CC BY
6Динамические системы, 2020, том 10(38), №2, 140-150 MSC 2010: 37D15
On a stable arc connecting Palis diffeomorphisms on a surface1
E. Nozdrinova
Higher School of Economics
Nizhny Novgorod E-mail:maati@mail.ru
Abstract. In this paper, a class of gradient-like diffeomorphisms f on a closed orientable surface is considered, under the assumption that all non-wandering points of f are fixed and have a positive orientation type. The main result is a construction of a stable arc joining two such diffeomorphisms. The diffeomorphisms under the consideration are Palis diffeomorphisms, who highlights their as only surface diffeomorphisms included in topological flows. By S. Newhouse, M. Peixoto, and J. Fleitas result, all Morse-Smale flows on a given manifold are joined by a stable arc. However, this fact cannot be used directly to construct an arc between cascades, since Palis diffeomorphisms are included only in the topological flow. An idea of a stable arc construction between Palis diffeomorphisms is based on the construction of a bifurcation-free arc joining a Palis diffeomorphism with a diffeomorphism that is a one-time shift of a generic gradient flow of a Morse function. Keywords: gradient-like diffeomorphism, stable arc, saddle-node bifurcation.
1. Introduction and formulation of results
The problem of the existence of an arc with no more than a countable (finite) number of bifurcations connecting structurally stable systems (Morse-Smale systems) on manifolds is on the list of fifty Palis-Pugh problems [14] under number 33.
In 1976, S. Newhouse, J. Palis, F. Takens [7] introduced the concept of a stable arc connecting two structurally stable systems on a manifold. Such an arc does not change its quality properties with a small perturbation. In the same year, S. Newhouse and M. Peixoto [9] proved the existence of a simple arc (containing only elementary bifurcations) between any two Morse-Smale flows. It follows from the result of G. Fleitas [3] that a simple arc constructed by Newhouse and Peixoto can always be replaced by a stable one [8].
xThe construction of a stable arc was supported by RSF (Grant No. 17-11-01041), the construction of a Morse energy function was supported by Laboratory of Dynamical Systems and Applications NRU HSE, by Ministry of Science and Higher Education of the Russian Federation (ag. 075-15-2019-1931) and by Foundation for the Advancement of Theoretical Physics and Mathematics "BASIS" (project 19-7-1-15-1).
© E. NOZDRINOVA
For Morse-Smale diffeomorphisms given on manifolds of any dimension, examples of systems that cannot be connected by a stable arc are known. Obstruction appear already for orientation-preserving diffeomorphisms of the circle S1, which are connected by a stable arc only if the rotation numbers coincide [10]. Beginning with dimension two, additional obstruction appear to the existence of stable arcs between isotopic diffeomorphisms. They are associated with the existence of periodic points [1], [11], heteroclinic intersections [6].
Recall that a diffeomorphism f is gradient-like if its non-wandering set Qf consists of a finite number of hyperbolic points and the invariant manifolds of different saddle points do not intersect (the diffeomorphism f has no heteroclinic intersections). In this paper, we consider the class G(M2) of gradient-like diffeomorphisms f on a closed orientable surface M2, under the assumption that all non-wandering points are fixed and have positive orientation type. The main result of this work is the proof of the following theorem.
Theorem 1. Any diffeomorphisms f,f' E G(M2) can be connected by a stable arc with a finite number of generically unfolding non-critical saddle-node bifurcations.
The proof of this result is based on the construction of an arc without bifurcations connecting the diffeomorphism f E G(M2) with the diffeomorphism E G(M2), which is a one-time shift of a generic gradient flow of some Morse function. By virtue of the works [9], [3], [8], any two such flows are connected by an arc with a finite number of saddle-node bifurcations.
2. Proof of the main result
In this section, we outline the proof of theorem 1 with references to the statements that will be proved in the following sections. Let us first give the necessary definitions.
Consider a family of diffeomorphisms (an arc) pt : M ^ M,t E [0,1]. An arc pt is called smooth, if map F : M x [0,1] ^ M, defined by the formula F(x,t) = pt(x) is smooth.
A smooth arc pt is called a smooth product of smooth arcs and ^t such that
Ф2т (t), 0 ^ t ^
ф1 = ф0, if Pt = ^ где t : [0,1] ^ [0,1] is a smooth monotone
Ф2т(t)-1, 1 ^ t ^ 1,
map such that t(t) = 0 for 0 ^ t ^ | and t(t) = 1 for | ^ t ^ 1. We will write
Pt = 4>t * Фг-
Following [8], 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 are called conjugate if there are
\ / ^ К
V, Л \
Fig. 1. Saddle-node bifurcation
homeomorphisms h : [0,1] ^ [0,1], Ht : M ^ M such that Htpt = p'h{t)Ht,t e [0,1] and Ht continuously depend on t.
In [8] also established that the arc {pt}, consisting of diffeomorphisms with a finite limit set, is stable iff all its points are structurally stable diffeomorphisms with the exception of a finite number of bifurcation points, pbi,i = 1,... ,q such that pbi:
1) has no cycles;
2) has a unique non-hyperbolic periodic orbit, which is a non-critical saddle-node or flip;
3) the invariant manifolds of all periodic points of the diffeomorphismpbi intersect transversally;
4) the transition through pbi is a generically unfolded saddle-node or period doubling bifurcation, wherein the saddle-node point is non-critical.
Recall the definition of generically unfolding non-critical saddle-node bifurcations for the case of a fixed saddle-node. An arc {pt} e Q unfolds generically through a saddle-node bifurcation pbi (Fig. 1), if in some neighborhood of the nonhyperbolic point (p, hi) the arc pt is conjugate to
(
±x2+nu
xi + 0, 5x2 + t, ±2x2, • • •, ±2xi+nu
2
0
where (x1,...,xn) e Rn, \xi\ < 1/2, \t\ < 1/10.
In the local coordinates (x1,..., xn, t) the bifurcation occurs at time t = 0 and the origin O e Rn is a saddle-node point. The axis Ox1 is called a central manifold WO, the half-space {(x1,x2,... ,xn) e Rn : x1 > 0, x2+nu = • • • = xn = 0} is the unstable manifold WU, half-space {(x1,x2,..., xn) e Rn : x1 < 0, x2 = • • • = x1+nu = 0} is the stable manifold WS of the point O.
If p is a saddle-nodal point of the diffeomorphism , then there exists a unique pbi-invariant foliation Fs with smooth leaves such that dWp, is a leave of this foliation [5]. Fs is called a strongly stable foliation (Fig. 2). A similar strongly unstable foliation is
2
denoted by Fuu. A point p is called s-critical, if there exists some hyperbolic periodic point q such that WU non-transversally intersect some leaf of the foliation F£s; u-criticality is defined similarly. Point p is called
- semi-critical if it is either s- or u-critical;
- bi-critical if it is s- and u-critical;
- non-critical if it is not semi-critical.
Fig. 2. Strongly stable and unstable foliations
Let f, f' E G(M2). Let us prove that the diffeomorphisms f, f' are connected by a stable arc pt : M2 ^ M2,t E [0,1], whose diffeomorphisms are gradient-like except for a finite number of generically unfolding non-critical saddle-node bifurcations.
Proof. In section 4 we construct an arc without bifurcations rf)t, which connects the diffeomorphism f E G(M2) with a diffeomorphism 0f E G(M2) being a one-time map of a generic gradient flow 0f of some Morse function. According to [9], [3], [8] any two such flows 01 ,0\ are connected by an arc r^^t = ilT,t E [0,1]} with a finite number of saddle-node bifurcations. Denote by 01, 02,Yt the one-time shift of the flows 01,02,Yt respectively. By construction, the arc r^1;^2;t = {^t,t E [0,1]} connects the diffeomorphisms 01 and 02. Then the desired arc is pt = r^t * r$ftt * rf,1-t. □
3. An energy function for canonical diffeomorphisms of the class G(M2)
Let us give necessary definitions, following to [2], [4], for example.
If Mn is a smooth n-manifold and Ф : Mn ^ R is a Cr-smooth (r > 2) function then a point p £ Mn is critical for Ф if grad Ф(р) = 0, that is дф (p) = • • • = §хг (p) = 0 in local coordinates x\,... ,xn of the point p. A point p is called a non-degenerate if the matrix of the second derivatives (Hessian matrix) X") \p is non-degenerate,
otherwise the point p is called a degenerate. A function $ is called a Morse function if all its critical points are non-degenerate.
A diffeomorphism f : Mn ^ Mn is called a Morse-Smale diffeomorphism if
1) the non-wandering set Qf is finite and hyperbolic;
2) for every two distinct periodic points p, q the manifolds Ws, W^ intersect transversally.
A continuous function $ : Mn ^ R is called Lyapunov function for a Morse-Smale diffeomorphism f : Mn ^ Mn if
1) $(f (x)) < $(x) for every x E Qf;
2) $(f (x)) = $(x) for every x E Qf.
A Lyapunov function $ : Mn ^ R for a Morse-Smale diffeomorphism f : Mn ^ Mn is called a Morse-Lyapunov function if every periodic point p is a non-degenerate maximum (minimum) of the restriction of $ to the unstable (stable) manifold Wpu (Wp). Morse-Lyapunov function $ is called an energy function for Morse-Smale diffeomorphism f if the set of critical points of $ coincides with the set Qf.
D. Pixton [15] proved the existence of an energy function for any Morse-Smale diffeomorphism given on a smooth closed two-dimensional manifold. However, for gradient-like surface diffeomorphisms we need an energy function with more subtle properties. In more detail.
Let f E G(M2). Denote by Qq, q E {0,1, 2} the set of fixed points p of the diffeomorphism f such that dim Wpu = q. Let Lp be a frame of saddle separatrices going to the node p, denote kp their number. Denote Lk C R2 a frame of rays l1,... ,lk, which in polar coordinates (p, 9) has a form li = {(p,0) E R2 : 9 = 9}, 9i E [0, 2n).
A diffeomorphism f E G(M2) is called a canonical if every fixed point p of a diffeomorphism f has a local chart (Up,^p) such that p E Up,^p(p) = O and
1) ^pf^-1(x, y) = (1 ^ 2y)for pe ^ ^pf^-l(x,v) = (I^ 2y) for Pe Af%l(x,y) = (2^ 2v) for pe q2;
2) Vp(Lp) C Lkp for any nodal point p.
Denote by G0(M2) C G(M2) a class of all canonical diffeomorphisms.
Theorem 2. For any diffeomorphism g E G0(M2) there is an energy function $; whose level lines intersect every saddle separatrices at 'most one point.
Proof. We split the construction into steps.
Step 1. Construction of local energy functions in the neighborhood of fixed points.
Define functions Oq : R2 ^ R, q E {0,1, 2} by the formulas:
©0(x, y) = -x2 — y2, 6i(x, y) = 1 + x2 — y2 and ©2(x, y) = 2 + x2 + y2. Then, in the neighborhood Up of p E Qq a local energy function $p : Up ^ R is defined by the formula
$p = ©q o ^p.
It follows from the definition of a canonical diffeomorphism that the level lines of local energy functions intersect every saddle separatrices at most one point.
Step 2. Confluece of local level lines in the neighborhoods of sinks and saddles.
Denote by $q a function composed by the functions $p,p E Hq. Let Uq = (J Up
P&Qq
and L0 = (J Lp. We choose £0 E (0,1/2) so that $o 1([0, 2£0]) C U0. Let
peQo
Sr = $01(r), r E [0, 2£o].
We choose £1 E (0,£0/4) so that i-1 ([1 — £1, 2£1]) n U1 = 0. Let
Sr = $01(1 + r), r E [—£1,£1].
By construction, every connected component of the set S£o transversally intersects the set L0, and for each separatrix l C L0 the intersection l n S£0 is either empty or consists of exactly one point. According to A-lemma [13], there exists k E N, such that S_£1 C g~k($0"1([0, £0/2]), the intersection g~k(S£o) n S0 is transversal and each connected component of the set S0 \ contains exactly one point of this intersection. Since $1 is an energy function for the diffeomorphism g with the set of critical points, then $1(g_1(S0 \ Q1)) > $1(S0) = 1. According to this inequality and the transversality of the intersection g_k(S£o) n E0, there is £ E (0,£1) such that:
(1) the intersection g_k(S£o+r) n Er, r E [—£,£] is transversal, and each connected component of the set E0 \ Q1 contains exactly one point of this intersection and each connected component of the set Er \ Q1, r = 0 contains exactly two points of this intersection;
(2) for K£ = U g_k (S£o+r), T£ = U Sr ,E£ = K£ n T£ the inequality
r£[_£,£] r£[_£,£]
$1(g_1(E£)) > 1 + £ holds.
Let Pr = g_k($_1([0,£o + r])), Hr = $_1([1 — £1, 1 + r]), Qr = Pr U Hr}r E [—£; £]. Without loss of generality we assume Qr smooth, since one always makes it so by an arbitrarily small smoothing of the corners.
Step 3. Construction of energy functions on the set Q_£. By construction P£ C Q_£ and each connected component of the set L0 intersects dP£ and dQ_£ at exactly one point. According to annulus hypothesis K = Q_£ \ int P£
Fig. 3. Illustration for step 2
is diffeomorphic to S1 x [0,1]. Let us construct for g an energy function : Q-£ — [0,1 — e], the level lines of which intersect the saddle separatrices at most one point. Consider two possibilities:
1) dQ-£ П g-nd(P£) = 0 for any n£ N;
2) there exists n£ N such that dQ-£ П g-nd(P£) = 0 .
In the case 1), denote by m the smallest of the natural number for which gm(Q-£) С int P£. If m =1 (Fig. 4), then we define a diffeomorphism vK : K — S1 x [0,1] so that vK(dP£) = S1 x {0}, vK(dQ-£) = S1 x {1} and each connected component of the intersection vK (K П L0) has the form {s^} x [0,1]. Define the function OSix[0;1] : S1 x [0,1] — [0,1] by the formula ©Six[0)1](s,x) = x and the function p : [0,1] — [e, 1—e] by the formula p(x) = (1 — 2e)x + e. Let Фк = p о 6Six[0>1] о vK : K — [e, 1 — e]. Then the required function ФQ_£ has a form
т ,, \ Фо, x £ P£, (x)=
I Фк, x £ K.
Since g(Q-£) С intP£, the constructed function is energy.
If m > 1 (Fig. 5), then the required function ФQ_£ is constructed on the set g1-m(P£) as Фс^™-1 and is defined similarly to the case m =1 on the annulus Q-£ \int g1-m(P£).
In the case 2), without loss of generality we assume dQ-£ is transversal to the set (J g-n(dP£). Then the set J = dQ-£ ^ U g-n(dP£) is finite. We now show the way
n>0 n>0
to decrease the number of the points in the intersection by an isotopic modification of P£ while providing that it remains the energy function on it. From the condition of intersection transversality and homotopy of curves dP£,dQ-£ в WS0 \ П0, the sum of the intersection indices of points in the set J is zero. Thus, the set J contains an even number of points. Then, since the set J is finite, among them there is a pair of points
jii 32 ^ 9~n(dP£) for some /?. such that they bound the arc 8 C <9Q_e, the interior of
which does not contain points of the set J. Denote by d C g~n(dP£) an arc bounded
by the points ji, j-2 such that the closed curve 8 U d bounds a 2-disc D C (W^ \ Qo)-
There are two possibilities for the disk D: (a) gn(D) C P£ (Fig. 6) and (b) gn(D) C
g~1(P£) (Fig. 7). Define P'E to be cl(P£ \ gn(D)) in the case (a) and define it to be
cl{P£ U gn{D)) in the case (b). Then g(P£) C P'£ C P£ in the case (a) and P£ C P'£ C
g~1(P£) in the case (b). In both cases there is a smoothing P£ of the set P'£ such that
g(P£) C inl I' C P£ in the case (a) and we have P£ C inl I' C g~1(P£) in the case (b)
and the cardinality of the set J = dQ-£ fl (J g~n(dP£) is less than the cardinality of
n> o
the set J.
Let us show how to construct for g an energy function $p£ with the level line dP£. In the case (a), on the set g(Pe) energy function $0 is defined, to the annulus P£ \g(P£) this function is continued similarly to the case 1). In the case (b), on the set P£ energy function $0 is defined, to the annulus g-1(P£) \ P£ this function is continued similarly to the case 1). We will repeat the described process until the set J becomes empty, after which we construct the desired function according to the algorithm of the case 1).
Step 4.Construction of energy functions on the set Q£.
On the set Q-£ there exists an energy function $q_£ such that $q_£ (dQ-£) = 1 — e. Define on the set Q£ a function $0 : Q£ ^ R by the formula:
(x), if x e Q-e, Л 1 + r, if x e dQr.
(X)
We now check that is an energy function for g. Represent the set Q£ as the union of subsets with mutually disjoint interiors: Q£ = A U B U C, where A = Q-£, B = K£ \ Q-£ and C = Q£ \ (P£ U Q-£). By construction \a = \a, the function
\b has no critical points and \c = \c. We now check that the function decreases along the trajectories. If x £ A then g(x) £ A and (g(x)) < (x) because $Q ^ is an energy function for g. If x £ B then (x) > 1 — e. Due to the choice of e, g(x) £ A and, therefore, (g(x)) < 1 — e, whence (g(x)) < (x). If x £ C then from the condition (2) of the choice of e either g(x) £ A and the decrease can be proved similarly to the case x £ B, or g(x) £ C and the decrease follows from the fact that $1 is a Lyapunov function.
Step 5. Construction of energy functions on M2
The set of sources is an attractor for the diffeomorphism g-1. Then the energy function on the set M2 \ Q£ is constructed similarly to Step 3. □
4. Construction of the arc rf,t
In this section we construct an arc without bifurcations rf,t, which joins the diffeomorphism f £ G(M2) with a diffeomorphism £ G(M2), which is the onetime shift of a generic gradient flow of some Morse function.
Proof. According to [11] there is an arc ^t without bifurcations connecting ff £ G(M2) with some diffeomorphism g £ G0. By theorem 2 for the diffeomorphism g £ G0 there exists an energy function $, whose inverse gradient vector field generates a gradientlike flow $>tf. Consider 0f = 0]- and construct an arc without bifurcations connecting g with 0f. Throughout what follows we will use the notation of section 3. Recall that L0 = U Lp is the set of the unstable separatrices of the diffeomorphism g. Denote by
peQo
Ьф the set of the unstable separatrices of the diffeomorphism . By construction, the sets bo and Ьф coincide outside the set Q1-£. Define the diffeomorphism (Q : Q1-£ ^ Qi-£ such that:
- Zo = id on dQi-£]
- Zo(Lo П dQr) = Ьф n dQr for r e [0; 1 - e].
By construction, the diffeomorphism Zo is isotopic to the identity by means of some isotopy Zo,i, which is identity on dQ1-£. Moreover, Z0 sends the unstable separatrices of the diffeomorphism g into the unstable separatrices of the diffeomorphism . The isotopy Z2,t : M2 \ Q1+£ ^ M2 \ Q1+£ is constructed in a similar way, it transforms the stable separatrices of the diffeomorphism g into the stable separatrices of the diffeomorphism ф^. Thus, on the manifold M2 it is correctly defined an isotopy
iZo,t(x), if x e Ql-£; x, if x e Qi+£ \ Qi-£; Z2,t(x), if x e M2 \ Qi+£.
Then the arc xt = ^g^-1 connects the diffeomorphism g with some diffeomorphism gf e Go, the closures of the invariant saddle manifolds of which coincide with the analogous manifolds of the diffeomorphism фf. According to [12, Lemmas 6.2, 6.3], the diffeomorphism gf is connected by an arc without bifurcations with the diffeomorphism фf. Then the required arc rf, t has a form:
rf,t = ^t * Xt * 6-
□
Acknowledgments. The author thanks professor O. V. Pochinka for careful reading the manuscript.
References
1. Blanchard P. (1980). Invariants of the NPT isotopy classes of Morse-Smale diffeomorphisms of surfaces, Duke Mathematical Journal 47, No. 1, 33-46.
2. Grines V. Z., Laudenbach F, Pochinka O. V. (2012). Dynamically ordered energy function for Morse-Smale diffeomorphisms on 3-manifolds, Proceedings of the Steklov Institute of Mathematics 278, 27-40.
3. Fleitas G. (1977). Replacing tangencies by saddle-nodes, Bol. Soc. Brasil. Mat. 8, No. 1, 47-51.
4. Grines V., Medvedev T, Pochinka O. (2016). Dynamical Systems on 2- and 3-Manifolds, Springer International Publishing Switzerland.
5. Hirsh M. W., Pugh C.C., Shub M. (1977). Invariant manifolds, Springer Lecture Notes in Math. 583.
6. Matsumoto S. (1979). There are two isotopic Morse-Smale diffeomorphisms which cannot be joined by simple arcs, Inventiones mathematical 51, 1-7.
7. Newhouse S., Palis J., Takens F. (1976). Stable arcs of diffeomorphisms, Bull. Amer. Math. Soc. 82, No.3, 499-502.
8. Newhouse S.E., Palis J., Takens F. (1983). Bifurcations and stability of families of diffeomorphisms, Publications mathematiques de l' I.H.E.S. 57, 5-71.
9. Newhouse S., Peixoto M. (1976).There is a simple arc joining any two Morse-Smale fows, Asterisque 31, 15-41.
10. Nozdrinova E. (2018). Rotation number as a complete topological invariant of a simple isotopic class of rough transformations of a circle, Russian Journal of Nonlinear Dynamics 14, No. 4, 543-551.
11. Nozdrinova E, Pochinka O. (2020). Solution of the 33rd Palis-Pugh problem for gradientlike diffeomorphisms of a two-dimensional sphere, Discrete Continuous Dynamical Systems - A doi: 10.3934/dcds.2020311
12. Nozdrinova E, Pochinka O. (2020). Stable arcs connecting polar cascades on a torus, Cornell University Series arxive "math". arXiv:2012.01140 [math.DS]
13. Palis J., de Melo W. (1982). Geometric theory of dynamical systems: An introduction, New York: Springer 198.
14. Palis J., Pugh C. (1975). Fifty problems in dynamical systems, Lecture Notes in Math. 468, 345-353.
15. Pixton D. (1977). Wild unstable manifolds, Topology 16, No.2, 167-172.
Получена 01.06.2020
