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5Russian Journal of Nonlinear Dynamics, 2019, vol. 15, no. 2, pp. 199-211. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd190209
MATHEMATICAL PROBLEMS OF NONLINEARITY
MSC 2010: 37D15
On a Class of Isotopic Connectivity of Gradient-like Maps of the 2-sphere with Saddles of Negative Orientation Type
T. V. Medvedev, E. V. Nozdrinova, O. V. Pochinka, E. V. Shadrina
We consider the class G of gradient-like orientation-preserving diffeomorphisms of the 2-sphere with saddles of negative orientation type. We show that the for every diffeomorphism f G G every saddle point is fixed. We show that there are exactly three equivalence classes (up to topological conjugacy) G = Gi U G2 U G3 where a diffeomorphism f1 G Gi has exactly one saddle and three nodes (one fixed source and two periodic sinks); a diffeomorphism f2 G G2 has exactly two saddles and four nodes (two periodic sources and two periodic sinks) and a diffeomorphism f3 G G3 is topologically conjugate to a diffeomorphism f-1. The main result is the proof that every diffeomorphism f G G can be connected to the "source-sink" diffeomorphism by a stable arc and this arc contains at most finitely many points of period-doubling bifurcations.
Keywords: sink-source map, stable arc
Received June 05, 2019 Accepted June 20, 2019
The construction of a stable arc (Theorem 2) is supported by RSF (Grant no. 17-11-01041), the splitting G into equivalence classes (Theorem 1) is supported by the Basic Research Program at the National Research University Higher School of Economics (HSE) in 2019.
Timur V. Medvedev [email protected]
National Research University Higher School of Economics ul. Rodionova 136, Niznhy Novgorod, 603093 Russia
Elena V. Nozdrinova
Olga V. Pochinka
Elena V. Shadrina
National Research University Higher School of Economics
ul. Bolshaya Pecherckaya 25/12, Niznhy Novgorod, 603155 Russia
1. Introduction
In 1976 S. Newhouse, J. Palis and F. Takens [1] introduced the notion of a stable arc connecting two structurally stable systems on a manifold, the arc being stable in the sense that any nearby arc exhibits the same type of behavior. Soon S. Newhouse and M. Peixoto [2] showed the existence of a stable arc between any two Morse-Smale flows. On the other hand, there are examples of Morse-Smale diffeomorphisms on manifolds of any dimension that cannot be joined by a stable arc. Therefore, it is important to find an invariant which defines the equivalence class of Morse-Smale diffeomorphisms with respect to relation of connectivity by a stable arc (stable connectivity component).
For orientation-preserving structurally stable maps of the circle the Poincare rotation number is this invariant [3]. P. Blachard found [4] some necessary conditions for existence of a stable arc connecting two Morse-Smale diffeomorphisms on surfaces. These conditions, in particular, imply that even on the 2-sphere there are infinitely many stable connectivity components. The simplest example of a 2-dimensional diffeomorphism is the "source-sink" system on the 2-sphere. The main result of this paper is the proof that a gradient-like orientation-preserving diffeomor-phism of the 2-sphere with saddles of negative orientation type can be joined by a stable arc to the "source-sink" diffeomorphism.
2. Dynamics of G diffeomorphisms
Let Mn be a smooth closed orientable manifold with a metric d and let f : Mn — Mn be an orientation-preserving diffeomorphism. A point x £ Mn is said to be wandering for f if there is an open neighborhood Ux of x such that fn(Ux) n Ux = 0 for all n £ N. Otherwise the point x is called nonwandering. The set of all nonwandering points of f is called the nonwandering set, denoted by Q/.
If Q/ is finite, then each p £ Q/ is periodic and its period mp £ N. A point p £ Q/ is
idfmP\
called hyperbolic if the Jacobian matrix ( Jdx J \p has no eigenvalues on the unit circle. If all
the eigenvalues lie inside (outside) the unit circle, then p is a sink (a source). Sinks and sources are called nodes. If a hyperbolic periodic point is not a node, then it it called a saddle.
(a) i (b) j (c)
X1/ xV î v
t; : A .yf i >. S ! .....
source sink saddle
Fig. 1. (a) source, (b) sink, (c) saddle.
The hyperbolic structure of a periodic point p implies the existence of the stable Ws and the unstable Wpu manifolds defined by
W s = {x G S g : lim d(f k'p er(p)(x),p) = 0},
p
= {x e Sg : ? lim d(f-k'per(p)(x),p) = 0};
rs \JTU
k^+œ
lim
k^+œ
W ^ and W^ are diffeomorphic to Rn_qp, Rqp, respectively, where qp is the number of the eigenvalues of the Jacobian matrix outside the unit circle.
Stable and unstable manifolds are called invariant manifolds. A path-component of the set
\ p (Wp \ p) is called an unstable (stable) separatrix.
Periodic data of the periodic orbit Op of a periodic point p is the collection (mp, qp, vp) where mp is the period of p, qp = dim W'U and vp is the type of orientation, meaning p: p = +1 (p = —1) if fmppreserves (reverses) the orientation. For orientation-preserving diffeomorphisms the orientation type of each node is +1, whereas the orientation type of a saddle can be +1 as well as — 1.
A diffeomorphism f : Mn — Mn is called a Morse-Smale diffeomorphism if
1) the nonwandering set Qf consists of a finite number of hyperbolic orbits;
2) the manifolds Wp, WU of each two distinct nonwandering points p and q intersect transver-sally.
A Morse - Smale diffeomorphism is called gradient-like if from WS1 = 0 for two distinct points (Ti,a2 E Qf it follows that dimWU1 < dimWU2■
A Morse-Smale flow on a manifold Mn is defined in the same way. A Morse-Smale flow is said to be gradient-like if it has no periodic trajectories.
For n = 2 the dynamics of gradient-like diffeomorphisms is closely interconnected with the dynamics of periodic homeomorphisms. Recall that a homeomorphism 0 : M2 M2 is periodic of order m E N if = id and = id for each natural ¡i <m.
Proposition 1 ([5], Theorem 3.3). Every orientation-preserving gradient-like diffeomorphism f : M2 — M2 is topologically conjugate to the composition of a periodic homeomorphism and the time-1 map of a gradient-like flow.
According to the classification by B. von Kerekjarto [6], periodic points of an orientation-preserving periodic homeomorphism of period m on the 2-sphere can only be of periods 1 and m, whereas the set of fixed points of this diffeomorphism is not empty. Hence, he have the following corollary of Proposition 1.
Proposition 2. Periodic points of an orientation-preserving gradient-like diffeomorphism of the 2-sphere can only be of period 1 or of period m (the case m = 1 is possible). The set of fixed points of this diffeomorphism is not empty.
For an orientation-preserving gradient-like diffeomorphism f : M2 - M2 the following proposition is also true.
Proposition 3 ([5], Lemmas 3.1, 3.3). Let f : M2 — M2 be an orientation-preserving gradient-like diffeomorphism and let mf be the least natural for which Qfmf consists of the fixed points of positive orientation type. Then the period of each saddle separatrix of f equals mf.
Thus, from Propositions 2 and 3 we have
Proposition 4. Let f be an orientation-preserving gradient-like diffeomorphism of the 2-sphere. Then
1) m/ = m;
2) each saddle with negative orientation type is fixed.
Recall that the 2-sphere is a manifold diffeomorphic to S2 = {(xi,x2,x3) £ R3 : x2 + x"^ + + x2 = 1}. Denote by G the set of gradient-like orientation-preserving diffeomorphisms of the 2-sphere with saddles of negative orientation type.
In this paper we give a topological classification of diffeomorphisms of G, namely, we prove the following theorem.
Theorem 1. There are three equivalence classes G = G1 U G2 U G3 up to topological conjugacy, where a diffeomorphism f1 £ G1 has exactly one saddle and three nodes: one fixed source and two periodic sinks (Fig. 2); a diffeomorphism f2 £ G2 has exactly two saddles and four nodes: two periodic sources and two periodic sinks (Fig. 3); a diffeomorphism f3 £ G3 is topologically conjugate to a diffeomorphism f-1 (Fig. 4).
3. Stable arcs in the space of diffeomorphisms
Consider a 1-parametric family of diffeomorphisms (an arc) pt : Mn — Mn,t G [0,1]. Denote by Q the set of arcs {pt} whose end points are Morse-Smale diffeomorphisms and such that:
1) every diffeomorphism has a finite limit set for every t G [0,1];
2) the arc {pt} contains a finite set of bifurcation points bi,...,bm G (0,1).
Following [7], an arc {pt} is called stable if it is an inner point of the equivalence class with respect to the following relation: two arcs {pt}, {p't} G Q are called conjugate if there are homeomorphisms h : [0,1] — [0,1],Ht : Mn — Mn such that h(bi) = b'i,i G {1, ...,m}, Htpt = p'h(t)Ht,t G [0,1] and Ht continuously depend on t.
It is shown in [7] that an arc {pt} gQ is stable if and only if
each diffeomorphism ,i £{1, ..m} has no cycles and it has exactly one nonhyperbolic periodic orbit, this orbit being a flip or noncritical saddle-node, and the arc passes through the bifurcation point in the typical way;
stable and unstable manifolds of any two periodic points of the diffeomorphism , t £ [0,1] intersect transversally (Fig. 5).
<pi = r
saddle-node or flip
fo — f
Fig. 5. An arc from Q.
We say that two diffeomorphisms f0, fi are in the same class of stable isotopic connectivity if they can be joined in the space of diffeomorphisms by an arc of Q.
An arc {<£t} £Q is said to pass through a saddle-node bifurcation in the typical way if in some neighborhood of the nonhyperbolic point (p, bi) the arc is conjugate to
(Pt(x 1 ,x2, ■ ■ ■,x1+nu ,x2+n
xi + -xx +1, ±2x2, • • •, ±2xi+n.u
i Xn ) —
±X2+r
±Xr
where (xi, ■■■,Xn) e Rn, \xi\ < 1/2, \t\ < 1/10.
\ "X \ ''X
\ > \
¥
>
—>
A
Fig. 6. Saddle-node.
An arc {<£t} £ Q is said to pass through period-doubling bifurcation (flip) in the typical way if in some neighborhood of the nonhyperbolic point (p, bi) the arc is conjugate to the arc
<Pt(Xl ,X2, . . .,Xl+nu ,X2+Uu, ...,Xn) =
-Xl{l ± i) + xl, ±2X2, ±2x1+nu, • • •, ^
where (xi, ...,Xn) £ Rn, \xi\ < 1/2, \t\ < 1/10.
The main result of this paper is the following theorem.
Theorem 2. Every diffeomorphism f £ G is connected to the "source-sink" diffeomorphism by a stable arc.
u
u
2
2
Fig. 7. A period-doubling bifurcation (flip).
4. Proof of Theorem 1
In this section we prove that up to topological conjugacy there are exactly three equivalence classes G = G1 U G2 U G3 where a diffeomorphism /1 G G\ has exactly one saddle and three nodes — one fixed source and two periodic sinks; a diffeomorphism /2 g G2 has exactly two saddles and four nodes — two periodic sources and two periodic sinks (Fig. 3); a diffeomorphism /3 G G3 is topologically conjugate to a diffeomorphism /-1 (Fig. 4).
Proof.
Let / g G. It follows from the properties of G that any saddle a of / is of negative orientation type. Then, according to Proposition 4, the point a is fixed. Denote by £J,££J the stable separatrices of a and denote by YJ,yJ the unstable separatrices of a. Consider two cases
1) there is a saddle a such that cl(£j)\£J = cl(£j)\£J or cI(yJ)\yJ = cI(yJ)\yJ;
2) there are no saddles satisfying 1).
In the case 1) to be definite consider cl(£j)\£J = cl(£j)\£J (Fig. 2). Since / is gradient-like, the set cl(£j)\£J = cl(£j)\£J is the source; denote it by a. Thus,
a = cl(£j )\£J = cl(£j )\£J
and then the set K = a U £^ U £2a U a is the circle which is invariant under /. On the ambient 2-sphere this circle is the border of two 2-disks D1 ,D2 such that yj C D1 c D2. Since a is of negative orientation type, we have /(D1) = D2, /(D2) = D1.
Hence, the disk D1 is of period two and, consequently, cannot contain saddle points because they should be fixed. The Euler characteristic of the 2-sphere equals 2, therefore, there are exactly two sinks G D1 ,^2 g D2 lying in the closures of the unstable separatrices y1 , respectively, and such that / (u1) = u2, /(<u2) = u1.
So, in this case the nonwandering set of / is Qf = {a, u, /(u), a}. Having applied the same reasoning for the case cI(yj)\yj = cI(y2)\y^, we come to a diffeomorphism conjugate to /-1.
Consider case 2). Let a1 be a saddle of /. Then all the four saddle separatrices have four distinct nodes in their closures. Let cl(£^j1 )\£^T1 = a,cl(£^ )\£^J1 = /(a) and cI(yJ1 )\yJi =
= u,cl(Yl)\y2i = /(u) (Fig. 8).
Since the Euler characteristic of the 2-sphere equals 2, the nonwandering set of / contains at least one saddle distinct from a1. Since the set S2 \ P is a connected manifold (here S2 is the 2-sphere and P is the set of sources of /), the union of the unstable manifolds of all saddles is connected. Therefore, there exists a saddle a2 such that
cl(Yj2 )\yJ2 = cl(YJi )\yJi = u.
wi ll
7cti W2
Fig. 8. The saddle o^.
Thus, the set K = w U f (w) U U y2x U U is an f -invariant circle. By the same reasoning as in case 1) we find that there are no saddles of f distinct from a1,a2 and, hence,
5. Reduction of diffeomorphisms to the standard form
To decrease the number of periodic orbits of a diffeomorphism f £ G, we are going to construct an arc that passes through a period-doubling bifurcation in the typical way. In order to do this, the diffeomorphisms must be reduced to the standard form. To that end we reduce the dynamics in the neighborhood of a sink to the canonical contraction (Lemmal) and we map the unstable saddle separatrix in the basin of the canonical sink to a smooth arc (Lemma 2). An important tool for all the constructions is
Proposition 5 (Thom's isotopy extension theorem, [8], Theorem 5.8). Let Y be a
smooth manifold without border, let X be a smooth submanifold of Y and let {ft : X ^ Y, t £ [0,1]} be a smooth isotopy such that f0 is the inclusion map of X into Y. Then for every compact set A c Y containing the support1 supp{ft} there is a smooth isotopy {gt £ Diff (Y), t £ [0,1]} such that g0 = id, gt\x = ft\x for every t £ [0,1] and supp{gt} lies in A.
The details of the construction.
Denote by 0(0,0) the origin of coordinates on the plane R2. For every r > 0 let Br = = {(x,y) £ R2 : x2 + y2 < r2}. Denote by g : R2 ^ R2 the diffeomorphism defined by g(x,y) = (x/2,y/2).
Lemma 1. Let a diffeomorphism ^>0 : M2 ^ M2 have a hyperbolic sink w0 of period m and let (U0 ,^0) be a local chart of the manifold M2 such that w0 £ U0, ^0(w0) = O and ) c U0.
1 The support supp{ft} of an isotopy {ft} is the closure of the set {x £ X : ft(x) = fo(x) for some t £
□
[0,1]}.
Then there are neighborhoods U1,U2 of u0 such that U2 C Uj C U0 and there is a bifurcation free arc pt : M2 — M2, t G [0,1] such that
m— 1
1) the diffeomorphism pt, t G [0,1] coincides with p0 outside the set [J pk(U1) and the set
k=0
m— 1
U pQ(u0) is the hyperbolic sink orbit of period m for every pt; k=00
2) the diffeomorphism Pm 1 coincides with g on the set (U2).
Proof.
Let (q = Pm and (q = : R2 — R2. To prove the lemma it suffices to construct
neighborhoods (lj, U2 of O such that U2 C Uj C R2 and a bifurcation free arc (t : R2 — R2,t G [0,1] such that
1') the diffeomorphism ()t,t G [0,1] coincides with (0 outside the set Uj and O is the hyperbolic sink for all diffeomorphisms (f>t;
2') the diffeomorphism (j coincides with g on the set U2.
Indeed, one can get the arc pt : M2 — M2 from the arc ( in the following way. Let Uj = = l(U1), U2 = ^—J(U2) and (t = (on Uj. Then for every t G [0,1] pt coincides with p0
m— 1
outside U Pk (Uj), Pt(z) = Po (z) for z g p'k (Uj), k g{0, ...,m- 2} and pt (z) = (t (pj—m(z)) k=0
for z G Pm—j(Uj).
Now we construct the arc (t.
Denote by Q : R2 — R2 the differential of the map ()Q at the point O. Since O is a hyperbolic sink of (¡)Q, there exists a number 0 < A < 1 such that for every v G R2
||Q(v)|| < A \\v\\,
where ||-|| is the Lyapunov norm in R2 (see, for example, [9]). For every r > 0 let Br =
= {v G R2 :\\v\\ < r}. Since (q is a diffeomorphism with the fixed point O and since all norms
in R2 are equivalent, there is ej > 0 such that for every v G B£1 the equation (0(v) = Q(v) +
+ ||v||a(v) holds, where lim a(v) = 0. Therefore, there exists 0 < e2 < ej such that for every
IMI^o
nonzero v G B£2 the inequality W^(v)|| < ||v|| is true. Let Uj = B£2. Since g(clU{) C (Jj, the family of the maps Xt : U — R2 is well-defined on the set Uj by
Xt = (1 - t)(o + tg.
By construction, Hxt(v)|| < ||v|| for every nonzero v G Uj. Notice that the origin of coordinates is a fixed point of Xt, Xt(clUi) C Uj for every t G [0,1] and the isotopy £t = <P—JXt joins the identity map with jg. Let U2 = (0(UJ) (Fig. 9). Then £t(clU2) C U for all t G [0,1]. By Proposition 5, there is an isotopy St : R2 — R2 which coincides with £t on U2 and which is the identity outside (Jj. The desired arc is defined by
(t = (q St. □
Lemma 2. Let a diffeomorphism p0 : M2 — M2 have a hyperbolic sink w0 and a hyperbolic saddle a0 such that the unstable separatrix of lies in the basin of the sink and let a0 and w0 be of the same period m. Let (U0,^0) be a local chart of the manifold M2 for which w0 E U0, (w0) = O and (U0) C U0. Then there are neighborhoods V1,V2 of w0 such that V2 C V1 C U0 and there is a bifurcation free arc pt : M2 — M2,t E [0,1] with the following properties:
m— 1
1) the diffeomorphism pt, t E [0,1] coincides with p0 outside the set [J pk(V1) and the set
k=0
m— 1
U pO (w0) is the hyperbolic sink orbit of period m for every pt; k=o o
2) H V2) C OX1 where is the unstable separatrix of a0 with respect to p1 (Fig. 10).
Fig. 10. Smoothing of the separatrix.
Proof.
Let = POO and <0 = : R2 ^ R2. By Lemma 1, without loss of generality,
suppose <0 = g on the disk B2r0 for some r0 > 0. Let K0 = B2r0 \ Br0 and Y<t>0 = ^0(y^0).
Denote by Eg the set of contractions < : R2 — R2 coinciding with <0 outside B2ro and coinciding with g on Br^ where r^ ^ 2r0. For every < £ Eg let
Y4> = U <n Ko).
kez
By construction, the ^-invariant curve 7^ coincides with the <0-invariant curve 7^0 outside the disk Bro. Then to prove the lemma it suffices to construct a curve of contractions <t : R2 — R2, t £ [0,1] such that
1') the diffeomorphism <t,t £ [0,1] coincides with <0 outside the set Bro; 2') Yi n Br^) C OXi.
The arc pt : M2 — M2 is constructed from the arc <t in the same way as in Lemma 1 if one sets Vi = ^-1(Br0) and V2 = ^-1(Br^ ).
In order to construct the arc <t, we now introduce the following notations for an arbitrary diffeomorphism < £ Eg.
Represent the 2-torus T2 as the orbit space of the diffeomorphism g on the set R2 \ O and denote by p : R2 \ O — T2 the natural projection. Let a = p(OX1) and b = p(Sr) be generators of T2, let K ^ = Br^ \ Br^/2, 70 = p(Y ^ n K and 7^ = p(Y ^ n K^). Then the curve 7^ is the knot
on the torus T2 with coordinates < 1, —n^ >, n^ £ Z in the basis a, b (see, for example, [5]). The arc <t is the smooth product of the arcs nt and (t where
I) the arc nt,t £ [0,1] consists of contractions, it coincides with <0 outside the set Bro and it joins the diffeomorphism n0 = <0 with some diffeomorphism ni £ Eg such that the knot 7V1 has coordinates < 1,0 > in the basis a, b;
II) the arc Zt £ Eg ,t £ [0,1] joins the diffeomorphism Z0 = ni to the diffeomorphism Zi such
that 7z1 = a.
Fig. 11. Illustration to Lemma 2, Part I.
I) If n$ Ot,t £ [0,1] :
0, then let nt = <0 for every t £ [0,1]. Otherwise define the diffeomorphism ; — R2 in such a way that dt(O) = O and
pei!f, p > r0,
0t(pelif) =
pe pe
i((p+2n$irt) n < m.
, y \ 2 •
f < P < r0;
Then nt = dt<p0 : R2 — R2 is the desired arc (Fig. 11).
II) By construction, the diffeomorphism nJ G Eg and the knot Yni has the coordinates (1,0) in the basis a,b. According to [10], there is a diffeomorphism h : T2 — T2 which is smoothly isotopic to the identity and such that h(Afv1) = a. For r > 0 let Kr = Br \ Br/2. Pick an open cover D = {Dj, ... ,Dq} of the torus T2 such that the connected component Di of the set p~1(Di) is a subset of Kri for some i\ < and r\ ^ ro/2. According to [11, Lemma de fragmentation] there are diffeomorphisms W^ ... ,Wq : T2 — T2 smoothly isotopic to the identity, with the following properties:
i) for each i g{1, .. .,q} there exists a smooth isotopy {wi,t} which is the identity outside Di and which joins the identity and W;
ii) h = wj... wq.
Let wiit : R2 — R2 be a diffeomorphism which coincides with (p\Kri)—jwi,tp on Kri and which coincides with the identity outside Kri (Fig. 12). Then the desired arc is defined by
Zt = VjWj,t . . . Wq,t.
□
Fig. 12. Illustration to Lemma 2, Part II.
6. Proof of Theorem 2
In this section we construct a stable arc that joins the diffeomorphism / G G and "source-sink" diffeomorphism on the 2-sphere. The key technical tool to do this is the following lemma.
Lemma 3. Let the nonwandering set of a gradient-like diffeomorphism p : M2 — M2 contain a fixed saddle a of negative orientation type and such that cl (WU) = WU U u U p(u),
where u is a sink of period two. Then there exists a stable arc H^^u,t joining p to a gradientlike diffeomorphism pU whose nonwandering set is the same as Q/ but with the points a,u, f (u) removed.
Then the desired arc is: !) H/i,(/i)2,tfor fi e Gi;
2) H/2Mi t * H,t where ^ = (f )£ )-1 for f2 e G;
3) f/i,t for f3 e
Proof of Lemma 3.
From Lemmas 1 and 2 it follows that there exists a local chart (U, of the manifold M2 such that u e U, ^(u) = O, f 2(U) C U and ^(7 n U) C OX1. Let A/ = Wu U u U f (u) and l = WU U (OX1) U f (^~1(OX1)). Then l is a smooth curve which contains A/, for which the points oj,f(oj) are interior and f(l) C I. Let IIi = {(xi,x2) G R2 : \xi\ ^ Define the diffeomorphism p : n ^ R2 by
11.. , 3 ^2
By construction, p(n1) C int n1. The diffeomorphism p has the saddle O and the periodic sink orbit {Po,p(Po)} where Po(-xo, 0), p(Po) = (xo, 0), xo e (0,1/2). Let n2 = p(n^. Pick a closed neighborhood V of the set A/ and pick a diffeomorphism 3 : V ^ n1 such that f (V)_C intV, 3(l n V) = OX1 n n1, 3(f (V)) = n2, 3(u) = Po and 3(f (u)) = p(Po) (Fig. 13). Let f = 3f3-1 : n1 ^ n2. Then the family of maps xt : n2 ^ R2 is well-defined on the set n2 by
Xt = (1 - t)f + tp.
By construction, Xt(n2) C int n2 for every t e [0,1]. Notice that the origin of coordinates is the fixed saddle for the diffeomorphism xt and the points Po, p(Po) form the sink orbit. The isotopy £t = f-1xt|n2 joins the identity map and the diffeomorphism f-1p and £t(n2) C intn1.
By Proposition 5, there is an isotopy : n ^ ni which coincides with on n2 and which is the identity on 9n1. Let
ht = fa.
Notice that h1 = p on n2. Let n3 = <p(n2). Define the arc nt : n3 ^ R2 by
Vt(xl,x2) = (~Xl + i)) + *?,-?) •
By construction, nt(n3) C int n3 for every t G [0,1] and the isotopy (t = p-1 nt joins the identity map with the diffeomorphism p-1n1 and (t(n3) C intn2. By Proposition 5, there is an isotopy dt : R2 ^ R2, which coincides with (t on n3 and which is the identity outside n2. Let
Ôt = pBf
Then the desired arc t is the product of the arcs ht, Ôt : M2 ^ M2 where ht coincides with f outside V, ht(z) = /-1(ht(P(z))) for z G V and f3t coincides with h1 outside h1(V), Ôt(z) = /-1(Ôt(3(z))) for z G h1(V). □
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