Topological conjugacy of cascades generated by gradient flows on the two-dimensional sphere

POLONICI MATHEMATICI LXXIII (2000)

Topological conjugacy of cascades generated by gradient flows on the two-dimensional sphere

by Andrzej Bielecki (Krak´ow)

Abstract. This article presents a theorem about the topological conjugacy of a gra- dient dynamical system with a constant time step and the cascade generated by its Euler method. It is shown that on the two-dimensional sphere S

²

the gradient dynamical flow is, under some natural assumptions, correctly reproduced by the Euler method for a suffi- ciently small time step. This means that the time-map of the induced dynamical system is globally topologically conjugate to the discrete dynamical system obtained via the Euler method.

1. Introduction. In recent years several papers have been devoted to studying the qualitative properties of discrete-time dynamical systems ob- tained via discretization methods. The basic question is whether the quali- tative properties of continuous-time systems are preserved under discretiza- tion. Various concepts of differentiable dynamics were investigated. Re- sults on stability and attraction properties ([KL]), the saddle-point structure about equilibria ([AD], [Bey1], [Bey2]), invariant manifolds ([BL], [Fec1]), averagings ([Fec2]) and algebraic-topological invariants ([MR]) can be men- tioned as examples. A number of applications have been studied as well ([Gar4]). The investigations are concerned with both local (see, for instance, [Gar1], [Fec3]) and global conjugacy ([Gar2], [Gar3], [Gar5]).

This paper is devoted to the problem of topological conjugacy between the discretization of a gradient dynamical system and the cascade generated by its Euler method. Similar problems have been solved in recent years for numerical methods of order greater than one (see [Gar2], [Li]).

2. Topological conjugacy of gradient cascades.As mentioned above we consider a gradient differential equation and its Euler method. The time- map of the induced solution is compared to the cascade obtained via the

2000 Mathematics Subject Classification: 34C35, 34A50.

Key words and phrases: topological conjugacy, gradient dynamical system, Euler method.

[37]

Euler method. We show that on the two-dimensional sphere S

²

a gradient dynamical system is, under some natural assumptions, correctly reproduced by the Euler method for a sufficiently small time step. This means that the time-map of the induced dynamical system is globally topologically conju- gate to the discrete dynamical system obtained via the Euler method. This fact can be expressed as follows.

Theorem 2.1. Let S

²

be the unit sphere in R

³

and let φ : S

²

× R → S

²

be the dynamical system generated by a differential gradient equation

(1) x = − grad E(x), ˙

where E ∈ C

²

(S

²

, R), having a finite number of singularities, all hyperbolic.

Let , furthermore, the dynamical system φ have no saddle-saddle connec- tions. Moreover , let φ

h

: S

²

→ S

²

be the discretization of φ, i.e. φ

h

(x) :=

φ(x, h), and let ψ

h

: S

²

→ S

²

be generated by the Euler method for (1).

Then, for sufficiently small h > 0, there exists a homeomorphism α = α

h

: S

²

→ S

²

globally conjugating the cascades generated by φ

h

and ψ

h

, i.e.

(2) φ

h

◦ α = α ◦ ψ

_h

.

Remarks. 1. Axiom A and the strong transversality condition are known to be equivalent to the structural stability of a dynamical system (see [PM], p. 171, and [Man]). On the other hand, for gradient dynamical systems, Axiom A implies that the system has only a finite number of singularities, all hyperbolic, whereas the strong transversality condition implies that the system has no saddle-saddle connections. Thus, the structural stability of a dynamical system (S

²

, φ) implies the assumptions of Theorem 2.1. Moreover, the set of structurally stable systems is open and dense in the space of gradient dynamical systems (see [PM], p. 116).

2. A dynamical system generated by the equation (1), having only a finite number of singularities, all hyperbolic, without saddle-saddle connections is called a gradient Morse–Smale system.

3. Estimation of the Euler method on S

ⁿ

. Let n

0

∈ N and let a = n

0

h denote the length of the time interval on which the error e

n

:=

%

M

(φ

ⁿ_h⁰

(x), ψ

_hⁿ⁰

(x)) is estimated (%

M

denotes the Riemannian metric on M). We will show that on the n-dimensional sphere

(3) e

n

< ξ(a)h,

where, for a given problem, ξ(a) is a constant value which only depends on a.

Consider the problem

(4) x = f (x), ˙ x(0) = x

0

, 0 ≤ t ≤ a,

on a compact manifold M, where f is a vector field on M. The Euler iterative rule for the problem (4) is of the form

(5) x

n

= exp

_xn

(−hf (x

n−1

)), where f (x

n−1

) is a vector of the tangent space T

xn−1

M.

The Euler method in R

ⁿ

is a first order numerical method ([Kru], p. 31).

For x, y ∈ M in the domain of a chart ϑ of a manifold M, we have (6) m

1

%(ϑ(x), ϑ(y)) ≤ %

_M

(x, y) ≤ m

2

%(ϑ(x), ϑ(y)),

where m

1

, m

2

are constant for a given chart ϑ (see [Rob], p. 453, formula (2.2)); % is the euclidian metric on R

ⁿ

. By (6), the Euler method on a compact manifold is also a first order method.

The error in a single step of a numerical method is defined by r(x, h) := %

_M

(ψ

h

(x), φ

h

(x))

and is a continuous function of x for a constant h. Therefore, on a compact manifold, it reaches its maximum. Set

r(h) := max

x∈M

r(x, h).

Let us estimate the error of the Euler method for a gradient equation on the sphere S

ⁿ

in R

ⁿ⁺¹

. By the assumptions of Theorem 2.1 a gradient dynamical system on a compact manifold has at least one attracting singu- larity. Change coordinates in R

ⁿ⁺¹

so that this fixed point is the south pole of the sphere. The sphere can be covered by two charts ϑ

1

, ϑ

2

with

(7) ϑ

1

: S

ⁿ

\ p

_north

→ R

ⁿ

, ϑ

2

: S

ⁿ

\ p

_south

→ R

ⁿ

, where p

north

, p

south

are the poles of the sphere.

If the south pole is the starting point of the Euler method then

%

M

(φ

ⁿ_h

(x), ψ

ⁿ_h

(x)) = 0

for each n ∈ N as the gradient is zero at the south pole. In this case the error is zero.

In the other case, if the starting point x is different from p

south

then

φ

ⁿ_h

(x) 6= p

south

for each n ∈ N, because ψ

h

is invertible for h sufficiently

small. Thus, for each n ≤ n

0

, both φ

ⁿ_h

(x) and ψ

ⁿ_h

(x) lie in the domain

of the chart ϑ

2

. Therefore, the point x and the systems φ

h

and ψ

h

can

be transformed into R

ⁿ

in order to perform the iterations e ψ

ⁿ_h⁰

(ϑ

2

(x)) and

φ e

ⁿ_h⁰

(ϑ

2

(x)). Afterwards, we return to the sphere via ϑ

⁻¹₂

. The dynamical

systems e ψ

h

and e φ

h

are the systems ψ

h

and φ

h

transformed from the sphere

to R

ⁿ

by the maps of the atlas. Notice that if ψ

h

is generated by the Euler

method of the gradient equation generating φ

h

on the sphere then e ψ

h

is

generated by the Euler method of the equation generating e φ

h

in R

ⁿ

because

the atlas preserves the differential structure.

The error estimate with step h for the equation

˙

x = f (x),

where f is lipschitzian with a constant L in R

ⁿ

, is given by the formula (see [Kru], p. 32, formula 1.22)

(8) e

n

≤



e

0

+ rh L

 e

^La

,

where a = n

0

h is the time interval on which the solution is considered, rh

²

is the maximal error in a single step and e

0

is the initial error. Since the right side of (1), being a differentiable map on a compact space, is a lipschitzian map, so is the right side of the equation transformed to R

ⁿ

, by (6). Therefore, the estimate (8) can be applied to the systems e ψ

h

and e φ

h

. The flow and its Euler method start from the same point, thus e

0

= 0 and

%( e ψ

_hⁿ⁰

(ϑ

2

(x)), e φ

ⁿ_h⁰

(ϑ

2

(x))) ≤ rh L e

^La

. By (6) we get the following estimate on S

ⁿ

:

(9) %

M

(ψ

_hⁿ⁰

(x), φ

ⁿ_h⁰

(x)) ≤ m

2

rh L e

^La

,

which is of the same form as in R

ⁿ

. Thus we have obtained (3).

4. Lemmas. One of the key points in the proof of Theorem 2.1 is a construction of proper homeomorphisms conjugating the cascades φ

h

and ψ

h

in a neighbourhood of an attracting singularity (see Lemma 4.6). This construction is based on the following geometric lemma.

Lemma 4.1. Let γ

1

, γ

2

, δ

1

, δ

2

be curves in R

²

, each parametrized by τ ∈ [0, 1], homeomorphic to a line segment and such that γ

1

(0) = δ

1

(0), γ

1

(1) = δ

2

(0), γ

2

(0) = δ

1

(1) and γ

2

(1) = δ

2

(1). Assume that their union is the boundary of a simply connected domain D. Then there exists a homeo- morphism Λ : D → [0, 1]

²

such that Λ(γ

1

) = [0, 1] × {1}, Λ(γ

2

) = [0, 1] × {0}, Λ(δ

1

) = {1} × [0, 1], Λ(δ

2

) = {0} × [0, 1].

The proof of this well known fact is omitted. It can be found in [Bie].

A great number of theorems concerning the topological conjugacy near hyperbolic singularities are known. We will need the following theorem (see [Bey], [Gar2], [Gar3]).

Theorem 4.2. For each equilibrium point x

⁰

of the cascade φ

h

and for

sufficiently small h there exists an equilibrium point x

h

of the cascade ψ

h

,

a neighbourhood V

x0

of x

0

, and a homeomorphism α

h

: V

x0

→ α

_h

(V

x0

) such

that α

h

(x

0

) = x

h

and (α

h

◦ φ

_h

)(x) = (ψ

h

◦ α

_h

)(x), whenever x and φ

h

(x)

are in V

x0

.

In our case, since x

h

is an equilibrium point of ψ

h

iff gradE(x

h

) = 0, we have x

h

= x

0

for every h > 0.

Throughout this section M denotes a compact differentiable manifold of dimension greater than one, φ denotes a Morse–Smale gradient system on M, and φ

h

and ψ

h

are the cascades defined as in Theorem 2.1. Let, furthermore,

g

ci

: V

ci

→ V

_c^∗_i

be a homeomorphism locally conjugating φ

h

and ψ

h

in the neighbourhood V

ci

of the singularity c

i

and V

_c^∗_i

= g(V

ci

).

Lemma 4.3. Let c

i

be an attracting or saddle singularity. Then there exists r

i

> E(c

i

) such that for every x 6= c

i

in W

_φ^s

(c

i

) ∩ V

ci

the following implication holds: if E(x) ≤ r

i

then there exists t > 0 such that

E(φ(x, −t)) > r

i

and φ(x, −t) ∈ W

_φ^s

(c

i

) ∩ V

i

. Furthermore the sets

K

ci,0

: = {x ∈ W

_φ^s

(c

i

) : E(x) ≤ r

i

},

K

ci,1

: = {x ∈ W

_φ^s

(c

i

) : E(x) ≤ r

i

}, where E(c

i

) < r

i

< r

i

, are nonempty.

P r o o f. The point c

i

is attracting on the stable manifold W

_φ^s

(c

i

). Choose R < min{%

M

(c

i

, c

j

) : i 6= j, c

j

is a fixed point}.

Let B

rel

(c

i

, R) be the closed ball and S

rel

(c

i

, R) the sphere in W

_φ^s

(c

i

). By compactness there exists x

0

∈ S

_rel

(c

i

, R) such that

E(x

0

) = inf{E(x) : x ∈ S

rel

(c

i

, R)} =: e

i

.

The following proposition is necessary to complete the proof of Lemma 4.3. Its proof follows simply from the Morse lemma and therefore is omitted (it can be found in [Bie]).

Proposition 4.4. Let e

i

> r

^∗

> E(c

i

). Then the set B

lev

(c

i

, r

^∗

) := {x ∈ W

_φ,c^s _i

: E(x) ≤ r

^∗

} and the ball B

rel

(c

i

, R) are homeomorphic.

Since the potential E decreases along each orbit, every r

i

< r

^∗

(where r

^∗

is defined in Corollary 4.4) satisfies the implication in Lemma 4.3. That implies that the sets K

ci,m

, m = 0, 1, are nonempty.

Lemma 4.5. The sets K

^ci,m

, m = 1, 2, have the following properties:

(i) c

i

∈ int

_rel

K

ci,m

,

(ii) ∂

rel

K

ci,0

= {x ∈ W

_φ,c^s _i

: E(x) = r} and ∂

rel

K

ci,1

= {x ∈ W

_φ,c^s _i

: E(x) = r},

(iii) {φ(x, t) : t > 0} ⊂ K

ci,m

for every x ∈ K

ci,m

,

(iv) K

ci,m

is arcwise connected ,

(v) inf{%

_M

(x, y) : x ∈ K

ci,1

, y ∈ K

ci,0

} > 0.

The boundaries and interiors are considered in the relative topology of W

_φ,c^s _i

and therefore are marked by the subscript “rel”.

P r o o f. (i) This is clear since K

ci,m

is homeomorphic to a ball B

rel

(c

i

, r).

(ii) Let x

0

∈ ∂

_rel

K

ci,0

. Then, for every neighbourhood V

x0

of x

0

, V

x0

∩ {x ∈ W

_φ,c^s _i

: E(x) ≤ r} 6= ∅ and V

x0

∩ {x ∈ W

_φ,c^s _i

: E(x) > r} 6= ∅.

Since the map E is continuous, E(x

0

) is equal to r.

Let x

0

∈ W

_φ,c^s _i

. The definition of K

ci,m

implies that x

0

∈ K

ci,0

. Assume, by contradiction, that x

0

∈ int

_rel

K

ci,0

. Then there exists t

0

> 0 such that φ(x

0

, −t) ∈ K

ci,0

for every t ≤ t

0

. However, for those t,

E(φ(x

0

, −t)) > E(φ(x

0

, 0)) = E(x

0

) = r.

This means that φ(x

0

, t) 6∈ K

ci,0

, a contradiction.

The property of K

ci,1

can be proved in the same way.

(iii) Let x

0

∈ ∂

rel

K

ci,0

. Since the potential decreases along orbits, E(φ(x

0

, t)) < E(x

0

) for every t > 0. Hence φ(x

0

, t) ∈ K

ci,0

by the defi- nition of K

ci,m

.

If x

0

∈ int

_rel

K

ci,0

, then E(x

0

) < r. Since the set K

ci,0

= B

lev

(c

i

, r) is homeomorphic to a ball B

rel

(c

i

, R) such that K

ci,0

⊂ B

_rel

(c

i

, R) (see Corollary 4.4), the set {φ(x

0

, t) : t > 0} would intersect the boundary

∂

rel

K

ci,0

if the semiorbit φ(x

0

, t), t > 0 were not included in K

ci,0

. However this is impossible because the potential on the boundary of the set K

ci,0

is greater than in the interior.

The property of K

ci,1

can be shown in the same way.

(iv) Define τ : R 3 t 7→ τ (t) ∈ [−1, 1] by τ (t) =

( −1 if t = −∞,

t/(1 + |t|) if t ∈ R,

1 if t = ∞.

Let x

1

, x

2

∈ K

_ci,m

, where m equals 1 or 2. Then the formulas γ

¹₂

τ (t)
= φ(x

1

, t) and γ 1 −

¹₂

τ (t)
= φ(x

2

, t)

define an arc from x

1

to x

2

; the arc is included in K

ci,m

by Lemma 4.5(ii).

(v) The assertion follows from the compactness of M. Assume, by con- tradiction, that inf{%

M

(x, y) : x ∈ K

ci,1

, y ∈ K

ci,0

} = 0. Then there exists a sequence {x

n

} ⊂ K

_ci,1

which converges to a point x

0

∈ ∂

_rel

K

ci,0

. But E(x

0

) = r and E(x

n

) ≤ r, which is a contradiction because r < r and E is continuous.

The following lemma is necessary for the construction of a global hom-

eomorphism conjugating φ

h

and ψ

h

.

Lemma 4.6. Let a be an attracting fixed point of the flow φ on a two- dimensional manifold M. Let , furthermore, s

k

, k ∈ {1, . . . , K}, be saddle points whose stable manifolds intersect the stable manifold of a. Then there exists a neighbourhood V

a

of a and a homeomorphism g

a

defined on this neighbourhood locally conjugating the flows (M, φ

h

) and (M, ψ

h

) such that for every k ∈ {1, . . . , K},

(10) g

a

(W

_φ^u_h

(s

k

) ∩ V

a

) ⊂ W

_ψ^u_h

(s

k

).

P r o o f. We first record some facts and introduce a few definitions. As M is two-dimensional, the components of the unstable manifolds W

_φ^u

h

(s

k

) of the saddle points are curves with a common end at a. Choose a level set of E on which E > E(a) and such that its connected component is included both in W

_φ^s_h

(a

i

) and in a sufficiently small neighbourhood of a. Denote this component by F

1

. Since near an attracting singularity a level is homeo- morphic to a circle (see Corollary 4.4), this level can be parametrized as F

1

= F

1

(τ ) with F

1

(0) = F

1

(1), τ ∈ [0, 1], where the map F

1

is continuous.

The set W

_φ^u

(s

k

) \ {s

k

, a} has two connected components, each being an orbit of φ. As already mentioned, the curve F

1

is homeomorphic to a circle and the value of the potential decreases along an orbit. Thus, every connected component of S

K

k=1

(W

_φ^u

h

(s

k

) \ {s

k

, a}) intersects F

1

in exactly one point.

Let there be M such components and let they be numbered from 0 to M − 1 according to the increasing τ, say W

_φ^u_h,m

, m = 0, . . . , M − 1.

Set F

2

:= φ

h

(F

1

). The curve F

2

does not intersect F

1

and is homeomor- phic to F

1

. The point a lies in the interior of the domain bounded by F

2

. Therefore the set

P

F1,F2

:= {φ(x, t) : x ∈ F

1

, t ∈ [0, h]}

is homeomorphic to a closed annulus B(a, R) \ int B(a, r), where r < R.

Define

δ

m

:= W

_φ^u_h,m

∩ P

_F1,F2

, m = 0, . . . , M − 1, (11) Λ : [0, 1] × [0, h] 3 {τ, t} 7→ Λ(τ, t) = φ(F

1

(τ ), t) ∈ P

F1,F2

.

The mapping Λ is a homeomorphism because it is a superposition of homeo- morphisms and there exists a finite sequence τ

0

= 0, τ

1

, . . . , τ

M −1

such that Λ(τ

m

, [0, h]) = δ

m

.

Now, we begin the proof of Lemma 4.6.

Step 1: Construction on fragments of W

φ^uh

(s

k

). Let g

sk

denote a home- omorphism locally conjugating φ

h

and ψ

h

near s

k

and let φ

−h

be the inverse of φ

h

. The formula

(12) g

sk,a

(x) := (ψ

_h^n0(x)

◦ g

_sk

◦ φ

ⁿ_−h^0(x)

)(x)

defines a homeomorphism conjugating φ

h

and ψ

h

on δ

m

. The natural number

n

0

(x) is chosen in such a way that φ

ⁿ_−h^0(x)

(x) ∈ K

sk,1

and φ

ⁿ_−h^0(x)−1

(x) ∈ int

rel

(K

sk,0

\K

sk,1

). Note that the parameter m uniquely identifies the saddle point s

k

.

The mapping

g

a

:= g

sk,a

, x ∈ δ

m

,

defines a homeomorphism conjugating φ

h

and ψ

h

on S

M −1

m=0

δ

m

. The map- pings ψ

h

, g

sk

and g

a

are continuous in h and each is the identity for h = 0.

Step 2: Construction on a fragment of the annulus. Set

γ

1,m

:= {x = F

1

(τ ) : τ ∈ [τ

m

, τ

m+1

]}, γ

2,m

:= {x = F

2

(τ ) : τ ∈ [τ

m

, τ

m+1

]}, where τ

m

is the value of τ for which

F

j

(τ

m

) = F

j

∩ W

_φ^u_h,m

=: x

j

(m), j ∈ {1, 2}.

Note that γ

j,m

, j = 1, 2, m = 0, . . . , M − 1, is a curve with end points x

j

(m), x

j

(m + 1) whereas δ

m

is a curve with end points x

1

(m), x

2

(m). All these curves are homeomorphic to line segments. This implies that γ

1,m

∪ γ

2,m

∪ δ

_m

∪ δ

_m+1

bounds a simply connected domain in R

²

. By the Riemann Theorem this domain is homeomorphic to a ball and hence to a rectangle.

By Lemma 4.1 there exists a homeomorphism Λ

m

:= Λ|

[τm,τm+1]×[0,h]

from [τ

m

, τ

m+1

] × [0, h] onto the closed domain Sq

_{γ1,m,γ2,m,δm,δm+1}

bounded by the curves γ

1,m

, γ

2,m

, δ

m

and δ

m+1

. The mapping Λ has the following properties:

Λ

m

(τ

m

, [0, h]) = δ

m

, Λ

m

([τ

m

, τ

m+1

], 0) = γ

1,m

, Λ

m

([τ

m

, τ

m+1

], h) = γ

2,m

.

Perform a similar construction for the cascade ψ

h

. Set x

^∗_j

(m) := g

a

(x

j

(m)).

Let a curve F

₁^∗

:= g

a

(F

1

) parametrized by τ be homeomorphic to a circle and such that a is in the interior of the simply connected domain bounded by F

₁^∗

. Let, furthermore, the points x

^∗₁

(m), m ∈ {0, . . . , M − 1}, lie in F

₁^∗

and let the parametrization have the property

F

₁^∗

(τ

m

) = x

^∗₁

(m), m = 0, . . . , M − 1.

Let F

₂^∗

:= ψ

h

(F

₁^∗

) and suppose ψ

h

(F

₁^∗

(τ )) = F

₂^∗

(τ ) for all τ. Set δ

^∗_m

:= g

a

(δ

m

),

and let δ

_m^∗

be parametrized in such a way that

δ

_m^∗

(h

0

) := g

a

(δ

m

(h

0

)), h

0

∈ [0, h].

The homeomorphism g

a

is a continuous function of h and is the identity for h = 0, so for sufficiently small h, the curve δ

^∗_m

lies near δ

m

and does not intersect any other δ

m⁰

, m

⁰

6= m. Thus, if m increases then so does the value of τ which parametrizes F

₁^∗

. Therefore, F

₁^∗

can be reparametrized in such a way that the values of τ at x

j

(m) equal those at x

^∗_j

(m), m = 0, . . . , M − 1.

Define

γ

_1,m^∗

:= {x ∈ F

₁^∗

(τ ) : τ ∈ [τ

m

, τ

m+1

]}, γ

_2,m^∗

:= {x ∈ F

₂^∗

(τ ) : τ ∈ [τ

m

, τ

m+1

]}.

Arguing as for φ

h

, we can show that the domain bounded by F

₁^∗

and F

₂^∗

is homeomorphic to an annulus whereas the domain Sq

_γ^∗

1,m,γ_2,m^∗ ,δ_m^∗,δ^∗_m+1

bounded by γ

_1,m^∗

, γ

_2,m^∗

, δ

_m^∗

and δ

_m+1^∗

is homeomorphic to a rectangle. By Lemma 4.1 the homeomorphism can be chosen to map the vertices of the rectangle to the points x

^∗₁

(m), x

^∗₂

(m), x

^∗₁

(m + 1) and x

^∗₂

(m + 1). Denote it by

ξ

m

: Sq

_γ^∗

1,m,γ_2,m^∗ ,δ^∗_m,δ^∗_m+1

→ [τ

_m

, τ

m+1

] × [0, h].

It has the following properties:

ξ

m

(γ

_1,m^∗

) = [τ

m

, τ

m+1

] × {0}, ξ

m

(γ

_2,m^∗

) = [τ

m

, τ

m+1

] × {h},

ξ

m

(δ

^∗_m

) = {τ

m

} × [0, h].

Furthermore, ξ

m

can be constructed in such a way that on the curves δ

_m^∗

it is consistent with the mapping g

m,a

(the parameter m determines the saddle point s

k

uniquely):

(13) g

m,a

(φ(x

m

, t)) = ξ

m

(τ

m

, t), t ∈ [0, h], and

(14) g

m+1,a

(φ(x

m

+ 1, t)) = ξ

m

(τ

m+1

, t), t ∈ [0, h].

This can be shown in the following way. Equations (13) and (14) imply that on the vertical sides τ

m

×[0, h] and τ

m+1

×[0, h] the parametrization is settled by increasing homeomorphisms f

1

: [0, h] → [0, h] and f

2

: [0, h] → [0, h].

We will show that the mapping g : [τ

m

, τ

m+1

] × [0, h] 3 {τ, t} 7→

g(τ, t) = (g

1

(τ, t), g

2

(τ, t)) ∈ [τ

m

, τ

m+1

] × [0, h]

defined by

g

1

(τ, t) = τ, g

2

(τ, t) = τ

m+1

− τ

τ

m+1

− τ

_m

· f

1

(t) + τ − τ

m

τ

m+1

− τ

_m

· f

2

(t)

is a homeomorphism with the required properties. We will prove that it

transforms the rectangle [τ

m

, τ

m+1

] × [0, h] onto itself. Fix τ

0

∈ [τ

_m

, τ

m+1

]

and t

0

∈ [0, h]. Without losing generality we can assume that f

₁

(t

0

) ≤ f

2

(t

0

).

Then

g

2

(τ

0

, t

0

) ≥ τ

m+1

− τ

₀

τ

m+1

− τ

_m

· f

₁

(t

0

) + τ

0

− τ

_m

τ

m+1

− τ

_m

· f

₁

(t

0

) = f

1

(t

0

) ∈ [0, h]

and

g

2

(τ

0

, t

0

) ≤ τ

m+1

− τ

0

τ

m+1

− τ

_m

· f

2

(t

0

) + τ

0

− τ

m

τ

m+1

− τ

_m

· f

2

(t

0

) = f

2

(t

0

) ∈ [0, h].

Furthermore, g

2

(τ

m

, ·) = f

1

(·) and g

2

(τ

m+1

, ·) = f

2

(·). The jacobian jac g =  ∂g

1

/∂τ ∂g

1

/∂t

∂g

2

/∂τ ∂g

2

/∂t



= τ

m+1

− τ τ

m+1

− τ

m

· df

1

(t)

dt + τ − τ

m

τ

m+1

− τ

m

· df

2

(t) dt is positive at each point because both components are nonnegative (f

1

and f

2

are increasing) and do not equal zero simultaneously.

The superposition

Υ

m

= Λ

m

◦ g ◦ ξ

m

is a homeomorphism transforming the closure of Sq

_γ^∗

1,m,γ^∗_2,m,δ^∗_m,δ_m+1^∗

onto the closure of Sq

_{γ1,m,γ2,m,δm,δm+1}

in such a way that for each x ∈ δ

m

∪ δ

_m+1

,

(15) Υ

m

(x) = g

m,a

(x).

Step 3: Construction on the annulus. By (13) and (14), Υ

m

(x)=Υ

m+1

(x) for each x ∈ δ

m+1

and Υ

0

(x) = Υ

M −1

(x) for x ∈ δ

0

. Therefore the mapping defined as

Υ (x) = Υ

m

(x) for x ∈ Sq

_γ^∗

1,m,γ^∗_2,m,δ^∗_m,δ_m+1^∗

and m ∈ {0, 1, . . . , M − 1}

is a homeomorphism transforming the “annulus” P

F₁^∗,F₂^∗

onto the “annulus”

P

F1,F2

.

Step 4: Construction on the neighbourhood of a. Extend Υ to the whole neighbourhood of a. Let y ∈ P

F₁^∗,F₂^∗

, and define x ∈ V

a

by

x = ψ

^k(x)_h

(y), k ∈ N.

Set

(16) Υ e

a

(x) :=







(ψ

^−k(x)_h

◦ Υ ◦ φ

^k(x)_h

)(x)

if ∃y ∈ P

F₁^∗,F₂^∗

: x = ψ

_h^k(x)

(y), k ∈ N,

a for x = a.

For points of the curve F

₂^∗

the mapping e Υ

a

is defined in two ways. First,

F

₂^∗

⊂ P

_F1^∗_,F2^∗

so we can take zero as the value of k. On the other hand we

can take k = 1 because every point of F

₂^∗

is the image of a point of F

₁^∗

.

However, both φ

h

and ψ

h

preserve values of τ so for each x ∈ F

₂^∗

both the

ways give the same image. This also implies that e Υ

a

is continuous on F

₂^∗

.

To prove the continuity of e Υ

a

at a consider a sequence {x

n

} converging to a. For each n there exists a natural number k

n

such that ψ

_h^−kn

(x

n

) ∈ P

F₁^∗,F₂^∗

. Furthermore, k

n

→ ∞. Thus e Υ

a

(x

n

) → a. Hence, the function

Υ e

a

: P

F₁^∗,F₂^∗

→ P

F1,F2

is a homeomorphism.

Since on the sets δ

m

, m = 0, . . . , M − 1 (see (11)) the mapping Υ is defined by the local conjugating homeomorphism g

sk

(see (12) and (15)) which transforms W

_φ^u_h

(s

k

) onto W

_ψ^u_h

(s

k

), the inclusion (10) is satisfied.

Introduce the following notations:

D

_φh

— the set of all repelling points,

S

_φh

— the set of all saddle points and the attracting points which are not contained in the closure of the unstable manifold of any saddle point, P

_φh

— the set of attracting points which are contained in the closure of the

unstable manifold of a saddle point.

Define

W

_φ^s_h

(S

φh

) := [

c∈S_φh

W

_φ^s_h

(c), W

_φ^s_h

(P

φh

) := [

c∈P_φh

W

_φ^s_h

(c),

and Θ : W

_φ^s_h

(S

φh

) → M by

(17) Θ(x) =  (ψ

_−hⁿ⁰

◦ g

_c

◦ φ

ⁿ_h⁰

)(x) for x ∈ W

_φ^s_h

(c) \ {c},

x for x = c,

where c ∈ S

φh

, g

c

is a local homeomorphism conjugating the flows φ

h

and ψ

h

on a neighbourhood of c and ψ

−h

is the inverse of ψ

h

. If x is in one of the sets K

c,1

, then the natural number n

0

= n

0

(x) is zero. In the other case it is chosen in such a way that φ

ⁿ_h⁰

(x) ∈ K

c,1

and φ

ⁿ_h⁰⁻¹

(x) ∈ int

rel

(K

c,0

\ K

_c,1

).

Define α : M → M as follows:

(18) α(x) =







Θ(x) for x ∈ W

_φ^s_h

(S

φh

),

Υ e

a

(x) for x ∈ W

_φ^s_h

(P

φh

), a ∈ P

φh

, x for x ∈ D

φh

.

Every nonrepelling point on a compact manifold is contained in a stable manifold of an attracting or saddle singularity. Thus (18) defines α on the whole M. In the next section it is shown that α is a homeomorphism globally conjugating the flows (M, φ

h

) and (M, ψ

h

).

Lemma 4.7. For all positive constants ε, there exists a positive constant h

0

such that for all x ∈ M, t ∈ R and 0 < h < h

0

,

%

_M

(φ(x, t), φ(x, t + h)) < ε.

The proof of this simple lemma is omitted (it can be found in [Bie]).

Lemma 4.8. Let a be an attracting fixed point of the flow φ, and p

1

, p

2

repelling fixed points, p

1

6= p

₂

. Assume that p

1

, p

2

∈ cl W

_φ^s

(a). Then there exist saddle points q

1

, q

2

, not necessarily different , such that p

1

∈ cl W

_φ^s

(q

1

) and p

2

∈ cl W

_φ^s

(q

2

).

P r o o f. Since p

1

, p

2

∈ ∂W

_φ^s

(a), there exist y

1

and y

2

such that

t→∞

lim φ(y

1

, t) = a, lim

t→−∞

φ(y

1

, t) = p

1

,

t→∞

lim φ(y

2

, t) = a, lim

t→−∞

φ(y

2

, t) = p

2

.

Let z

1

∈ orb(y

₁

) ∩ V

a

and z

2

∈ orb(y

₂

) ∩ V

a

, where orb(y) denotes the orbit of y, and let V

a

⊂ W

_φ,a^s

be a neighbourhood of a. The manifold M is locally arcwise connected, so let

l : [0, 1] 3 τ 7→ l(τ ) ∈ X

be a closed arc included in V

a

, avoiding a and such that l(0) = z

1

, l(1) = z

2

. Since z

1

and z

2

lie in the unstable manifolds of p

1

and p

2

respectively, the boundaries of the unstable manifolds intersect l in points w

1

and w

2

, not necessarily different. As the unstable manifolds of repelling points are open and M is compact, the points w

1

, w

2

are contained in the unstable manifolds of saddle points q

1

, q

2

, not necessarily different. From the λ-lemma it follows that for every neighbourhood V

qi

of q

i

there exists t > 0 such that

φ(l ∩ W

_φ^u

(p

1

), −t) ∩ V

q1

6= 0.

We have

(W

_φ^s

(q

1

) \ {q

1

}) ∩ V

_q1

6= 0.

Thus, for every ε > 0 there exists t > 0 such that φ(l∩W

_φ^u

(p

1

), −t) intersects the ε-envelope of W

_φ^s

(q

1

). This implies that there exists u ∈ W

_φ^s

(q

1

) such that

t→−∞

lim φ(u, t) = p

1

.

Therefore, the point p

1

∈ cl W

_φ^u

(q

1

). The same can be said about the points q

2

and p

2

.

Corollary 4.9. Let p be a repelling point which is not contained in the closure of the stable manifold of any saddle point. Then there exists only one attracting point a such that p ∈ cl W

_φ^s

(a).

This follows easily from Lemma 4.8.

Corollary 4.10. Let M = S

ⁿ

, n > 1. Then, for every h > 0, the cascade (S

ⁿ

, φ

h

) has no saddle-saddle connections and , for sufficiently small h, the cascade (S

ⁿ

, ψ

h

) has no such connections either.

P r o o f. By the assumptions of Theorem 2.1, the flow (S

ⁿ

, φ) has no

saddle-saddle connections. Thus each saddle point q

i

, i = 1, . . . , I, has a

neighbourhood U

qi

such that every point x

0

∈ U

_qi

lies in W

_φ^s

(q

i

) or in W

_φ^s

(a

j

) for an attracting point a

j

. For every x ∈ S

ⁿ

and h > 0,

t→∞

lim φ(x, t) = lim

n→∞

φ

ⁿ_h

(x), hence x

0

∈ W

_φ^s

h

(q

i

) or x

0

∈ W

_φ^s

h

(a

j

).

Define

P

i

:= {x ∈ U

qi

: ∀

n∈{1,2,...}

ψ

_hⁿ

(x) 6∈ U

qi

}

and let U

qi

⊂ int V

_qi

, where V

qi

is a neighbourhood on which the cascades φ

h

and ψ

h

are locally conjugate. Choose real numbers R and h

i

such that B(q

i

, R) ⊂ int U

qi

and P

i

∩ B(q

_i

, R/2) 6= ∅ for every h ∈ [0, h

i

].

Step 1: We will show that P

ⁱ

∩ W

_φ^s

(q

i

) = ∅ for sufficiently small h

i

. Define

t

i

:= sup

x∈U_qi∩W_φ^s(qi)

{t : φ(x, t) ∈ B(q

_i

, R/4) \ B(q

i

, R/8)}.

It is obvious that t

i

∈ (0, ∞). Suppose h

_i

is such that there exists a natural number n

1

with t

i

= n

1

h

i

and

ξh

i

< R/4,

where ξ is the constant from (3). Since there are only a finite number of singularities, we can take ξ as the maximum value of the constants of all saddle points.

Let x ∈ U

qi

∩ W

_φ^s

(q

i

). Then

φ

ⁿ_hi¹

(x) ∈ B(q

i

, R/4) and by (3) and the choice of h

i

,

%

M

(φ

ⁿ_hi¹

(x), ψ

_hⁿ_i¹

(x)) < R/4, or equivalently

ψ

ⁿ_hi¹

(x) ∈ B(q

i

, R/2).

Since P

i

∩ B(q

_i

, R/2) = ∅, the point x is not in P

i

.

Step 2: Proof of lemma. Let x ∈ P

i

. The first step implies that x 6∈ W

_φ^s

(q

i

). Since there are no saddle-saddle connections, the point x is in W

_φ^s

(a

j

) for an attracting point a

j

. Let V

aj

be the neighbourhood of a

j

on which φ

h

and ψ

h

are topologically conjugate. Decompose the set P

i

into disjoint components in the following way:

P

ij

:= {x ∈ P

i

: x ∈ W

_φ^s

(a

j

)}.

Let, furthermore,

t

ij

:= sup{t : φ(x, t) ∈ B(a

j

, r/2) \ B(a

j

, r/4), x ∈ P

ij

},

where r is chosen in such a way that B(a

j

, r) ⊂ V

aj

. It is obvious that

t

ij

∈ (0, ∞). Denote by e

_n

the error after the nth step of the Euler method.

Choose h

ij

so small that φ

h

and ψ

h

are topologically conjugate on V

aj

and ξh

ij

< r/2. Moreover, let t

ij

= n

2

h

ij

, n

2

∈ {1, 2, . . .}. Then φ(x, t

_ij

) ∈ B(a

j

, r/2) for every x ∈ P

ij

. By (3) and the choice of h

ij

,

e

n

:= %

M

(φ

ⁿ_h²_ij

(x), ψ

_hⁿ_ij²

(x)) < r/2.

Hence

ψ

_hⁿ²

ij

(x) ∈ B(a

j

, r).

Assuming that there are I saddle points and J attracting points, set

(19) h

0

:= min

i∈{1,...,I}, j∈{1,...,J }

{h

_ij

}.

As V

aj

⊂ W

_φ^s

h0

(a

j

) and the cascades are locally topologically conjugate, we have V

aj

⊂ W

_ψ^s

h

(a

j

) for every h ∈ [0, h

0

]. Hence ψ

ⁿ_h²

(x) ∈ W

_ψ^s_h

(a

j

).

Therefore, every x ∈ U

qi

is either in W

_ψ^s_h

(q

i

) or in W

_ψ^s_h

(a

j

), where a

j

is an attracting point. This implies that ψ

h

has no saddle-saddle connections.

Remark. By Lemma 4.10 and Theorem 4.2, if (S

ⁿ

, φ) is a gradient Morse–Smale system, then so are the cascade (S

ⁿ

, φ

h

) (for every positive h) and (S

ⁿ

, ψ

h

) (for sufficiently small h).

Corollary 4.11. Let U

qi

be a neighbourhood of saddle point q

i

on which the cascades φ

h

and ψ

h

are conjugate by a homeomorphism g

qi

: U

qi

→ g

qi

(U

qi

). Then there exists a constant h

0

> 0 such that for every h ∈ (0, h

0

) and x ∈ U

qi

∩ W

_φ^u

h

(q

i

) if x ∈ W

_φ^s_h

(a

j

), where a

j

is an attracting fixed point , then g

i

(x) ∈ W

_ψ^s_h

(a

j

).

P r o o f. Step 1. Define

(20) H

ij

:= W

_φ^u_h

(q

i

) ∩ (B(q

i

, r

i

) \ int B(q

i

, r

i

/2)) ∩ W

_φ^s_h

(a

j

).

First, it will be shown that there exists h

0

> 0 so small that for every h ∈ (0, h

0

) and every x ∈ H

ij

we have g

i

(x) ∈ W

_ψ^s

h

(a

j

). The radius r

i

is chosen such that B(q

i

, r

i

) ⊂ U

qi

.

The definition (20) implies that H

ij

is closed in M, hence compact.

The stable manifold W

_φ^s_h

(a

j

) is open and H

ij

⊂ W

_φ^s

h

(a

j

). Thus, for every x ∈ H

ij

there exists r

x

> 0 such that B(x, r

x

) ⊂ W

_φ^s

h

(a

j

). The set K := {B(x, r

_x

) ∩ W

_φ^u_h

(q

i

)}

is a covering of H

ij

. Thus, we can choose a finite subcovering K

^∗

. Let r

ij

be the smallest radius of the balls B(x, r

x

) used to construct K

^∗

. Take h

0ij

> 0 such that g

qi

(x) ∈ B(x, r

ij

) for every point x ∈ H

ij

. Then g

qi

(x) ∈ W

_φ^s_h

(a

j

).

It can be shown (in the same way as in the second step of the proof of

Corollary 4.10) that there exists h

1ij

such that g(H

ij

) ⊂ W

_ψ^s_h

(a

j

) for every

h ∈ (0, h

1ij

). Set h

ij

:= min{h

0ij

, h

1ij

}.

Step 2. Suppose, by contradiction, that there exists y ∈ W

φ^uh

(q

i

) ∩ int B(q

i

, r

i

/2) such that y

1

∈ W

_φ^s

h

(a

j

) and z

1

:= g

i

(y

1

) ∈ W

_φ^s_h

(a

k

), where a

k

is an attracting fixed point and k 6= j. Corollary 4.7 implies that, for sufficiently small h, there exists a natural m

0

such that

φ

^m_h⁰

(y

1

) ∈ B(q

i

, r

i

) \ int B(q

i

, r

i

/2).

As y

1

∈ W

_φ^s

h

(a

j

), the point y

2

:= φ

^m_h⁰

(y

1

) is in W

_φ^s_h

(a

j

). Therefore, by Step 1, z

2

:= g

i

(y

2

) ∈ W

_ψ^s_h

(a

j

) as y

2

∈ P

_ij

. However

z

2

:= g(y

2

) = g(φ

^m_h⁰

(y

1

)) = (g ◦ φ

h

)(φ

^m_h⁰⁻¹

(y

1

))

= (ψ

h

◦ g ◦ φ

_h

)(φ

^m_h⁰⁻²

(y

1

))

= (ψ

h

◦ ψ

_h

◦ g)(φ

^m_h⁰⁻²

(y

1

)) = . . . = ψ

_h^m0

(g(y

1

)) = ψ

_h^m0

(z

1

), and ψ

^m_h⁰

(z

1

) ∈ W

_ψ^s_h

(a

k

) as z

1

∈ W

_ψ^s

h

(a

k

), which leads to a contradiction.

5. Proof of Theorem. Firstly, we prove that the map α defined in (18) is a bijection. Then we prove that it is continuous on M. This implies that α is a homeomorphism. The map α conjugates the cascades φ

h

and ψ

h

, which follows directly from its definition (see (16)–(18)).

Injectivity. Let x

1

, x

2

∈ S

²

. If at least one of them is a repelling singu- larity of the system φ, then α(x

1

) 6= α(x

2

) by the definition of α. Otherwise the following two cases have to be considered:

Case 1: The points lie in different orbits. Then y

1

:= φ

ⁿ_h^0(x1)

(x

1

) 6= φ

ⁿ_h^0(x2)

(x

2

) =: y

2

.

The system φ has only a finite number of singularities and every singularity c

i

is a fixed point of g

i

. The sphere S

²

is compact so every point is contained in the stable manifold of a stable point: x

1

∈ W

_φ^s

(c

i

), x

2

∈ W

_φ^s

(c

j

).

If i = j, then z

1

:= g

i

(y

1

) 6= g

i

(y

2

) = z

2

as g

i

is a bijection. Since g

i

is a local conjugating homeomorphism, the images of different orbits do not intersect. Thus ψ

ⁿ_−h^0(x1)

(z

1

) 6= ψ

ⁿ_−h^0(x2)

(z

2

), which implies that α(x

1

) 6= α(x

2

).

If i 6= j, then we can choose the domains V

i

of the homeomorphisms g

i

in such a way that V

i

∩ V

_j

= ∅ and g

i

(V

i

) ∩ g

j

(V

j

) = ∅ for i 6= j. Thus z

1

:= g

i

(y

1

) 6= g

j

(y

2

) =: z

2

. As x

1

∈ W

_φ^s

h,ai

and x

2

∈ W

_φ^s

h,aj

, the points z

1

and z

2

lie in the disjoint manifolds W

_ψ^s_h,ai

and W

_ψ^s_h,aj

respectively. This implies that α(x

1

) 6= α(x

2

).

Case 2: The points lie in the same orbit. We can assume that E(x

2

) >

E(x

1

). Then x

1

= φ

^m_h

(x

2

) for some positive natural number m = m(x

1

, x

2

).

Thus

α(x

2

) = (ψ

_−h^n0(x1)+m

◦ g

_ci

◦ φ

ⁿ_h^0(x1)+m

)(x

2

)

= (ψ

_−h^m

◦ ψ

_−h^n0(x1)

◦ g

ci

◦ φ

ⁿ_h^0(x1)

)(x

1

) = ψ

^m_−h

(α(x

1

)) 6= α(x

1

).

Surjectivity. If y ∈ S

²

is a repelling fixed point, then y = α(y) (see definition of α). Otherwise y ∈ W

_ψ^s_h,ci

for some fixed point c

i

(attracting or saddle). Let n

0

= n

0

(y) be such that

y

n0−1

:= ψ

_hⁿ⁰⁻¹

(y) ∈ int

rel

(g(K

ci,0

) \ g(K

ci,1

)) and y

n0

:= ψ

_hⁿ⁰

(y) ∈ g(K

ci,0

).

As g is a conjugating homeomorphism,

(φ

⁻¹_h

◦ g

⁻¹

◦ ψ

_h

)(y

n0−1

) = g

⁻¹

(y

n0−1

).

Furthermore, since ψ

h

(y

n0−1

) = y

n0

, we have

(φ

⁻¹_h

◦ g

⁻¹

)(y

n0

) = g

⁻¹

(y

n0−1

).

Set

x

n0

:= g

⁻¹

(y

n0

) and x

n0−1

:= g

⁻¹

(y

n0−1

) = φ

⁻¹_h

(x

n0

).

Since g

ci

is a homeomorphism, x

n0−1

∈ (int

_rel

K

ci,0

\ K

_ci,1

) and x

n0

∈ K

_ci,1

. This means that for x

^∗

= φ

ⁿ_−h⁰

(x

n0

) the natural number n

0

is the same as in the definition of α, which implies that y = (ψ

⁻ⁿ_h ⁰

◦ g ◦ φ

ⁿ_h⁰

)(x

^∗

) = α(x

^∗

).

Continuity

Case 1: Continuity on the stable manifold of an attracting point. Let x lie on the stable manifold of an attracting point a

i

. There exists n

0

= n

0

(x) such that

φ

ⁿ_h⁰

(x) ∈ K

ai,1

and φ

ⁿ_h⁰⁻¹

(x) ∈ int(K

ai,0

\ K

_ai,1

).

If φ

ⁿ_h⁰

(x) ∈ int K

ai,1

, then as int K

ai,1

and int(K

ai,0

\ K

_ai,1

) are open and the map φ(·, t) is continuous, there exists a neighbourhood U

x

of x such that

φ

ⁿ_h⁰

(U

x

) ⊂ int K

ai,1

and φ

ⁿ_h⁰⁻¹

(U

x

) ⊂ int(K

ai,o

\ K

_ai,1

).

As g

i

, ψ

−h

(·, t) and φ

h

(·, t) are continuous, the map α = ψ

ⁿ_−h⁰

◦ g

_i

◦ φ

ⁿ_h⁰

is continuous at x.

If y := φ

ⁿ_h⁰

(x) ∈ ∂K

ai,1

, then every neighbourhood U

y

of y intersects int(K

ai,0

\ K

_ai,1

). Let (y

n

)

^∞_n=1

⊂ int(K

_ai,0

\ K

_ai,1

) converge to y. Then there exists N such that for every natural n greater than N, φ

h

(y

n

) ∈ int K

ai,1

. Indeed, suppose otherwise. Then there exists a subsequence (y

nk

) such that φ

h

((y

nk

)

^∞_k=1

) 6⊂ int K

ai,1

. However, φ

h

(y) ∈ int K

ai,1

because the potential E is constant on ∂K

ai,1

and decreases along a trajectory. This means that for every k, %

_M

(φ

h

(y), φ

h

(y

nk

)) > inf{%

_M

(φ

h

(y), w) : w ∈ ∂K

ai,1

)} > 0.

This is a contradiction because φ is continuous.

We have demonstrated that for every y ∈ ∂K

ai,1

there exists a neigh- bourhood U

_y^?

with φ

h

(U

_y^?

) ⊂ K

ai,1

. The neighbourhood U

_y^?

has two disjoint components: the first one, U

_y^?1

, is included in int K

a1,1

whereas the second, U

_y^?2

, is not. The point x is transformed by the map α in such a way that U

_y^?1

is transformed by g

i

, and U

_y^?2

by ψ

−h

◦ g

_i

◦ φ

_h

. Afterwards U

_y^?1

∪ U

_y^?2

is transformed by the map ψ

_−hⁿ⁰

. But g

i

is a conjugating homeomorphism so

(ψ

−h

◦ g

_i

◦ φ

_h

)(U

_y^?2

) = (ψ

−h

◦ ψ

_h

◦ g

_i

)(U

_y^?2

) = g

i

(U

_y^?2

)

and therefore we can say that the whole neighbourhood U

_y^?

is transformed by g

i

. The continuity of φ

h

, g

i

and ψ

−h

implies that α is continuous at x.

Case 2: Continuity at repelling fixed points. There are two possibilities in this case: either

(i) the repelling point, say p, is in the closure of the stable manifold of a saddle point q, or

(ii) p is in the closures of the stable manifolds of attracting points.

Suppose first that (i) holds.

Step 1: Restriction to the stable manifold. Let {x

k

}

_k∈N

⊂ W

_φ,q^s

, p ∈ cl W

_φ^s

(q) and lim

k→∞

x

k

= p.

Let V

q

be a neighbourhood of q such that the homeomorphism g

h,q

conju- gating φ

h

and ψ

h

is defined on V

q

. For sufficiently small h and every natural N almost all elements of the sequence {x

k

}

_k∈N

have the following property:

φ

ⁿ_h^k

(x

k

) ∈ int K

q,1

, φ

ⁿ_h^k−1

(x

k

) ∈ int(K

q,0

\ K

_q,1

), n

k

> N, which follows from Lemma 4.7 (for the definition of the sets K

q,0

and K

q,1

, see Lemma 4.3). The same lemma also implies that there exists r > 0 such that y

k

= φ(x

k

, n

k

h) 6∈ B(q, r)∩W

_φ^s

(q) = K

q,2

for all k. In other words, since the step on the manifold M is small, all the y

k

lie near ∂K

q,1

. Since q is a fixed point of g

h,q

, g

h,q

(V

q

) is a neighbourhood of q and g

h,q

(V

q

) ∩ W

_φ^s_h

(q) ⊂ W

_ψ^s_h,q

as g

h,q

locally conjugates φ

h

and ψ

h

. Since p is also a repelling point of ψ

h

, it lies in W

_ψ^s_h

(q) (by Corollary 4.11). As g

h,q

is a homeomorphism, g

q

(y

k

) 6∈ g

h,q

(K

q,2

) for every k. Let W

_ψ^p,s_h

(q) be the maximal connected component of W

_ψ^s_h

(q) \ {q} containing p. Then, for every ε > 0, there exists N such that

ψ

_−hⁿ

(W

_ψ^p,s

h

(q) \ K

q,2

) ⊂ B(p, ε) for all n > N.

Since almost all n

k

are greater than N and every y

k

is in W

_ψ^p,s_h

(q) \ K

q,2

, almost all ψ

_−h^nk

(y

k

) lie in B(p, ε). This means that

k→∞

lim ψ

_−h^nk

(g

q

(y

k

)) = p.

Thus, we have shown that the map α restricted to W

_φ^s_h

(q)∪{p} is continuous

at p.

Step 2: Continuity at the repelling point. Let ε > 0. Since α is continuous on the stable manifold of every saddle point (see case 3), for every x

^∗

in such a manifold and ε

1

> 0, there exists δ

1

> 0 such that if x ∈ B(x

^∗

, δ

1

) then α(x) ∈ B(x

^∗

, ε

1

). On the other hand, p is a fixed point of α, and α|W

_φ^s

(q) is continuous at p (see step 1). Thus, for every ε

2

> 0 there exists δ

2

> 0 such that if x

^∗

∈ B(p, δ

2

), then α(x

^∗

) ∈ B(p, ε

2

). Let ε

1

and ε

2

be such that ε

1

+ ε

2

= ε. Then the triangle inequality implies that if x ∈ B(p, δ

1

+ δ

2

) then α(x) ∈ B(p, ε

1

+ ε

2

).

Suppose now that (ii) holds. Corollary 4.9 implies that there exists only one attracting singularity a such that p ∈ W

_φ^s

(a), whereas Lemma 4.8 implies that only one repelling point can lie in cl W

_φ^s

(a). This means that there exists a neighbourhood V

p

of p such that

t→∞

lim φ(x, t) = a

for all x ∈ V

p

\ {p} and there exists a neighbourhood V

_a

of a such that

t→∞

lim ψ(y, −t) = p for all y ∈ V

a

\ {a}.

We can now repeat the first step of subcase (i) to show that the map α is continuous at p.

Case 3: Continuity on the stable manifold of a saddle point. Let q

i

be a saddle point. If the restriction φ

h

|W

_φ^s

h

(q

i

) is considered, then q

i

is an attracting singularity and, repeating the argument from case 1, we have continuity of α in the relative topology on the stable manifold.

Thus, let x

0

∈ W

_φ^s

h

(q

i

) and x ∈ U

x0

\ W

_φ^s

h

(q

i

), where U

x0

is a neighbour- hood of x

0

. Since there are no saddle-saddle connections, there exist attract- ing points a

j

, a

k

, not necessarily different, such that W

_φ^u_h

(q

i

)

1

⊂ W

_φ^s

h

(a

j

) and W

_φ^u_h

(q

i

)

2

⊂ W

_φ^s

h

(a

k

), where W

_φ^u_h

(q

i

)

1

and W

_φ^u_h

(q

i

)

2

are the connected components of the manifold W

_φ^u_h

(q

i

).

For every x

0

∈ W

_φ^s

(q

i

), each neighbourhood V

qi

of q

i

and sufficiently small h > 0, there exists δ

1

> 0 and n

1

such that

φ

ⁿ_h¹

(B(x, δ

1

)) ⊂ V

qi

.

Suppose V

qi

∩ W

_φ^s

(q

i

) = int K

q1,1

, and h and n

1

are such that φ

ⁿ_h¹⁻¹

(B(x

0

, δ

1

) ∩ W

_φ^s

(q

i

)) ⊂ int(K

qi,0

\ K

_qi,1

).

Thus x

ⁿ¹

:= φ

ⁿ_h¹

(x) lies in the local stable manifold W

_φ,loc^s

(q

i

). Denote by D

^u_φ

a disc transversal to W

_φ,loc^s

(q

i

), containing x

ⁿ⁰

, x

ⁿ¹

:= φ

ⁿ_h^1(x)

and embedded in V

qi

. According to the λ-lemma, for every δ

2

> 0 and sufficiently small h, there exists n

2

such that

D

^u_n2

:= φ

ⁿ_h²

(D

_φ^u

)