On the centralizer of vector fields: criteria of triviality and genericity results

Abstract

In this paper we study the problem of knowing when the centralizer of a vector field is “small”. We obtain several criteria that imply different types of “small” centralizers, namely collinear, quasi-trivial and trivial. There are two types of results in the paper: general dynamical criteria that imply one of the “small” centralizers above; and genericity results about the centralizer. Some of our general criteria imply that the centralizer is trivial in the following settings: non-uniformly hyperbolic conservative $C^2$ flows; transitive separating $C^1$ flows; Kinematic expansive $C^3$ flows on 3 manifolds whose singularities are all hyperbolic. For genericity results, we obtain that $C^1$-generically the centralizer is quasi-trivial, and in many situations we can show that it is actually trivial.

Appendices

Appendix A: The separating property is not generic

In this section we prove that the separating property is not generic. Indeed we will see that it is not even $C^1$-dense. Let M be a compact, connected Riemannian manifold. Take any Morse function $f\in C^2(M,\mathbb {R})$ and let $X := \nabla f$ be the gradient vector field which is $C^1$. It holds that X has two hyperbolic singularities, $\sigma _s$ and $\sigma _u$ with the following properties:

• $\sigma _s$ is a hyperbolic sink and $\sigma _u$ is a hyperbolic source;

• $W^s(\sigma _s) \cap W^u(\sigma _u) \ne \emptyset$;

• for any $C^1$ vector field Y which is sufficiently $C^1$-close to X, then $W^s(\sigma _s(Y)) \cap W^u(\sigma _u(Y)) \ne \emptyset$, where $\sigma _*(Y)$ is the continuation of $\sigma _*$ for the vector field Y, for $*=s,u$.

We claim that X is $C^1$-robustly not separating. Let U be a compact ball inside $\left( W^s(\sigma _s) \cap W^u(\sigma _u)\right) -\{\sigma _s , \sigma _u \}$. Since compact parts of stable and unstable manifolds vary continuously with the vector field, it holds for any Y sufficiently $C^1$-close to X it holds that $U \subset \left( W^s(\sigma _s(Y)) \cap W^u(\sigma _u(Y)) \right) - \{\sigma _s, B_u \}$.

Take any $\varepsilon >0$ and consider the balls $B(\sigma _s , \frac{\varepsilon }{2})$ and $B(\sigma _u, \frac{\varepsilon }{2})$. Since U is compact, there exists $T_X= T(\varepsilon )>0$ such that any point $x\in U$ verifies

$$\begin{aligned} X_{-t}(x) \in B\left( \sigma _u, \frac{\varepsilon }{2}\right) \text { and } X_t(x) \in B\left( \sigma _s, \frac{\varepsilon }{2}\right) , \text { for all } t\ge T. \end{aligned}$$

(A.1)

Notice that for any two points $x,y\in B(\sigma _s, \frac{\varepsilon }{2})$ it holds that $d(X_t(x), X_t(y)) < \varepsilon$, for all $t\ge 0$. Similar statement is true for points in $B(\sigma _u, \frac{\varepsilon }{2})$ and the backward orbit.

Since T that verifies (A.1) is fixed, there exists $\delta >0$ such that for any $x\in U$ and any $y\in B(x, \delta )\subset U$, it holds that

$$\begin{aligned} d(X_t(x), X_t(y))< \varepsilon , \text { for any } t\in \mathbb {R}. \end{aligned}$$

In particular X is not separating. Also, observe that this holds for any Y sufficiently $C^1$-close to X. Thus we conclude that X is $C^1$-robustly not separating.

Remark A.1

It is easy to see that the same type of example proves that the hypothesis of Proposition 2.8 is not even $C^1$-dense. We conclude that the hypotheses of Propositions 2.4 and 2.8 are not $C^1$-dense as well.

Appendix B: Periodic orbits and chain-recurrent classes for flows: sketch of the proof of Item (3) in Theorem 5.2

In this appendix, we briefly explain the structure of the proof of Item (3) of Theorem 5.2 for diffeomorphisms, which was proved by Crovisier in [20]. Then we explain why the same proof works for vector fields as well.

As we will see, the proof of this result follows easily from Theorem B.1 below, which is an easy consequence of Proposition B.2 below. We emphasize that the proof of Proposition B.2 has two parts: a perturbation part, which only uses Hayashi’s connecting lemma, and a combinatorial part, which has no perturbation at all. Usually in these pertubation lemmas such as the connecting lemma, closing lemma and others, the combinatorial part is the hardest part in the proof. The idea of the combinatorial part is to find the right pieces of orbits that you will connect by some elementary $C^1$-perturbation. The same happens in the proof of Item (3) of Theorem 5.2. There is the combinatorial argument that will give which pieces of orbits are good to connect, and instead of some elementary perturbation you use Hayashi’s connecting lemma to connect these pieces of orbits. We remark that the combinatorial argument will be the same for diffeomorphisms and flows, and that Hayashi’s connecting lemma is available for flows, so the perturbative tool is also available for flows.

Let us give some of the dynamical background used in the proof. Below, we follow the notation in [20].

Let M be a compact, connected, Riemannian manifold, and let $\mathrm {Diff}^1(M)$ be the set of $C^1$-diffeomorphisms of M. A diffeomorphism $f\in \mathrm {Diff}^1(M)$ verifies Condition (A) if for any $n\in \mathbb {N}$, every periodic orbit of period n is isolated in M. Observe that by Kupka-Smale’s Theorem (see Theorem 3.1 in [38]), Condition (A) is a $C^r$-generic condition.

Let $f\in \mathrm {Diff}^1(M)$. We say that a compact and invariant set $\mathcal {X}$ is weakly transitive if for any non-empty open sets U and V that intersect $\mathcal {X}$, and any neighborhood W of $\mathcal {X}$, there exists a segment of orbit $\{x, f(x), \ldots , f^n(x)\}$ contained in W and such that x belongs to U and $f^n(x)$ belongs to V, and $n\ge 1$.

The main perturbation technique we want to describe is given by Theorem 3 from [20] which states the following:

Theorem B.1

(Theorem 3 in [20]) Let f be a $C^1$ diffeomorphism that satisfies Condition (A), let $\mathcal {U}$ be a $C^1$-neighborhood of f in $\mathrm {Diff}^1(M)$ and let $\mathcal {X}$ be a weakly transitive set of f. Then, for any $\eta >0$, there exists $g\in \mathcal {U}$ and a periodic orbit $\mathcal {O}$ of g such that $\mathcal {O}$ is $\eta$-close to $\mathcal {X}$ for the Hausdorff distance.

Theorem B.1 is the main perturbation technique used to prove Item (3) in Theorem 5.2 (which is given by Theorem 4 in [20]). The key ingredient in the proof of this theorem is the following proposition:

Proposition B.2

(Proposition 8 in [20]) Let f be a $C^1$ diffeomorphism and $\mathcal {U}$ a $C^1$-neighborhood of f. Then, there exists $N\ge 1$ with the following property:

if $W\subset M$ is an open set and $\mathcal {X}$ a finite set of points inside W such that:

• (1) the points $f^j(x)$ for $j\in \{1, \ldots , N\}$ are two-by-two distinct and contained in W, for $x\in \mathcal {X}$;

• (2) for any two points $x,x' \in \mathcal {X}$, for any neighborhood U and V of x and $x'$, respectively, there is a point $z\in U$ and $f^n(z) \in V$, with $n>1$, such that $\{z, \ldots , f^n(z)\} \subset W$;

then, for any $\eta >0$ there exists a perturbation $g \in \mathcal {U}$ of f with support in the union of the open sets $f^j(B(x,\eta ))$, for $x\in \mathcal {X}$ and $j\in \{0,\ldots , N-1\}$, and a periodic orbit $\mathcal {O}$ of g contained in W, which crosses all the balls $B(x,\eta )$, for $x\in \mathcal {X}$.

The proof of Theorem B.1 using Proposition B.2 follows from a short argument. Let us explain the main steps of the argument. Suppose that f is a diffeomorphism verifying Condition (A) and $\mathcal {X}$ is a weakly transitive set. We may suppose also that $\mathcal {X}$ is not a periodic orbit, otherwise there would be nothing to prove. We use Condition (A) to find a finite set $\hat{X} \subset \mathcal {X}$ such that any point $x \in \hat{X}$ is not a periodic point, any two different points in $\hat{X}$ have disjoint orbits, and any point $z\in \mathcal {X}$ belongs to $B(f^k(x),\eta _0)$, for some $k\in \mathbb {Z}$ and $x\in \hat{X}$. We then can fix some $\eta \in (0,\eta _0)$ to be small enough such that any compact invariant set K which is contained in a $\eta _0$-neighborhood of $\mathcal {X}$ and intersecting all the balls $B(x,\eta )$ for $x\in \hat{X}$, is $\eta _0$-close to $\hat{X}$ in the Hausdorff topology. One can then apply Proposition B.2 and obtain a periodic orbit that verifies the conclusion of the theorem (see Section 2.4 in [20]).

Let us now explain the structure of the proof of Proposition B.2. We also refer the reader to Section 4.0 in [20], where the structure and difficulties of the proof of Proposition B.2 are explained very clearly. The proof has two parts: the actual perturbation part, which only uses Hayashi’s connecting lemma; and a combinatorial part, which is the most delicate part.

Let us recall Hayashi’s connecting lemma. The original proof was given by Hayashi in [28]. Some other references are given in [3, 12, 55].

Theorem B.3

(Hayashi’s connecting lemma, [3, 12, 28, 55]) Let $f_0$ be a diffeomorphism of a compact manifold M, and $\mathcal {U}$ a $C^1$-neighborhood of $f_0$. Then there exists $N\ge 1$ such that for any $z\in M$ which is not a periodic point of period less than or equal to N, any two open neighborhoods V and U of z such that $V\subset U$ has the following property.

For any diffeomorphism f that coincides with $f_0$ in $U\cup \cdots \cup f^{N-1}_0(U)$, for any two points $p,q\in M - \left( U \cup \cdots \cup f^{N}_0(U)\right)$ and any integers $n_p, n_q \ge 1$ such that $f^{n_p}(p)$ belongs to V and $f^{-n_q}(q) \in V$ there is a diffeomorphism g in $\mathcal {U}$ such that:

• g coincides with f on $M - \left( U \cup \cdots \cup f^{N}_0(U)\right)$;

• there exists $m\ge 1$ such that $g^m(p) = q$;

• the piece of orbit $\{p, \ldots , g^m(p)\}$ can be cut into three parts:

  • the beginning $\{p, \ldots , g^{m'}(p)\}$, for some $m' \in \mathbb {N}$, is contained in

    $$\begin{aligned} \{p, \ldots , f^{n_p}(p)\} \cup U \cup \cdots \cup f^N_0(U); \end{aligned}$$

  • the central part $\{g^{m'}(p) , \ldots , g^{m'+N}(p)\}$ is contained in

    $$\begin{aligned} U \cup \cdots \cup f_0^N(U); \end{aligned}$$

  • the end $\{g^{m'+N}(p), \ldots , g^m(p)\}$ is contained in

    $$\begin{aligned} U \cup \cdots \cup f_0^N(U) \cup \{f^{-n_q}(q), \ldots , q\}. \end{aligned}$$

Let $\mathcal {X}$ be a finite set which is weakly transitive, and let W be a neighborhood of $\mathcal {X}$, and fix some $\eta >0$. Fix some order in $\mathcal {X}= \{p_1, \ldots , p_k\}$ and for each $p_i$ we can associate two neighborhoods $V_i\subset U_i$ contained in W, where we will apply the connecting lemma. Observe that we may consider these neighborhoods to be arbitrarily small. The weak transitivity implies that for each i, there is a point $z_i$ in $V_i$ whose future orbit intersects $V_{i+1}$ and it is contained in W. The first naive approach then would be to simply use the connecting lemma, and connect the orbit of each $z_i$ with $z_{i+1}$, where one would connect the future orbit of $z_k$ with $z_0$. Applying the connecting lemma k times, we would hope to have created a periodic orbit for a diffeomorphism $g\in \mathcal {U}$, and this periodic orbit contained in W. This is a perturbation of f whose support is contained in the union of the first $N-1$ iterates of $U_i$, for every $i=1, \ldots , k$. This approach would only work if we had the following “ideal” picture (see Fig. 7 below).

Fig. 7

“Ideal” picture

The problem is that with this approach one cannot guarantee that the piece of future orbit of $z_i$ connecting $V_i$ and $V_{i+1}$ does not intersect any other $U_j$, for $j\ne i,i+1$ (see Fig. 8 below).

Fig. 8

“General” picture

There is a delicate inductive and “combinatorial” argument that allows one to find a smaller set $\mathcal {X}' \subset \mathcal {X}$, with $\mathcal {X}' = \{p_1', \ldots , p_s'\}$, neighborhoods for the connecting lemma $V_i' \subset U_i'$, and points $\{z_1, \ldots , z_s\}$ that verify the following: each $z_i$ connects $V'_i$ to $V'_{i+1}$, the piece of future orbit of $z_i$ connecting these two neighborhoods does not intersect any other $U'_j$, for $j\ne i, i+1$, also the union of these pieces of future orbits of the $z_i$’s intersects every $B(p, \eta )$, for every $p\in \mathcal {X}$. Hence, one can apply Hayashi’s connecting lemma around the points of the set $\mathcal {X}'$, connecting the orbits of the points $z_i$ mentioned above, and obtain a perturbation g of f with a periodic orbit that verifies the conclusion of Proposition B.2. We refer the reader to Section 4.0 in [20] for more details.

There is no perturbation in the combinatorial part, even though it is the most delicate part. Observe that the points in $\mathcal {X}$ are not fixed points, and if they are periodic they must have the period larger than N.

Since Hayashi’s connecting lemma is also available for flows [28], and the points where we are using the connecting lemma are far from the singularities (it is a finite fixed set of non-singular points), one can obtain the following result for flows:

Proposition B.4

Let X be a $C^1$ vector field on M and let $\mathcal {U}$ be a $C^1$-neighborhood of X. There exists $T\ge 1$ with the following property: if $W\subset M$ is an open set and $\mathcal {X}$ is a finite set of points inside W such that:

• (1) for each $x\in \mathcal {X}$ the map $t\mapsto X_t(x)$ is injective for $t\in [0,T]$ and $X_{[0,T]}(x)$ is contained in W;

• (2) for any two points $x,x' \in \mathcal {X}$, for any neighborhood U and V of x and $x'$, respectively, there is a point $z\in U$ and $T'>1$ such that $X_{T'}(z) \in V$ and $X_{0,T'}(z)$ is contained in W;

then, for any $\eta >0$ there exists a perturbation Y of X in $\mathcal {U}$ with support in the union of the open sets $\cup _{t\in [0,T-1]}X_t(B(x,\eta ))$, with $x\in \mathcal {X}$, and a periodic orbit $\mathcal {O}$ for Y contained in W and intersects all the balls $B(x,\eta )$, for $x\in \mathcal {X}$.

Obseve that for flows we can also define an analogous of Condition (A), which we will call (Af). We say that a vector field verifies Condition (Af) if for any $T > 0$ every periodic orbit of period less than or equal to T is isolated in M. As explained above, this proposition implies the following theorem:

Theorem B.5

Let X be a $C^1$ vector field verifying Condition (Af), let $\mathcal {U}$ be a $C^1$-neighborhood of X and let $\mathcal {X}$ be a weakly transitive set for X. Then, for any $\eta >0$, there exists Y in $\mathcal {U}$ and a periodic orbit $\mathcal {O}$ for Y which is $\eta$-close to $\mathcal {X}$ for the Hausdorff topology.

Below, we explain how Crovisier uses Theorem B.1 to conclude Item (3) in Theorem 5.2. The same proof works for flows.

First, given a metric space H we denote by $\mathcal {K}(H)$ the set of all compact sets of H with the Hausdorff distance. For each diffeomorphism f we let $K_{per}(f)$ be the closure in $\mathcal {K}(M)$ of the set of periodic points of f. This is a set in $\mathcal {K}(\mathcal {K}(M))$. It is known that the the points of continuity of the function $f\mapsto K_{per}(f)$ contains a $C^1$-residual subset of $\mathrm {Diff}^1(M)$, see for instance [53]. Let us denote this residual set by $\mathcal {R}_1$.

For each diffeomorphism f, consider also $K_{WT}(f)$ to be the closure in $\mathcal {K}(M)$ of the set of weak transitive sets of f. Also, let $K_{CT}(f)$ to be the closure in $\mathcal {K}(M)$ of the set of chain transitive sets of f. Both of these sets belong to $\mathcal {K}(\mathcal {K}(M))$. It is known that there exists a $C^1$-residual set $\mathcal {R}_2$ such that for any $f\in \mathcal {R}_2$ we have $K_{WT}(f) = K_{CT}(f)$.

We claim that if $f\in \mathcal {R}_1 \cap \mathcal {R}_2$ then $K_{per}(f) = K_{WT}(f) = K_{CT}(f)$. Suppose that $K_{per}(f) \subsetneq K_{WT}(f)$. Then there exists a weak transitive set $\mathcal {X}$ and some open neighborhood W of $\mathcal {X}$ in $\mathcal {K}(M)$, such that $K_{per}(f)$ does not intersect W. By the continuity of the function $f\mapsto K_{per}(f)$, at the point f, we know that this property holds in a neighborhood of f. However, Theorem B.1 implies that there is a diffeomorphism g which is $C^1$-close to f such that g has a periodic orbit $\mathcal {O}$ which is close in the Hausdorff distance to $\mathcal {X}$. This is a contradiction.

This implies that $C^1$-generically any chain transitive set is the Hausdorff limit of periodic orbits. Since a chain recurrent class is a chain transitive set, we conclude Item (3) of Theorem 5.2 for diffeomorphisms. We remark that the same proof also holds for flows. As we mentioned before, the main perturbative tool used is given by Theorem B.1.