M and for each x ∈ M, pick an

arbitrary curve γx from ∗ to x, then

un(x) :=

∫

γx

(nHn − Hn) = vn(x) − vn(∗).

(The definition of un(x) is independent of the choice of the curve γx). Hence osc(u
n − um) =

osc(vn − vm). Since osc(f) ≤ 2|f|, where |.| is the uniform sup norm, we need to show that

∫1

0

|

∫

γx

(nHn − Hn) − (mHm − Hm)|dt ≤

∫1

0

|

∫

γx

(nHn − mHm)|dt

+

∫1

0

|

∫

γx

(Hn − Hm)|dt, (1.2)

goes to zero , when n, m are sufficiently large.

The last integral tends to zero when n, m are large: indeed,

∫1

0

|

∫

γx

(Hn − Hm)|dt =

∫1

0

|

∫1

0

(Hn − Hm)(γx(u))(γ
′

x(u)du)|dt|

≤ A

∫1

0

|Hn − Hm|dt, (1.3)

where A = supu |γ
′

x(u)|. This goes to 0 since H
n is a Cauchy sequence.

To prove that
∫1
0
|
∫
γx

(nHn − mHm)|dt tends to zero when n, m → ∞, we write:

|

∫

γx

(nHn − mHm)| ≤ |

∫

γx

(nHn − mHn)|

+ |

∫

γx

(m(Hn − Hm) − n0(H
n
− Hm))|

+ |

∫

γx

(n0)(H
n
− Hm)|, (1.4)

for some large n0.



CUBO
14, 3 (2012)

Erratum to “on the group of ...” 61

The integral
∫1

0

|

∫

γx

(n0)(H
n
− Hm)|dt =

∫1

0

|

∫1

0

(Hn − Hm)(γn
0
(u))(Dµ

n0γ′x(u)du)|dt|

≤ B

∫1

0

|Hn − Hm|dt, (1.5)

where B = supu|Dµ
n0γ′x(u)| goes to zero when n, m → ∞ since H

n is a Cauchy sequence and

Dµn0 is bounded. ( Here γk = µ
k(γx)).

We now show that
∫
γx

(nHn − mHn) =
∫
γn

Hn −
∫
γm

Hn tends to zero when n, m → ∞

Let d0 be a distance induced by a Riemmanian metric g and let r be its injectivity radius.

For n, m large enough, supxd0(µ
n
t (x), µ

m
t (x)) ≤ r/2. It follows that there is a homotopy F :

[0, 1] × M → M between µn and µm , i.e F(0, y) = µn(y) and F(1, y) = µm(y) and we may define

F(s, y) to be the unique minimal geodesic v
y
mn(s) joining µ

n(y) to µm(y). See [[3]] ( Theorem

12.9). Let �(s, u) =: {F(s, γn(u)), 0 ≤ s, u ≤ 1}

Since by Stokes’ theorem,
∫
∂�

Hn = 0,
∫
γn

Hn−
∫
γm

Hn =
∫
L
Hn−

∫
L′

Hn where L, and L′ are

the geodesics vxmn and v
∗

mn. The integral over L is bounded by sups|H
n(vxmn(s)|d0(µ

n
t (x), µ

m
t (x)),

because the speed of the geodesics L, L′ is bounded by d0(µ
n
t (x), µ

m
t (x)). This integral tends to

zero when n, m → ∞ since Hn is also bounded . Analogously for the integral over L′.

The same argument apply to Hn − Hm with the geodesics L, L′ replaced by vxmn0 and v
∗

mn0
.

This finishes the proof of (1.1).

Remark : We will show in a forthcoming paper [2] that (1.1) is the main ingredient in the

proof of the main theorem of [1].

Acknowledgments

I would like to thank Mike Usher to have pointed out to me that Proposition 2 in [1] does not yield

a finite constant E.

Received: January 2012. Revised: January 2012.

References

[1] A. Banyaga, On the group of strong symplectic homeomorphisms, Cubo, Vol 12, No 03

(2010), 49-69

[2] A. Banyaga, On the group of strong symplectic homeomorphisms, II, preprint

[3] T. Brocker and K. Janich, Introduction to differential topology, Cambrigde University Press,

1982