Zero topological entropy for C1 generic vector fields

Chaos, Solitons and Fractals 108 (2018) 104–106

Contents lists available at ScienceDirect

Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos

Zero topological entropy for C1 generic vector fields Manseob Lee Department of Mathematics, Mokwon University, Daejeon 302–729, Republic of Korea

a r t i c l e

i n f o

Article history: Received 4 July 2017 Revised 23 January 2018 Accepted 23 January 2018 Available online 3 February 2018 MSC: 37A35 37C10

a b s t r a c t In this paper, we show that C1 generically, a vector field X has zero topological entropy if there are a C1 neighborhood U of X and d > 0 such that for every Y ∈ U and a periodic point p of Y,

PπY ( p) |Np  < π ( p)d , where π (p) is the period of p, and PY is the linear Poincaré flow associated to Y. This result is a generalization of Arbieto and Morales [1]. © 2018 Elsevier Ltd. All rights reserved.

Keywords: Topological entropy Linear Poincaré flow Scaled linear Poincaré flow Morse–Smale

1. Introduction Let M be a closed smooth Riemannian manifold, and let d be the distance on M induced from a Riemannian metric  ·  on the tangent bundle TM, and denote by X(M ) the set of C1 -vector fields on M endowed with the C1 -topology. Then every X ∈ X(M ) generates a C1 -flow Xt : M × R → M; that is a C1 -map such that Xt : M → M is a diffeomorphism satisfying X0 (x ) = x and Xt+s (x ) = Xt (Xs (x )) for all s, t ∈ R and x ∈ M. We say that σ is a singular point of X if X (σ ) = 0. Denote by Sing(X) the set of all singular points of X. A point x ∈ MࢨSing(X) is called a regular point of X. Denote by Reg(X) the set of all regular points of X. A point p ∈ M is called periodic if there is π (p) such that Xπ ( p) ( p) = p. Let Per(X) be the set of periodic orbits of X. Denote by Crit (X ) = Per (X ) ∪ Sing(X ). Let t = DXt : T M → T M be the tangent flow. For x ∈ Reg(X), let N ⊂ TM be the subbundle such that the fiber Nx at x is the orthogonal linear subspace of X(x) in Tx M; i.e., Nx = X (x )⊥ . Here X(x) is the linear subspace spanned by X(x). Let π : T N → N = x∈Reg(X ) Nx be the projection along X, and let

Px,t (v ) = π (t (Xt (v ))), for v ∈ Nx and x ∈ Reg(X). It is well known that Pt : N → N is a one parameter transformation group. Let μ be a Borel probability measure of M. For any Borel set A, we say that μ is invariant if μ(Xt (A )) = μ(A ) for every Borelian A E-mail address: [email protected] https://doi.org/10.1016/j.chaos.2018.01.034 0960-0779/© 2018 Elsevier Ltd. All rights reserved.

and every t ∈ R. We say that μ is ergodic if any invariant Borel set has measure 0 or 1. Denote by EX the set of all ergodic Xt invariant probability measures. Let I be a closed interval which contains 0. A map h : I → R is called a reparametrization if it is an increasing homeomorphism and h(0 ) = 0. Denote by Rep(I ) = {h : I → R : h is an increasing homeomorphism and h(0 ) = 0}. For any x ∈ M, and  > 0, we set (X, t, x,  ) = {y ∈ M : there exists h ∈ Rep[0, t] such that d(Xh(s) (x), Xs (y)) <  , 0 ≤ s ≤ t} and it is called a (X, t,  )-ball. We introduce the definition of topological entropy for flows [9]. Given μ ∈ EX and 0 < δ < 1, let N(δ , X, t,  ) denote the smallest number of (X, t,  )-balls needed to cover a set whose μ-probability is bigger than 1 − δ. Then the measure entropy of X is defined by

entμ (X ) = lim lim sup  →0

t→∞

1 log N (δ, X, t,  ). t

The topological entropy of X is defined by

ent (X ) = sup{eμ (X ) : μ ∈ EX }. In the paper we study a dynamical system which has zero topological entropy. It is well known that all homeomorphisms of the circle have zero topological entropy. Let M be a closed smooth Riemannian manifold, and let Diff(M) be the space of diffeomorphisms of M endowed with the C1 topology. We say that f ∈ Diff(M) satisfies Axiom A if the nonwandering set ( f ) = P ( f ) is hyperbolic. For Axiom A diffemorphisms, the topological entropy is zero if and only if the nonwnadering set (f) is finite (see [6]). We say that a diffeomorphism f is MorseSmale if the nonwandering set (f) is consists of only finitely many hyperbolic periodic points and their stable and unstable manifolds

M. Lee / Chaos, Solitons and Fractals 108 (2018) 104–106

intersect transversely. It is clear that Morse–Smale systems have zero topological entropy. Pujals and Smabarino [8] proved that for dimM = 2, the C1 closure of the set of Morse–Smale diffeomorphisms coincides with the closure of the interior of the set of diffeomorphisms with zero topological entropy. Arbieto and Morales [1] proved that if a periodic point is uniformly contracting then it has zero topological entropy, that is, if there are d ∈ N and C > 0 such that Dgπ (x) (x) ≤ Cπ (x)d for every diffeomorphism g which is C1 close to f and every periodic point x with period π (x) of f then it has zero topological entropy. For vector fields, Auslander and Be.g. [2] proved that if a flow is minimal distal and it has the shadowing property then it has zero topological entropy. We say that a vector field X ∈ X(M ) is C1 robust zero topological entropy if there is a C1 neighborhood U of X such that for any Y ∈ U, Y has zero topological entropy. In [3], Gan and Yang proved that if a three dimensional vector field can be accumulated by C1 robust zero topological entropy vector fields then it can be C1 accumulated by Morse–Smale vector fields. We say that a subset G ⊂ X(M ) is residual if G contains the intersection of a countable family of open and dense subsets of X(M ); in this case G is dense in X(M ). A property ”P” is said to be (C1 )-generic if ”P” holds for all vector fields which belong to some residual subset of X(M ). In the paper, we prove that the following result for C1 generic vector fields. Theorem A. For C1 generic X ∈ X(M ), if there are d ∈ N and a C1 neighborhood U of X such that

PπY ( p) |Np  <∞ π ( p )d Y ∈U,p∈Per (Y ) sup

105

W u (Orb( p)) = {x ∈ M : d (Xt (x ), Orb( p)) → 0 as t → −∞}. Then Ws (Orb(p)) is called the stable manifold of Orb(p), and Wu (Orb(p)) is called the unstable manifold of Orb(p). Lemma 2.1 [3, Lemma 19]. be a compact set with ∩ Sing(X ) = ∅. For any η > 0 and T > 0, there is  > 0 such that if x ∈ is (η, T ) − Pt∗ -contracting then B (x) ⊂ Ws (Orb(x)). Lemma 2.2 [3, Theorem 2.20]. Given a hyperbolic singularity σ , for any η > 0 and T > 0 there is  > 0 such that there is no (η, T ) − Pt∗ contracting point in B (σ ). For a vector field X ∈ X(M ), we say that X satisfies (∗ ) if there are d ∈ N and a neighborhood U of X such that

PπY ( p) |Np 

sup

π ( p)d

Y ∈U,p∈Per (Y )

< ∞,

where PtY is the linear Poincaré flow associated to Y. Then using the property (∗ ), we can rewrite Theorem A as following. Theorem 2.3. For C1 generic X ∈ X(M ), if X satisfies (∗ ) then X has zero topological entropy. The following is Franks’ lemma which is a flow version of diffeomorphisms (see [3, Lemma 2.5]). Lemma 2.4. Let X ∈ X(M ) and let U be a C1 neighborhood of X. Then there are a neighborhood V ⊂ U of X and  > 0 such that for any Y ∈ U, for any periodic orbit Orb(x) of Y with period T ≥ 1, any neighborhood U of Orb(x) and any partition of [0, T]:

then X has zero topological entropy, where π (p) is the period of p, and PY is the linear Poincaré flow associated to Y.

0 = t0 < t1 < · · · < tl = T , 1 ≤ ti+1 − ti ≤ 2, i = 0, 1, . . . , l − 1,

2. Proof of Theorem A

with Li − PtY

In this section, we introduce another linear Poincaré flow which is called scaled linear Poincaré flow. Let x ∈ Reg(X). We define scaled linear Poincaré flow Pt∗ : N → N as follows.

(a) PtY

Pt∗ (v ) =

|X ( x )| Pt (v ) P (v ) = , |X (Xt (x ))| t t | 

for any v ∈ Nx (see [3]). Let be a closed Xt -invariant set and E ⊂ T M be a t invariant bundle. We say that E is contracting if there are constants C > 1 and 0 < λ such that t |E (x )  ≤ Ce−λt for all x ∈ and t ≥ 0. We say that E is expanding if it is contracting for −X. A point x ∈ Sing(X ) is called a (η, T ) − Pt∗ -contracting point for some η > 0 and T > 0 if there is a time partition 0 = t0 < t1 < t2 < · · · , with T < tn+1 − tn < 2T such that n −1  i=0

Pt∗i+1 −ti |NXt (x)  ≤ e−ηtn , i

for all n ∈ N. Let be a closed Xt -invariant set. The set is called hyperbolic for Xt if there are constants C > 0, λ > 0 and a splitting Tx M = Exs  X (x )  Exu such that the tangent flow t = DXt : T M → T M leaves the invariant continuous splitting and

t |Exs  ≤ Ce−λt and −t |Exu  ≤ Ce−λt for t > 0 and x ∈ . We say that X ∈ X(M ) is Anosov if M is hyperbolic for Xt . Note that if x is a singularity then dim(X (x ) ) = 0 and if x ∈ Reg(X) then dim(X (x ) ) = 1. Note that if the orbit of Orb(x) is hyperbolic then x is hyperbolic or Orb(x) is hyperbolic. For any hyperbolic periodic orbit Orb(p), we define

W s (Orb( p)) = {x ∈ M : d (Xt (x ), Orb( p)) → 0 as t → ∞}, and

and any linear isomorphisms Li : NYt (x ) → NYt (x ) , i = 0, 1, . . . , l − 1 i i+1 i+1 −ti

i+1 −ti

|NX

ti ( x )

|NX

ti ( x )

 ≤  , there exists Z ∈ U such that

= Li , and

(b) Z = Y on (MࢨU) ∪ Orb(x). Lemma 2.5. If X satisfies (∗ ) then there are η > 0 and T > 0 such that for any p ∈ Per(X) with π (p) > T, we have

Pπ ( p) |Np  ≤ e−ηπ ( p) . Proof. Suppose that X satisfies (∗ ). Let U be a C1 neighborhood of X. Then there is a neighborhood V of X and a constant δ 0 > 0 such that for any Y ∈ V, the δ 0 ball of Y is contained in U. Then by Lemma 2.4, we can find a constant δ > 0 (depending on δ 0 ) such that if there is Y ∈ V such that Y has a periodic point p which PπY ( p) |N p has an eigenvalue 1, then by using Lemma 2.4, we can

choose Z ∈ U such that Z has a periodic point p that PπZ ( p) |N p has an eigenvalue with modulus bigger than eδπ (p) . Here PtZ is the linear Poincaré flow associated to Z. Let

M=

sup

Z ∈U,p∈Per (Z )

PπZ ( p) |Np  π ( p)d

.

Then we can find T > 0 such that eδ t > Mtd for any t ≥ T. This is a contradiction since X satisfies (∗ ). Thus we consider that for any Y ∈ V and any periodic point p of Y with period π (p) > T, the eigenvalue of PπY ( p) |N p has modulus less than 1, or equivalently speaking, p should be uniformly contracting in U. Then by Mañé’s result on uniformly contracting family (see [4]), we know that there is η > 0 such that for any p ∈ Per(X) with π (p) > T, we have

Pπ ( p) |Np  ≤ e−ηπ ( p) . The following lemma is Pliss type for flows.



106

M. Lee / Chaos, Solitons and Fractals 108 (2018) 104–106

Lemma 2.6. Let p ∈ Per(X). If Pπ ( p) |N p  < e−ηπ ( p) for some η > 0, then there is q ∈ Orb(p) such that q is (η, T ) − Pt∗ -contracting for some T > 0. Proof. The proof is similar to [3, Lemma 2.21].



Lemma 2.7. There is a residual set G ⊂ X(M ) such that for any X ∈ G, one has (a) X is Kupka–Smale, that is, every periodic orbit and singularity of X are hyperbolic and the corresponding invariant manifolds intersect transversely (see [5]). (b) (X ) = Per (X ) ∪ Sing(X ) (see [7]). Lemma 2.8. Let X ∈ G. If X satisfies (∗ ) then (X ) = Per (X ) ∪ Sing(X ). Proof. Let X ∈ G and let X satisfying (∗ ). Suppose, by contradiction, that there is x ∈ (X)ࢨ(Per(X) ∪ Sing(X)). Since X ∈ G, there is a pn ∈ Per(X) such that pn → x. Since π (pn ) → ∞, we may assume that π (pn ) > T for some T > 0. For any n ∈ N, there is qn ∈ Orb(pn ) such that

Pπ (qn ) |Nqn  ≤ e

−ηπ (qn )

for some η > 0. By Lemmas 2.5 and 2.6, we know that qn is (η, T ) − Pt∗ -contracting. Then assume that there is y ∈ M such that qn → y. Case 1. y ∈ Sing(X ) Take a compact neighborhood U of y such that U ∩ Sing(X ) = ∅. By Lemma 2.1, there is  > 0 such that B (qn ) ⊂ Ws (Orb(qn )). Then there is L > 0 such that ql = qm for l, m > L which is a contradiction. Case 2. y ∈ Sing(X) By Lemma 2.2, there is no (η, T ) − Pt∗ -contracting point in B (σ ). This is a contradiction since qn → y. 

A closed set ⊂ M, we say that μ is supported on if supp(μ) ⊂ , where supp(μ) denotes the support of μ. Proof of Theorem A. Let X ∈ G and let X satisfy (∗ ). Suppose, by contradiction, that ent(X) > 0. Then there is an ergodic measure μ such that entμ (X) > 0. X ∈ G and X satisfies (∗ ), by Lemma 2.8 we have (X ) = Per (X ) ∪ Sing(X ). Since supp(μ) ⊂ (X), we know that supp(μ ) = Orb( p) or {σ }, where p ∈ Per(X) and σ ∈ Sing(X). Then we have entμ (X ) = 0 which is a contradiction.  Acknowledgment The author would like to give thanks to X. Wen and M. Li for the comments and suggestions. This work is supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (No. 2017R1A2B4001892). References [1] Arbieto A, Morales C. A sufficient condition for zero entropy. Monatsh Math 2014;175:323–32. [2] Auslander J, Berg K. A condition for zero entropy. Isael J Math 1990;69:59–64. [3] Gan S., Yang D. Morse-smale systems and horseshoes for three dimansional singular flows. arXiv:1302.0946v1. [4] Hayashi S. Connecting invariant manifolds and the solution of the c1 -stability and ω-stabolity conjecture for flows. Ann Math 1997;145:81–137. [5] Kupka I. Contribution à la théorie des champs génériques. Contrib Diff Equ 1963;2:457–84. 3(1964), 411–420 [6] Pails J, Takens F. Hyperbolicity and sensitive chaotic dynamics at homoclinic bifurcations. fractal dimensions and infinitely many attractors. Cambridge studies in advanced mathematics, vol. 35. Cambridge: Cambridge University Press; 1993. [7] Pugh C. The closing lemma. Amer J Math 1967;89:956–1009. [8] Pujals E, Sambarino M. Homoclinic tangencies and hyperbolicity for surface diffeomorphisms. Ann of Math 20 0 0;51:961–1023. [9] Sun W, Vargas E. Entropy of flows, revisited. Bol Soc Bras Mat 1999;30:315–33.