$C^0$-robustness of topological entropy for geodesic flows

Abstract

In this paper, we study the regularity of topological entropy, as a function on the space of Riemannian metrics endowed with the $C^0$ topology. We establish several instances of entropy robustness (persistence of positive entropy after small $C^0$ perturbations). A large part of this paper is dedicated to metrics on the two-dimensional torus, for which our main results are that metrics with a contractible closed geodesic have robust entropy (thus, generalizing and quantifying a result of Denvir–Mackay) and that metrics with robust positive entropy on the torus are $C^{\infty }$ generic. Moreover, we quantify the asymptotic behavior of volume entropy in the Teichmüller space of hyperbolic metrics on a punctured torus, which bounds from below the topological entropy for these metrics. For general closed manifolds of dimension at least 2, we prove that the set of metrics with robust and large positive entropy is $C^0$-large in the sense that it is dense and contains cones and arbitrarily large balls.

Appendix A. Robustness of non-degenerate length spectrum

Here, we prove Proposition 7. The aim is to use only a low amount of technology.

Remark 45

Unfortunately, one cannot say anything about the position of the geodesic that is found by Proposition 7. $\square$

Before we start the proof, let us fix the setup: Let (M, g) be a closed Riemannian manifold. Denote by $\Omega =H^1(S^1,M)$ the Hilbert manifold⁶ of closed loops in M. The non-constant critical points of the energy functional ${\mathcal {E}}_g:\gamma \mapsto \frac{1}{2}\int _0^1 g(\dot{\gamma },\dot{\gamma })\;{\text {d}}t$ are exactly the closed geodesics. The negative gradient flow $\varphi _g^t$ of ${\mathcal {E}}_g$ has the Palais–Smale property in this space. Denote the sublevel set $\{\gamma \in \Omega \mid {\mathcal {E}}_g(\gamma )\leqslant a\}$ by $\Omega ^a_g$.

That $\gamma$ is non-constant and non-degenerate means that the connected component of ${\text {Crit}}{\mathcal {E}}_g$ containing $\gamma$ is a circle and Morse–Bott. If all geodesics are non-degenerate, then the energy spectrum is discrete. The following statement describes what happens topologically at a critical energy level.

Proposition 46

([9], see also [35]) Assume that $c\in (a,b)$ is the only critical value in [a, b]. Denote $N_1,\ldots ,N_r$ the components of ${\text {Crit}}({\mathcal {E}})$ with ${\mathcal {E}}(N_i)=c$ and with indices $\lambda _1,\ldots ,\lambda _r$. Assume they are Morse–Bott. Then

• Each manifold $N_i$ carries a well defined vector bundle $\nu ^-N_i\subset T\Omega |_{N_i}$ of rank $\lambda _i$ consisting of negative directions of $d^2 {\mathcal {E}}_g$.

• The sublevel set $\Omega ^b_g$ retracts onto a space homeomorphic to $\Omega ^a_g$ with the disc bundles $D\nu ^-N_i$ disjointly attached to $\Omega ^a_g$ along their boundaries.

• The retraction $r:\Omega ^b_g\rightarrow \Omega ^a_g\bigcup _{\partial D\nu ^-N_i} D\nu ^-N_i$ can be chosen such that ${\mathcal {E}}_g\circ r\leqslant {\mathcal {E}}_g$ and such that $r|_{N_i}=id$ and $r|_{\Omega ^a_g}=id$.

Remark 47

This proposition gives inductive instructions to build a CW-complex homotopy equivalent to $\Omega$. The building blocks are disk bundles, which are cell complexes. The retraction maps inductively provide the attaching maps.

Since we are only interested in the topology, we use the term topologically non-degenerate for a curve for which the conclusions of Proposition 46 hold:

Definition 48

We assume that $c\ne 0$ is the only critical value in (a, b). Denote $N_i$ the components of ${\text {Crit}}({\mathcal {E}})$ and assume that they are all isolated circles representing reparametrizations of non-constant closed geodesic $\gamma _i$ with energy c.

Then, we call $\gamma _i$ topologically non-degenerate if there are vector bundles $\nu ^-N_i\subseteq T\Omega |_N$ such that the sublevel set $\Omega _g^b$ retracts onto a space homeomorphic to $\Omega _g^a$ with the disc bundles $D\nu ^-N_i$ attached to $\Omega _g^a$ along the boundary via a retraction r with ${\mathcal {E}}_g\circ r\leqslant {\mathcal {E}}_g$ and such that $r|_{N_i}=id$ and $r|_{\Omega _g^a}=id$. $\square$

Remark 49

The assumption that the spectral value is isolated is actually too strong for our purpose; It would suffice to demand in Theorem 5 that a topologically non-degenerate $\gamma$ be isolated in the space of loops. The proof below would then work by localizing the gradient flow. One can do this by multiplying the gradient vector field with a bump function around a neighborhood of $N_i$ that is flow-invariant in the intended energy interval, and that separates $\gamma$ from other geodesics. The argument would become much more complicated as the resulting flows only locally transport the respective sub-level sets into each other.

We shall use a minimax principle. We use the following formulation from Klingenberg [26]. A flow-family ${\mathcal {A}}$ for ${\mathcal {E}}_g$ is a collection of subsets of $\Omega$ such that ${\mathcal {E}}_g|_{A}$ is bounded for all $A\in {\mathcal {A}}$ and such that $A\in {\mathcal {A}}$ implies $\varphi _g^t A\in {\mathcal {A}}$ for $t\geqslant 0$.

Proposition 50

([26, Theorem 2.1.1]) Let ${\mathcal {A}}$ be a flow-family for ${\mathcal {E}}_g$. Then

$$\begin{aligned} \inf _{A\in {\mathcal {A}}}\sup _{A} {\mathcal {E}}_g \end{aligned}$$

is a critical value of ${\mathcal {E}}_g$.

Proof of Proposition 7

We use Proposition 46 to define a suitable flow-family. For simplicity, assume that there is only one critical component. For Proposition 7 it is enough to consider the case $N_1=N\cong S^1$. The fundamental class of the transverse bundle relative its boundary $[D\nu ^-N;\partial D\nu ^-N]$ has nonempty intersection with the core N since it has nonempty intersection with any interior point. By extension the same is true for the class $\omega :=[\Omega ^a_g\bigcup _{\partial D\nu ^-N}D\nu ^-N;\Omega ^a_g]$. Denote by $r^*\omega$ the set of maps $u:(D\nu ^-N;\partial D\nu ^-N)\rightarrow (\Omega ^b_g,\Omega ^a_g)$ such that $[r\circ u]=\omega$. Then, the set of images of $u\in r^*\omega$ defines a flow-family.

The minimax value for $r^*\omega$ is the critical value c:

$$\begin{aligned} \inf _{u\in r^*\omega }\max {\mathcal {E}}_g\circ u \geqslant \inf _{u\in r^*\omega } \max {\mathcal {E}}_g\circ r\circ u \geqslant {\mathcal {E}}_g(N)=c. \end{aligned}$$

The other inequality is trivial since ${\mathcal {E}}_g$ restricted to the unstable disk bundle of N has maximum c.

The robustness statement now follows by using the very same retraction r to define a flow family for the perturbed metric $\widetilde{g}$: Let $\varepsilon >0$ be so small that c is the only critical value of $\mathcal E_g$ in $[(1-3\varepsilon )c,(1+3\varepsilon )c]$. Let $\widetilde{g}$ be a metric such that $\Vert v\Vert ^2_{\widetilde{g}}\in (1-\frac{1}{2}\varepsilon ,1+\frac{1}{2}\varepsilon )\Vert v\Vert ^2_{g}$ for all v. Note that for such $\varepsilon$ the following chain of inclusions holds

$$\begin{aligned} \Omega ^{(1-2\varepsilon )c}_{\widetilde{g}}\subseteq \Omega ^{(1-\varepsilon )c}_{g}\subseteq \Omega ^c_g\subseteq \Omega ^{(1+\varepsilon )c}_{\widetilde{g}}\subseteq \Omega ^{(1+2\varepsilon )c}_{g}. \end{aligned}$$

Let $r:\Omega ^{(1+2\varepsilon )c}_g\rightarrow \Omega ^{(1-\varepsilon )c}_g\bigcup _{\partial D\nu ^-N} D\nu ^-N$ be the retraction constructed with $\varphi _g^t$ and $r^*\omega$ the class described above. Define the subset $\widetilde{\omega }\subset r^*\omega$ by restriction of the target space $u:(D\nu ^-N;\partial D\nu ^-N)\rightarrow (\Omega ^{(1+\varepsilon )c}_{\widetilde{g}},\Omega ^{(1-2\varepsilon )c}_{\widetilde{g}})$. The set of images of maps in $\widetilde{\omega }$ is a flow-family for $\varphi _{\widetilde{g}}^t$ since it is defined through sub-level sets of ${\mathcal {E}}_{\widetilde{g}}$, and it is nonempty since it contains the $\varphi _g^t$-unstable disk bundle around N. We have

$$\begin{aligned} \inf _{u\in \widetilde{\omega }}\max {\mathcal {E}}_{\widetilde{g}}\circ u \geqslant \inf \max _{u\in \widetilde{\omega }} (1-\varepsilon ){\mathcal {E}}_{g} \circ u \geqslant (1-\varepsilon )c. \end{aligned}$$

On the other hand for u the $\varphi ^t_g$-unstable disk bundle around N we have $\max {\mathcal {E}}_{\widetilde{g}}\circ u\leqslant (1+\varepsilon )c$. Thus, the minimax principle produces some geodesic $\widetilde{\gamma }$ of ${\mathcal {E}}_{\widetilde{g}}$ with energy $|{\mathcal {E}}_{\widetilde{g}}(\widetilde{\gamma })-c|\leqslant \varepsilon c$.

Note that for any u in the flow-family, every path in the image of u is homotopic to a loop in N since the intersection of u and N is nonempty. Thus, also $\widetilde{\gamma }$ is homotopic to the unperturbed geodesic. $\square$