On the Existence of Smooth Lyapunov Functions for Arbitrary Closed Sets

Abstract

We give a simpler proof of the existence of isolating blocks due to Conley and Easton (Trans Am Math Soc 158:35–61, 1983) based on a generalization of the work of Wilson Jr. and Yorke (J Differ Equ 13:106–123, 1973) on Lyapunov functions and isolating blocks. This will also yield a proof of Massera’s (Ann Math 64:182–206, 1956) converse Lyapunov Theorem and a proof of a Theorem independently due to Duistermaat and Hörmander (Acta Math 128:183–269, 1972) and Sullivan (Invent Math 36:225–255, 1976).

Appendix: Continuous flows

In this appendix, we deal with flows which are not generated by a C$^1$ vector field.

We first consider the case of a continuous flow $\varphi _t: Z\rightarrow Z$, where Z is a non-empty metrizable locally compact and $\sigma$-compact space. Since Z is locally compact and $\sigma$-compact, we can find a sequence of (non-empty) compact subsets $(K_n)_{n\in {\mathbb {N}}}$ such that $X=\cup _{n\in {\mathbb {N}}} K_n$ and ${K_n\subset \mathring{K}_{n+1}}$.

The space $C^0(Z,{\mathbb {R}})$ of continuous maps from Z to ${\mathbb {R}}$ is endowed with the topology of uniform convergence on compact subsets is a Fréchet space, since its topology is defined by the sequence of semi norms

$$\begin{aligned} p_n(f)=\sup _{x\in K_n}|f (x)|, \end{aligned}$$

where $f\in C^0(Z,{\mathbb {R}})$ and $n\in {\mathbb {N}}$.

Recall the $f:Z\rightarrow {\mathbb {R}}$ is said to have a derivative in the direction of the flow $\phi _t$ at $x\in Z$ if $t\mapsto f\phi _t(x)$ is differentiable at $t=0$. When this limit exists, we set

$$\begin{aligned} D_{\{\phi _t\}}f(x)=\lim _{t\rightarrow 0}\frac{f\phi _t(x)-f(x)}{t} \end{aligned}$$

If $D_{\{\phi _t\}}f(x)$ exists for all $x\in Z$, we define $D_{\{\phi _t\}}f:Z\rightarrow {\mathbb {R}}, x\mapsto D_{\{\phi _t\}}f(x)$.

The following lemma is obvious.

Lemma A.1

Suppose $f_n:Z\rightarrow {\mathbb {R}}$ are continuous functions such that $D_{\{\phi _t\}}f_n$ exists everywhere and is continuous on Z. If both $f_n$ converges to $f:Z\rightarrow {\mathbb {R}}$ and $D_{\{\phi _t\}}f_n$ converges to $g:Z\rightarrow {\mathbb {R}}$ uniformly on every compact subset of Z, then $D_{\{\phi _t\}}f$ exists everywhere on Z and is equal to g.

We can now given the statement of the Main Theorem 1.4 in this case.

Theorem A.2

(Main Theorem for continuous flows) Assume that $\varphi _t: Z\rightarrow Z$ is a continuous flow on the non-empty metrizable locally $\sigma$-compact space Z.

If C is a closed subset of Z, we can find two continuous functions $f,g:M\rightarrow [0,+\infty [$ such that:

• (i) $f^{-1}(0)=\bigcap _{t\ge 0} \phi _{-t}(C)$.

• (ii) $D_{\{\phi _t\}}f$ exists everywhere and is continuous on Z, with $D_{\{\phi _t\}} f\ge 0$ everywhere on C and $D_{\{\phi _t\}} f(x)>0$ for every $x\in C{\setminus } \bigcap _{t\ge 0} \phi _{-t}(C)$.

• (iii) $g^{-1}(0)=\bigcap _{t\ge 0} \phi _{t}(C)$.

• (iv) $D_{\{\phi _t\}}g$ exists everywhere and is continuous on Z, with $D_{\{\phi _t\}} g \le 0$ on C everywhere and $D_{\{\phi _t\}} g(x)<0$ for every $x\in C{\setminus } \bigcap _{t\ge 0} \phi _{t}(C)$.

Proof

The proof replicates the first part of the proof of the Main Theorem 1.4.

We must start with a continuous function $\theta :Z\rightarrow [0,+\infty [$ such that $\theta ^{-1}(0)=C$. In this case, we can take

$$\begin{aligned} \theta (x)=\inf _{c\in C}d(x,c), \end{aligned}$$

where d is a distance defining the topology of Z. Then as we did in the proof of 1.4, for $t>0$, we define $\theta _t:Z\rightarrow [0,+\infty [$ by

$$\begin{aligned} \theta _t(x)=\int _0^t\theta (\phi _s(x))\,ds. \end{aligned}$$

As before

$$\begin{aligned} \theta _t(x)=0\iff x\in \bigcap _{0\le s\le t}\phi _{-s}(C). \end{aligned}$$

Moreover, the directional derivative $D_{\{\phi _t\}}\theta _t$ exists everywhere on Z and

$$\begin{aligned} D_{\{\phi _t\}}\theta _t(x)=\theta (\phi _t(x))-\theta (x), \text { for all } x\in Z. \end{aligned}$$

From which we conclude

$$\begin{aligned}\forall x\in C, D_{\{\phi _t\}}\theta _t(x)\ge 0\text { with } D_{\{\phi _t\}}\theta _t(x)=0\iff x\in \phi _{-t}(C). \end{aligned}$$

As above, we choose a sequence $t_n\in [0,+\infty [$ dense in $[0,+\infty [$. Again by Lemma 4.2, we can find a sequence $\epsilon _n>0$ such that both series $f=\sum _{n=0}^\infty \epsilon _n\theta _{t_n}$ and $\sum _{n=0}^\infty \epsilon _nD_{\{\phi _t\}}\theta _{t_n}$ converge uniformly on compact subsets of Z. By Lemma A.1, the directional derivative $D_{\{\phi _t\}}f$ exists everywhere on Z and

$$\begin{aligned} D_{\{\phi _t\}}f=\sum _{n=0}^\infty \epsilon _nD_{\{\phi _t\}}\theta _{t_n}(x), \text { for all }x\in Z. \end{aligned}$$

Like in the proof of the Main Theorem 1.4, we can conclude that f satisfies the required conditions (i) and (ii).

Again to obtain g, we apply what we just showed to the flow ${{\check{\phi }}}_t=\phi _{-t}$. $\square$

Even if Z is a manifold in the Main Theorem for continuous flows A.2, for an arbitrary flow $\phi _t$, we cannot require that f or g are even C$^1$, see the example in Fathi and Pageault (2019, §7 page 1697).

With this last remark in mind, it is not difficult to obtain the Duistermaat-Hörmander/Sullivan Theorem in its generalized form 2.3 and Massera’s converse to Lyapunov Theorem 1.1 for a continuous flow on a metrizable locally compact $\sigma$-compact: it suffices to replace the smoothness properties of the involved functions by the existence of the derivative in the direction of the flow. For the existence of an Isolating Block for a compact isolated invariant set, we could follow the proof of Theorem 1.3 finding $F: Z\rightarrow {\mathbb {R}}, x\mapsto (f(x),g(x))$. Then we find $r>0$ small enough such that $F(\partial V)\cap [0,r]\times [0,r]=\emptyset$. The inverse image ${\hat{N}}=f^{-1}([0,r]\times [0,r])\cap V$ is an Isolating Block. In fact, since $D_{\{\phi _t\}}f>0$ on $F^{-1}(\{r\}\times [0,r])$ and $D_{\{\phi _t\}}g<0$ on $F^{-1}([0,r]\times \{r\})$, we obtain that $\partial {\hat{N}}=F^{-1}(\{r\}\times [0,r])\cup F^{-1}([0,r]\times \{r\})$. Moreover, the flow is entering at every point of ${\hat{N}}^i=F^{-1}([0,r[\times \{r\})$, exiting at every point of ${\hat{N}}^o=F^{-1}(\{r\}\times [0,r[)$ and at every point $x\in {\hat{N}}^t=F^{-1}(\{r\}\times \{r\})$, the orbit of x is locally outside N except for x.

To obtain a more general case of the Main Theorem 1.4 where we could impose differentiability on f and g, we will consider the case where the flow is defined by a uniquely integrable vector field.

Recall that a continuous vector field X on a manifold M is uniquely integrable if it satisfies the uniqueness of the solutions through any given point in M. In this case, the local flow of X is well defined, we will assume that X is also complete (this is always the case when M is compact), which means that maximal solutions of X are defined on ${\mathbb {R}}$.

Note that when $\phi _t$ is the flow of a uniquely integrable vector field X and $f:M\rightarrow {\mathbb {R}}$ is differentiable at x, then $D_{\{\phi _t\}}f(x)=d_xf(X(x))=X\cdot f(x)$.

When $\phi _t$ is the flow of a uniquely integrable vector field it is therefore customary to denote the directional derivative, when it exists, of the continuous function $f:M\rightarrow {\mathbb {R}}$ at $x\in M$ by $X\cdot f(x)$ rather than by $D_{\{\phi _t\}}f(x)$.

Theorem A.3

The Main Theorem 1.4 is also true when the flow $\phi _t$ is the complete flow of a continuous uniquely integrable vector field X on the manifold M.

Proof

By applying The Main Theorem for continuous flows A.2, we can find a continuous function ${\hat{f}}: M\rightarrow {\mathbb {R}}$ such that

• (i) ${\hat{f}}^{-1}(0)=\bigcap _{t\ge 0} \phi _{-t}(C)$.

• (ii) $X\cdot {\hat{f}}$ exists everywhere and is continuous on M, with $X\cdot {\hat{f}}\ge 0$ everywhere on C and $X\cdot {\hat{f}}(x)>0$ for every $x\in C{\setminus } \bigcap _{t\ge 0} \phi _{-t}(C)$.

Now we cannot invoke, like in the proof of the Main Theorem 1.4, the usual approximation of C$^1$ function by C$^\infty$ functions to obtain ${\tilde{f}}:M\setminus \bigcap _{t\ge 0}\phi _{-t}(C)\rightarrow {} ]0,+\infty [$, because ${\hat{f}}$ is typically not C$^1$. Instead, we will invoke (Fathi and Pageault 2019, Theorem 5.1, page 1684) [or alternatively results contained in Kurzweil (1956) and Wilson (1969)].

Consider the subsets

$$\begin{aligned} {\hat{C}}=C{\setminus } \bigcap _{t\ge 0}\phi _{-t}(C)\rightarrow {} ]0,+\infty [\quad \text { and }\quad {\hat{M}}=M{\setminus } \bigcap _{t\ge 0}\phi _{-t}(C)\rightarrow {} ]0,+\infty [. \end{aligned}$$

Note that ${\hat{M}}$ is an open subset of M, hence a manifold. Moreover, the subset ${\hat{C}}$ is a closed subset of ${\hat{M}}$ (in the induced topology on ${\hat{M}}$).

On the set ${\hat{M}}$, the continuous ${\hat{f}}$ is everywhere $>0$, therefore we can find a C$^\infty$ function ${\tilde{f}}_1:{\hat{M}}\rightarrow {} ]0,+\infty [$ such that

$$\begin{aligned} |{\tilde{f}}_1(x)-{\hat{f}}(x)|<{\hat{f}}(x), \text { for all }\in {\hat{M}}. \end{aligned}$$

Consider next the set

$$\begin{aligned} V=\{x\in M| X\cdot f(x)>0.\} \end{aligned}$$

This subset V is open in M, by continuity of $X\cdot f$. Moreover, by condition (ii) above, it is contained in ${\hat{M}}$.

We can apply (Fathi and Pageault 2019, Theorem 5.1, page 1684) to obtain a C$^\infty$ function ${\tilde{f}}_2: V\rightarrow {} ]0,+\infty [$ such that

$$\begin{aligned} X\cdot {\tilde{f}}_2 (x)>0 \hbox { and } |{\tilde{f}}_2(x)-{\hat{f}}(x)|<{\hat{f}}(x), \text { for all}\ x\in V. \end{aligned}$$

Since ${\hat{C}}\subset V$, with ${\hat{C}}$ closed and V open in ${\hat{M}}$ (in the induced topology on ${\hat{M}}$), we can find a C$^\infty$ function $\varphi :{\hat{M}}\rightarrow [0,1]$, whose support is contained in V and such that $\varphi =1$ in a neighborhood of ${\hat{C}}$.

We can define a C$^\infty$ function ${\tilde{f}}:{\hat{M}} \rightarrow {} ]0,+\infty [$ by

$$\begin{aligned} {\tilde{f}}(x)= (1-\varphi )(x){\tilde{f}}_1(x)+ \varphi (x){\tilde{f}}_2(x). \end{aligned}$$

Obviously

$$\begin{aligned} |{\tilde{f}}(x)-{\hat{f}}(x)|<{\hat{f}}(x), \text { for all }\in {\hat{M}} \text { and } X\cdot {\tilde{f}} (x)>0, \hbox { for all } x\in {\hat{C}}. \end{aligned}$$

By the first condition ${\tilde{f}}$ extends continuously by 0 on $\bigcap _{t\ge 0}\phi _{-t}(C)$.

From here, we can carry out the rest of the argument like in the proof of the Main Theorem 1.4. $\square$

Remark A.4

From here, it is easy to show that both Sullivan’s Theorem in generalized form 2.3 and Massera’s converse to Lyapunov Theorem 1.1 hold for continuous uniquely integrable fields.

However, we cannot obtain the full Conley–Easton isolating block Theorem 1.3, since unfortunatly, we cannot guarantee that $X\cdot f$ has a continuous derivative in the flow direction.

Remark A.5

It is possible go beyond integrable vector fields, but in this case it will probably amount to consider multiple dynamics defined by a cone structure. This has been done for the generalization Massera’s converse to Lyapunov Theorem 1.1, see Siconolfi and Terrone (2007) and the references cited therein.

To obtain an analogue of the Main Theorem for multiple dynamics defined by a cone structure, the methods in Bernard and Suhr (2018a, 2018b), Fathi and Siconolfi (2012), Siconolfi and Terrone (2007) will certainly help.