Ogouyandjou, Carlos

English

IMHOTEP: African Journal of Pure and Applied Mathematics

Topological entropy of minimal geodesics on
surfaces
Carlos Ogouyandjou
Instutut de Mathematiques et de Sciences Physiques (IMSP)
Universite d'Abomey-Calavi
Republique du Benin
e-mail: ogouyandjou@imsp-uac.org
June 18, 2007
Abstract
Let (M; g) be a compact Riemannian manifold of hyperbolic type, i.e
M is a manifold admitting an another metric of strictly negative curva-
ture. In this paper we study the geodesic ow ~t restricted to the set of
geodesics which are minimal on the universal covering. In particular for
surfaces we show that the topological entropy coincides with the volume
entropy generalizing work of Freire and Ma~ne.
1 Introduction
Let (M; g) be a compact Riemannian manifold and X its universal Riemannian
covering. In [24], Manning introduced the volume entropy h(g) of M dened
by:
h(g) := lim
r!+1
1
r
log volBr(p);
where p 2 X and Br(p) denotes the ball with center p and radius r. He proved
that this limit exists and is independent of p. Let h(t) denotes the topological
entropy of the geodesic ow t on the unit tangent bundle SM . Manning proved
that the volume entropy is less or equal to the topological entropy. In the case
of nonpositive curvature he showed that equality holds. Subsequently this was
generalized by Freire and Ma~ne to metric without conjugate points (see [13] or
[24]). Let ~SM be the set of v 2 SM such that the lift of the geodesic cv with
_cv(0) = v is a globally minimizing geodesic. We denote by ~ffit the restriction on
~SM of the geodesic ow ffit. In [19] Katok and Hasselblatt stated the following
theorem:
Theorem 1.1. Let (M; g) be a compact Riemannian manifold, X be its uni-
versal Riemannian covering. Let ~SX dened as follows:
~SX := fv 2 SX j cv is a minimizing geodesic g
and ~SM := dp( ~SX), where p : X ! M is the covering map. Let ~ffit be the
restriction to ~SM of the geodesic ow ffit. Then,
h(~ffit)  h(g)
1
Let now (M; g) be a compact manifold of hyperbolic type. Then, there exists
a metric of strictly negative curvature g0 on M . The universal Riemannian
covering X0 of (M; g0) is a Hadamard manifold satisfying KX0   k20 for some
constant k0 > 0. Hence X0 and X are Gromov hyperbolic spaces. Therefore,
the distance function on X is 4-convex. This help us to justify the following
theorem.
Theorem 1.2. Let (M; g) be a compact Riemannian manifold of hyperbolic
type. There is some constant 0 depending only on (M; g) such that:
h(~ffit; 0)  h(g):
We will use the notion of entropy expansiveness for the following proposition:
Proposition 1.3. Let (M; g) be a compact Riemannian manifold of hyperbolic
type. If ~t is h-expansive for some constant   0, we have:
h(~ffit) = h(~ffit; 0) = hg:
As an application, we proof the following result:
Theorem 1.4. Let (M; g) be an orientable compact surface of genus  2. Let
~ffit be the restriction to ~SM of the geodesic ow ffit. Then,
h(~ffit) = h(g):
This paper is organized as follows. In the rst section we give a complete
proof using the ideas provided by Katok and Hasselblatt that the topological
entropy of the minimal geodesics is bounded below by the volume growth. In the
second section we study topological entropy of minimal geodesics on manifolds
of hyperbolic type and show that for surfaces the entropy equals the volume
growth.
2 Topological entropy of minimal geodesics
Let (V; d) be a compact metric space and ffit : V ! V be a continuous ow. For
each r > 0, we dene a new distance function
dr(v; w) := max
0tr
d(ffit(v); ffit(w)):
Let F be a subset of V . A set Y  F is called a (dr; )-separated set of F
if for dierent points y; y0 2 Y , dr(y; y0) > . Let sr(F; ) denotes the maximal
cardinality of a (dr; )-separated set.
A set Z  V is called a (dr; )-spanning set of F if for each y 2 F , there
exists z 2 Z such that dr(y; z)  : Let tr(F; ) denotes the minimal cardinality
of (dr; )-spanning set. It is easy to see that for all  > 0,
tr(F; )  sr(F; )  tr(F; 
2
):
Furthermore
h(ffit; F; ) := lim
r!1
1
r
log sr(F; ) = lim
r!1
1
r
log tr(F; ):
(see [27] or [28] for details).
2
Denition 2.1. The topological entropy h(ffit) of the ow ffit : V ! V is dened
by:
h(ffit) := lim
!0
h(ffit; V; ):
In the sequel we will need the following lemma which is similar to Lemma
2.1 in [3]
Lemma 2.2. Let (V; d) be a compact metric space, ffit : V ! V a continuous
ow and A be a subset of V . Then for all sequences 0 = t0 < t1 <    < tk = t
and  > 0
kY
i=1
sti ti 1(ffi
ti 1A; )  st(A; 2);
where sr(B; ) is the maximal cardinality of a (dr; )-separating set of B.
Proof. Let L = Et(A; 2) be a maximal (dt; 2)-separating set of A and for each
i 2 f1; : : : ; kg let Li = Eti ti 1(ffiti 1A; ) be a maximal (dti ti 1 ; )-separated
set of ffiti 1(A).
For each (x1; : : : ; xk) 2 L1      Lk consider the set
B(x1; : : : ; xk) = fz 2 L j d(ffis+ti 1z; fsxi)  ; 0  s  ti   ti 1g:
On the other hand, the set L is (dt; 2)-separated and, therefore, the triangle
inequality implies that the cardinality of each B(x1; : : : ; xk) is at most 1.
Therefore,
St(A; 2) = cardL 
kY
i=1
cardLi =
kY
i=1
cardSti ti 1(ffi
ti 1A; )
We need the following Theorem stated in the book of Katok and Hasselblatt
on the topological entropy of minimal geodesic on Riemannian manifolds. Even
though the main ideas of the proof they provide is correct, it contains some
inaccuracy and a mistake. Therefore and for the convenience of the reader we
will provide a complete proof of the result.
Theorem 2.3. Let (M; g) be a compact Riemannian manifold, X be its uni-
versal Riemannian covering. Let ~SX dened as follows:
~SX := fv 2 SX j cv is a minimizing geodesic g
and ~SM := dp( ~SX), where p : X !M is the covering map.
Let ~ffit be the restriction to ~SM of the geodesic ow ffit. Then,
h(~ffit)  hg
Proof. Fix x 2 X, T;  > 0, and a maximal 3-separated set N in the annulus
B(x; (1 + )T ) B(x; T ). If KT := supy2M vol(B(y; 3T )) then
#N  1
KT
(vol(B(x; T ))  vol(B(x; (1  )T )))  e(hg)(1 3)T
3
for sucient large T , where hg is the volume entropy of the manifold M .
Consider y 2 N and a minimizing geodesic cy joining x and y. If y1 and y2
are two distinct elements of N , and pi = cyi(T ), we have
d(p1; p2)  d(y1; y2)  d(y1; p1)  d(y2; p2)  T:
Thus the set
S := f _cy(0) j y 2 Ng
is (dT ; T )-separated in SX. Let us assume that T is at least as big as the
injectivity radius inj (M) of M . and consider the projections y :=   cy. Then
the set
f _y(0) j y 2 Ng
is (dT ;
1
2 inj(M))-separated in SM .
Consider the set
V (x;N; T ) := f _y(t) j y 2 N; t 2 [
p
T ; T  
p
T ]g =
[
p
TtT 
p
T
ffitS
This is a subset of
MT := fv 2 SM j 9s 2 [ T+
p
T ; 
p
T ]; such that cv is minimal on [s; s+T ]g;
since for each v = _y(t) 2 V (x;N; T ) the geodesic cv is minimal on [ t; T   t]
and  t 2 [ T + pT ; pT ]. Note, that for each v 2 MT the geodesic arc
cv : [ 
p
T ;
p
T ]! SM is minimal. Therefore,
\
T>0
MT = ~SM:
Let st(A; ) be the maximal cardinality of a (dt; )- separated set of A. Then
by lemma 3.2
sT 
p
T (ffi
p
TS; )  spT (S; )  sT (S; 2)  eh(g)(1 3)T
where  = 1=4inj (M)
Let h(ffit) be the topological entropy of the unrestricted geodesic ow. Then
there exists T0 > 0 such that for all T > T0
spT (S; )  spT (SM; )  e2h(ffi
t)
p
T
Therefore, choosing T such that
p
T  2h(ffit)
h(g) , we obtain
sT 
p
T (ffi
p
TS; )  eh(g)T (1 4)
Choose t0 2 (0; T   2
p
T ] and let m 2 N and k 2 [0; t0) be such that
T   2pT = mt0 + k. For j 2 f0; : : : ;m  1g consider the sets
Lj := ffi
jt0ffi
p
TS = f _y(
p
T + jt0) j y 2 Ng
By the considerations above they are all subsets of MT . For 0  j  m  1 let
lj be the maximal cardinality of a (dt0 ; =2)-separated subset of Lj . Applying
Lemma 3.2 again, we obtain
l0  l1  : : :  lm 1  sT pT (ffi
p
TS; )  eh(g)T (1 4):
4
Then, for some j 2 f0; 1; : : : ;m  1g we must have
lj = e
h(g)(1 4)T
m = eh(g)(1 4)(t0+
k+2
p
T
m
)  e(hg)(1 4)t0
Thus for all T large enough the setMT and hence ~SM contains a (dt0 ; inj(M)8 )-
separated set of cardinality at least e(h(g)(1 4)t0 . This implies that for all  > 0
the estimate h(~ffit)  hg   4 holds.
3 Manifolds of hyperbolic type
Denition 3.1. A Riemannian manifold (M; g) is called of hyperbolic type if
there exists another Riemannian metric g0 of strictly negative curvature (back-
ground metric) which is Lipschitz equivalent to g.
Note that, in dimension 2, a manifold M is of hyperbolic type if and only if
its genus  2.
Denition 3.2. Let (X1; d1) and (X2; d2) be two metric spaces. A map  :
X1  ! X2 is called a quasi-isometric map, if there exist constants A > 1 and
 > 0 with:
1
A
d1(x; y)    d2((x);(y))  Ad1(x; y) +  8x; y 2 X1:
In a metric space X, a quasi-geodesic (resp. quasi-geodesic ray) is a quasi-
isometric map  : R  ! X (resp.  : R+  ! X).
Denition 3.3. Let (X; d) be a metric space and A;B  X. The Hausdor
distance of A and B is dened as follow:
dH(A;B) = inffr j A  Tr(B); B  Tr(A)g;
where
Tr(A) := fx 2 X j d(x;A)  rg:
Theorem 3.4. (Morse Lemma) Let (X0; g0) be a Hadamard manifold with
sectional curvature KX0   k0 < 0 for some constant k0 > 0. Then for each
quasi-geodesic (resp. quasi-geodesic ray) c : R  ! X (resp. c : R+  ! X),
there exists a geodesic (resp. geodesic ray)  : R  ! X (resp.  : R+  ! X)
such that d(c(t); (t))  r0 (resp. d(c(R+); (R+))  r0); r0 depends only on
A,  and k0.
Proof. (see [20])
Denition 3.5. A function f : R ! R is called k-convex if for all x; y 2 R,
and t 2 [0; 1],
f(tx+ (1  t)y)  tf(x) + (1  t)f(y) + k:
Lemma 3.6. Let (X; g) be a Hadamard manifold of hyperbolic type and c1 :
[0; a] ! X and c2 : [0; b] ! X two minimizing geodesic arcs such that c1(0) =
c2(0) = 0. Then there exists a constant 0 depending only on (X; g) and the
background metric g0 such that
d(c1(ta); c2(tb))  td(c1(a); c2(b)) + 0
for all t 2 [0; 1].
5
Carlos: Could you give a proof of this. In the book of Coornaert
etc. they require that the geodesics are globally minimizing and not
only on an intervall.The proof should follow from the Morse lemma
and the fact that up to constants a triangle is a tripod.
Proof.
Using this Lemma one obtains:
Proposition 3.7. (See [7], [8] or [9]) Let (X; g) be a Hadamard manifold of
hyperbolic type and c1; c2 : [a; b] ! X be two minimizing geodesic arcs. Then
the function
f : R  ! R
t 7 ! d(c1(t); c2(t))
is -convex.
Proof. Is an easy application of the lemma above using 2 triangles.
Proof it!
Now we like to proof that in the class of manifolds of hyperbolic type the
topological entropy of the minimal geodesic ow is equal to the volume growth.
Let (M; g) be a Riemannian manifold of hyperbolic type and let g0 be a second
metric of negative curvature (background metric). Denote by SM and SX the
unit tangent bundles ofM and its universal coverX with respect to the metric g.
Let F be a fundamental domain in X of diameter a,  a small positive number.
For some large r consider the set Fr := fz 2 X j r   a  dg(z;F)  rg which
for each x 2 F is contained in the ball B(x; r + a) . Let F r be some maximal
(d; )-separated subset of Fr, where d is the metric induced by the Riemannian
metric g. Then
#F r  CvolB(x; r + a+

2
);
where C =
1
inf
y2M
volB(y; 2 )
Let F be some maximal -separated set of F . For any y 2 F, z 2 F r , let
yz be the geodesic with respect to the background metric g0 joining y to z.
Choose for each such y; z a vector v = vyz 2 SyX such that cv : [0;1)! R is a
minimizing g- geodesic with cv(+1) = yz(+1). Let Pr be the set of all such
vectors, i.e.
Pr := fvyz j y 2 F; z 2 F r g
Lemma 3.8. There exists a constant  > 0 such that for each minimizing
geodesic c : [0;1)! R with c(0) 2 F there exists v 2 Pr such that
d(cv(t); c(t))  
for all t 2 [0; r]. In particular, if ~SF is the set of the unit tangent vectors in F
corresponding to minimal geodesics
[
v2Pr
Bdr (v; ) ff ~SF
6
Proof. Let c : [0;1) ! R be a minimal geodesic with c(0) 2 F . This implies
that c(r) is contained in Fr. Since F and F r are maximal (d; )-separated
sets, there exists x 2 F and z 2 F r such that d(c(0); y) and d(c(r); z) are
at not bigger than . Consider the vector v = vyz 2 Pr. Since the function
f : [0; r] ! R with f(t) = d(c(t); cv(t)) is - convex and f(0)   as well as
f(r)   the proof follows for  = 2+ .
Now we study the cardinality of separated sets of minimal geodesics on an
innitesimal scale. For that we have to restrict ourself to surfaces.
Lemma 3.9. Let M be a surface and F as above. Then for all  > 0 there exists
0 such that for all (dr; )- separated sets E of ~Bdr (v; ) :=
~SF \Bdr (v; ) we
have that
#E  r0
Proof. The following proof does not work without modication. The
reason is that minimal geodesics might intersect once and therefore
the balls in the construction below might intersect. Can you give a
modication? For a given small  > 0 and some n 2 N, let E  Z(v)
be a (n; )-separated set (with respect to ~ffit), where v 2 ~SM . Let wk and
wk+1 be two distinct elements of E. Then, there is tk 2 [0; n + 1] such that
lk := d(cwk(tk); cwk+1(tk))  . Let  be the minimizing geodesic satisfying
(0) = cwk(tk) and (lk) = cwk+1(tk). Let us put Pk := (

5 ). Since wk and
wk+1 are elements of Z(v), d(cwk(t); cwk+1(t))  2 8t 2 R. Let B(Pk; 5 )
denote the geodesic ball of radius 5 about Pk.
1.case B(Pk;

5 )
T
cwk+1(]tk; +1[) 6= ;
Let s > 0 such that cwk+1(tk + s) 2 B(Pk; 5 ). Then:
d(cwk+1(0)); cwk+1(tk + s)  d(cwk+1(0); cwk(0)) +
+d(cwk(0); cwk(tk)) +

5
; hence,
tk + s  d(cwk+1(0); cwk(0)) + tk +

5
: Therefore,
d(cwk+1(0); cwk(0))  s 

5
:
Otherwise,
d(cwk+1(tk)); cwk+1(tk + s)  d(cwk+1(tk); cwk(tk))  d(cwk(tk); cwk+1(tk + s)):
Then, s     
5
:
Therefore, d(cwk+1(0); cwk(0)) 

5
:
2.case B(Pk;

5 )
T
cwk+1([0; tk]) 6= ;
Let s > 0 such that cwk+1(tk   s) 2 B(Pk; 5 ). In the same way as above, we
obtain:
d(cwk+1(0); cwk(0)) 

5
:
Rest of the proof of lemma 4.4 Since tk > 0, the ball B(pk;

5 ) lies in
the -tubular neighborhood T;v;n of the geodesic segment
fcv(t) such that t 2 [ 1;n+ 1]g:
7
Let know Pn; denote the number of disjoint geodesic balls of radius 5 lying in
T;v;n. Since dim(M) = 2, there is a constant C > 0 such that the volume of
T;v;n is smaller than C(n+ 2). Then,
inf
p2Z(v)
vol(B(pk;

5
))Pn;  C(n+ 2): Hence,
Pn;  C 0(n+ 2);
where C 0 is a constant depending on . Moreover, if P 0n; is the number of
wk 2 Z(v) such that d(cwk+1(0); cwk(0))  5 , we have #P 0n;  10 . Then,
#E  Pn; +P 0n;  C 0   (n+ 2) + 10 . Hence, if s(n; ; Z(v)) denotes the
maximal cardinality of a (n; )-separated subset of Z(v), we obtain,
s(n; ; Z(v))  C 0(n+ 2) +
10

:
Therefore,
lim
n!+1
(n) 1 log s(n; ; Z(v)) = 0
This implies that h(~ffit; ) = 0.
Corollary 3.10. For each   0 there exists a -separated set L of ~SF such
that
#L  #Prr0
From Proposition 4.3 and the previous Corollary, we obtain the following
result:
Theorem 3.11. Let (M; g) be an compact surface of genus  2, X be its
universal Riemannian covering. Let ~SX dened as follows:
~SX := fv 2 SX j cv is a minimizing geodesic g
and ~SM := dp( ~SX), where p : X !M is the covering map.
Let ~ffit be the restriction to ~SM of the geodesic ow ffit. Then,
h(~ffit) = h(g):
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10


Topological entropy of minimal geodesics on surfaces

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Topological entropy of minimal geodesics on surfaces

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