Polynomial Entropy and Polynomial Torsion for Fibered Systems

Abstract

Given a continuous fibered dynamical system, we first introduce the notion of polynomial torsion of a fiber, which measures the “infinitesimal variation” of the dynamics between the fiber and the neighboring ones. This gives rise to an (upper semicontinous) torsion function, defined on the base of the system, which is a new $C^{0}$ (fiber) conjugacy invariant. We prove that the polynomial entropy of the system is the supremum of the torsion of its fibers, which yields a new insight into the creation of polynomial entropy in fibered systems. We examine the relevance of these results in the context of integrable Hamiltonian systems or diffeomorphisms, with the particular cases of $C^{0}$-integrable twist maps on the annulus and geodesic flows. Finally, we bound from below the polynomial entropy of $\ell$-modal interval maps in terms of their lap number and answer a question by Gomes and Carneiro.

1 INTRODUCTION

The notions of metric and topological entropies play a central role in understanding the complexity of dynamical systems, mainly in the case of positive topological entropy. In the zero entropy case, in particular, for most integrable Hamiltonian systems, many questions about a good measurement of the complexity remain open. The notion of polynomial entropy was introduced in [7] (following several previous studies, see, in particular, [11]) to deal with the case of integrable Hamiltonian systems with “moderate” singularities.

The topological entropy of a continuous map on a compact metric space measures the exponential growth rate of the number of pairwise distinguishable¹ segments of orbits of length $n$ when $n$ tends to $\infty$, while the polynomial entropy measures the polynomial growth rate of the same quantity. We refer to [4] for precise definitions of the topological entropy and to [7] and the next section for the polynomial entropy. In particular, to define the polynomial torsion below, we crucially use of the possibility of defining in a natural way the entropies relative to a subset of the initial compact space (by considering segments of orbits issuing from the subset).

Both entropies share common properties (invariance by $C^{0}$ conjugacy, restriction, factors). However, a significant difference is that the topological entropy of the $n^{th}$ iterate of a function is $n$ times the topological entropy of the function, while the polynomial entropy of the $n^{th}$ iterate is equal to the polynomial entropy of the function. This remark (suitably generalized) will be at the core of our study of geodesic flows.

Another major difference occurs in the case of fibered dynamical systems, which are our main concern here. We say that $(X,\pi,B)$ is a (metric) compact continuous fibration when $X$ and $B$ are endowed with metrics $d_{X}$ and $d_{B}$ which make them compact and $\pi:X\rightarrow B$ is a continuous surjection. Given two continuous maps $f:X\rightarrow X$ and $g:B\rightarrow B$, we say that $(X,f)$ is fibered over $(B,g)$ relative to $\pi$ when $\pi\circ f=g\circ\pi$. We adopt the notation $(X,f,\pi,B,g)$ for such fibered systems (we always implicitly assume $X$, $B$ to be compact metric spaces and $f$, $\pi$, $g$ to be continuous).

Regarding the topological entropy of fibered systems, Bowen proved the following result.

The Bowen formula. Let $(X,f)$ be fibered over $(B,g)$ relative to $\pi$, where $f$ and $g$ are homeomorphisms. Then

$$h_{\mathrm{top}}(f)\leqslant\sup_{b\in B}h_{\mathrm{top}}\big{(}f,\pi^{-1}(b)\big{)}+h_{\mathrm{top}}(g),$$

(here $h_{\mathrm{top}}(f,\pi^{-1}(b))$ stands for the topological entropy of $f$ relative to the subset $\pi^{-1}(b)$). Moreover, when $g$ is the identity, the previous inequality becomes

$$h_{\mathrm{top}}(f)=\sup_{b\in B}h_{\mathrm{top}}\big{(}f,\pi^{-1}(b)\big{)}.$$

There is no hope to get a similar result for the polynomial entropy, as shown by the following simple example. Consider on $\mathbb{T}^{1}\times[0,1]$ the diffeomorphism $f:(\theta,r)\mapsto(\theta+r,r)$, naturally fibered over the identity relative to the projection map $\pi:\mathbb{T}^{1}\times[0,1]\rightarrow[0,1]$. The fibers $\mathbb{T}^{1}\times\{r\}$ are invariant under $f$, and the restriction of $f$ to a fiber is a rotation, so that the polynomial entropy of $f$ relative to any fiber is easily seen to be zero. Moreover, the polynomial entropy of the identity is zero. However, the polynomial entropy of $f$ is $1$ (see [7]).

To rule out this problem, we introduce in this paper the notion of polynomial torsion for a system $(X,f,\pi,B,g)$. The polynomial torsion of a fiber is obtained by “fattening the fiber” and taking the limit of the polynomial entropy relative to the fattened fiber when the radius of the fattening tends to zero (see the precise definition in Section 3). In this way, to each $b\in B$ we attach the polynomial torsion ${\tau_{\mathrm{pol}}}\big{(}f,\pi^{-1}(b)\big{)}\in[0,+\infty]$ of the fiber $\pi^{-1}(b)$.

Our first result is the following (easy) analogue of the Bowen formula for the polynomial entropy:

$$h_{\mathrm{pol}}(f)=\sup_{b\in B}{\tau_{\mathrm{pol}}}\big{(}f,\pi^{-1}(b)\big{)}.$$

The fact that the dependence of the polynomial entropy of $f$ on the dynamics of $g$ no longer appears here is natural since it is encapsulated in the fiber fattening process. The previous equality shows that the polynomial torsion can be seen as a good “local version” of the polynomial entropy for fibered systems, see the introduction of Section 3 for more details on our ultimate expectation with this notion.

Our second result is that the map $b\mapsto{\tau_{\mathrm{pol}}}\big{(}f,\pi^{-1}(b)\big{)}$ is upper semicontinuous on $B$. Since $B$ is compact, it is therefore bounded from above as soon as it is everywhere finite. As a consequence, the previous equality shows that the polynomial entropy of $f$ is finite if and only if the polynomial torsion of the fibers is everywhere finite.

It is therefore natural to introduce the torsion function $\tau_{f}:B\to[0,+\infty]$, which to each $b\in B$ associates the torsion of the fiber $\pi^{-1}(b)$. It turns out that $\tau_{f}$ is equivariant under $C^{0}$ fiber-conjugacies (see Section 3 for definitions), so that its main characteristics (number of discontinuities, jumps at the discontinuous points, etc) become new conjugacy invariants.

As a first application, we examine the case of $C^{2}$ completely integrable Hamiltonian systems and give some examples of the relevance of the previous notions and results, in particular, with a mechanism of creation of discontinuities of the torsion function. We also consider integrable systems or maps in the $C^{0}$ or $C^{1}$ setting, following the studies in [1, 2].

We then examine the case of geodesic flows on the tangent bundle of a compact Riemannian manifold $(M,\|\|)$, naturally fibered by the norm of the tangent vectors. Given $\nu\geqslant 0$, we denote by $T^{(\nu)}M$ the subset of $TM$ of all vectors of norm $\nu$ and, given $0<a<b$, we write

$$T(M,[a,b])=\cup_{\nu\in[a,b]}T^{(\nu)}M$$

for the “slice” comprised between $T^{(a)}M$ and $T^{(b)}M$. We prove the equality

$$h_{\mathrm{pol}}\big{(}\phi,T(M,[a,b])\big{)}={\tau_{\mathrm{pol}}}(\phi,T^{(1)}M)=h_{\mathrm{pol}}\big{(}\phi,T^{(1)}M\big{)}+1.$$

Finally, still using arguments from fibered systems, we obtain a lower bound of the polynomial entropy of modal maps of the interval as a function of the polynomial growth rate of turning points. This answers a question of [3].

2 POLYNOMIAL ENTROPY

We introduce the notion of polynomial entropy as presented in [7]. Let $(X,d)$ be a compact metric space. We consider a continuous map $f:X\rightarrow X$. For any integer $n>0$, we define the so-called Bowen metric $d_{n}^{f}$ on $X$ by

$$d_{n}^{f}(x,y)=\max_{0\leqslant k\leqslant n-1}d\big{(}f^{k}(x),f^{k}(y)\big{)}.$$

It turns out that, for any integer $n>0$, the metrics $d$ and $d_{n}^{f}$ are equivalent. Hence, $(X,d_{n}^{f})$ is a compact metric space and so, given $\varepsilon>0$, one can cover $X$ with a finite number of sets of $d_{n}^{f}$-diameter $\varepsilon$.

For $x\in X$ and $n\in\mathbb{N}$, we denote by $B_{n}(x,\varepsilon)$ the ball centered at $x$ and of radius $\varepsilon$ for the metric $d_{n}^{f}$ — in the following, such balls will be called $(n,{\varepsilon})$-balls. Given $Y\subset X$ (not necessarily $f-$invariant), we write $G(f,Y,n,\varepsilon)$ for the minimal number of balls $B_{n}(\cdot,\varepsilon)$ in a covering of $Y$² . When $X=Y$, we write $G(f,n,\varepsilon)$ instead of $G(f,X,n,\varepsilon)$.

Definition 1

The polynomial entropy of $f$ relative to $Y$, denoted by $h_{\mathrm{pol}}(f,Y)$, is defined by

$$h_{\mathrm{pol}}(f,Y)=\lim_{\varepsilon\rightarrow 0}\limsup_{n\rightarrow\infty}\dfrac{\log G(f,Y,n,\varepsilon)}{\log(n)}.$$

The polynomial entropy of $f$ is $h_{\mathrm{pol}}(f,X)$, which we denote by $h_{\mathrm{pol}}(f)$.

It is also useful to consider $(n,\varepsilon)$-separated sets in $Y$. Given $n\geqslant 0$ and $\varepsilon>0$, we say that a set $E\subset X$ is $(n,\varepsilon)$-separated if, for all $x,y\in E,d_{n}(x,y)>\varepsilon$. Denote by $S(f,Y,n,\varepsilon)$ the maximal cardinality of an $(n,\varepsilon)-$separated set contained in $Y$. The obvious inequalities

$$S(f,Y,n,2\varepsilon)\leqslant G(f,Y,n,\varepsilon)\leqslant S(f,Y,n,\varepsilon)$$

yield the equality³

$$h_{\mathrm{pol}}(f,Y)=\lim_{\varepsilon\rightarrow 0}\limsup_{n\rightarrow\infty}\dfrac{\log\big{(}S(f,Y,n,\varepsilon)\big{)}}{\log(n)}.$$

In order to control the behavior of the polynomial entropy from below, one can also introduce a Borel measure $\mu$ on $X$ which charges the open sets, and, given a subset $Y\subset X$, set

$$\mu_{\rm max}(Y,n,{\varepsilon})=\sup_{B\in{\mathscr{B}}(Y,n,{\varepsilon})}\mu(B),$$

(2.1)

where ${\mathscr{B}}(Y,n,{\varepsilon})$ is the set of $(n,{\varepsilon})$-balls which intersect $Y$. Then if $Y$ is a Borel subset, since $G(f,Y,n,\varepsilon)\geqslant\mu(Y)/\mu_{\rm max}(Y,n,{\varepsilon})$:

$$h_{\mathrm{pol}}(f,Y)\geqslant\lim_{\varepsilon\rightarrow 0}\limsup_{n\rightarrow\infty}\dfrac{-\log\big{(}\mu_{\rm max}(Y,n,{\varepsilon})\big{)}}{\log(n)}.$$

The following properties of $h_{\mathrm{pol}}$ have been proved in [7].

Proposition

Let $(X,d)$ be a compact metric space and $f$ a continuous map on $X$ .

• Independence of the metric. $h_{\mathrm{pol}}(f)$ does not depend on the choice of topologically equivalent metrics on X.

• Invariance. $h_{\mathrm{pol}}(f)=h_{\mathrm{pol}}(g)$ if $g:X\to X$ is $C^{0}$ conjugate to $f$.

• Factors. For any factor $f^{\prime}:X^{\prime}\to X^{\prime}$ of $f$, then

  $$h_{\mathrm{pol}}(f^{\prime})\leqslant h_{\mathrm{pol}}(f).$$

• Restrictions. For any $Y\subset X$,

  $$h_{\mathrm{pol}}(f,Y)\leqslant h_{\mathrm{pol}}(f).$$

  If, moreover, $Y\subset X$ is closed and invariant under $f$, then

  $$h_{\mathrm{pol}}(f_{|Y})=h_{\mathrm{pol}}(f,Y).$$

• Finite unions. If $X=\cup_{i=1}^{n}X_{i}$, where $X_{i}$ are subsets of $X$, then

  $$h_{\mathrm{pol}}(f)=\max h_{\mathrm{pol}}(f,X_{i}).$$

• Iterates. For $m\in\mathbb{N}^{*}$, $h_{\mathrm{pol}}(f^{m})=h_{\mathrm{pol}}(f)$ and if $f$ is invertible $h_{\mathrm{pol}}(f^{-m})=h_{\mathrm{pol}}(f)$.

• Products. Given another continuous map $f^{\prime}:X^{\prime}\to X^{\prime}$, where $X^{\prime}$ is a compact metric space,

  $$h_{\mathrm{pol}}(f\times f^{\prime})=h_{\mathrm{pol}}(f)+h_{\mathrm{pol}}(f^{\prime}).$$

Another major difference between the polynomial and the topological entropies is the fact that the finite union property does not extend to countable unions for the polynomial entropy, while it is still true for the topological entropy. Indeed, the polynomial entropy of time-one flow of a vector field on $[0,1]$ which vanishes only at $0$ and $1$ is equal to $1$, while it would be $0$ if the countable union property were satisfied, see [7] for a proof.

So far we have defined the polynomial entropy for discrete systems only. For a continuous flow $\Phi$ on a compact metric space $X$ the definitions are completely similar: one introduces the Bowen distances $d_{T}^{\Phi}$, the covering numbers $G(\Phi,Y,T,{\varepsilon})$ for a positive real number $T$ and set

$$h_{\mathrm{pol}}(\Phi,Y)=\lim_{{\varepsilon}\to 0}\limsup_{T\to\infty}\frac{\log G(\Phi,Y,T,{\varepsilon})}{\log T}.$$

It turns out that $h_{\mathrm{pol}}(\Phi,Y)=h_{\mathrm{pol}}(\phi,Y)$, where $\phi$ is the time-one map associated with $\Phi$.

3 FIBERED DYNAMICAL SYSTEMS AND THE POLYNOMIAL TORSION

Let us motivate our study by an example. Let $(X,f,\pi,B,g)$ be a fibered system. We say that $(X,f,\pi,B,g)$ has a contracting fibered structure when the following conditions hold true:

• $X$ is metrically fibered over $B$ : there exist a metric space $(F,d_{F})$ and a finite open covering $(U_{i})_{1\leqslant i\leqslant m}$ of $B$ such that for each $i$ there is an isometry $\phi_{i}:\pi^{-1}(U_{i})\to U_{i}\times F$ (this latter space being equipped with the product metric). We write $\phi_{i}(z)=(\pi_{i}(z),\varpi_{i}(z))\in U_{i}\times F$.

• If $z,z^{\prime}$ are two points of $X$ such that there exist $i$ and $j$ in $\{1,\ldots,n\}$ such that $z,z^{\prime}\in\pi_{i}^{-1}(U_{i})$ and $f(z),f(z^{\prime})\in\pi_{j}^{-1}(U_{j})$, then

  $$d_{F}\big{(}\varpi_{j}(f(z)),\varpi_{j}(f(z^{\prime}))\big{)}\leqslant d_{F}\big{(}\varpi_{i}(z),\varpi_{i}(z^{\prime})\big{)}.$$

The following result was proved in [6].

Theorem. Assume that $(X,f,\pi,B,g)$ has a contracting fibered structure. Then

$$h_{\mathrm{pol}}(f)=h_{\mathrm{pol}}(g).$$

A simple example is that of a diffeomorphism $f$ of a manifold $M$ which admits a normally hyperbolic compact invariant manifold $N$. Then its stable and unstable manifolds $W^{\pm}(N)$ admit an invariant foliation by the stable manifolds of the points of $N$, and there exists a projection $\pi^{\pm}$ from a neighborhood $E$ of $N$ in $W^{\pm}(N)$ to $N$ which associates with each point $x$ the unique point $a\in N$ such that $x\in W^{\pm}(a)$. It is not difficult to see that one can choose in the neighborhood of each invariant manifold $W^{\pm}(N)$ a metric, topologically equivalent to the initial one, such that $(W^{\pm}(N),f_{|W^{\pm}},\pi^{\pm},N,f_{|N})$ has a contracting fibered structure. Consequently,

$$h_{\mathrm{pol}}\big{(}f,W^{\pm}(N)\big{)}=h_{\mathrm{pol}}(f,N).$$

This immediately yields the following remark, thanks to the finite union property for the polynomial entropy.

Corollary 1

Let $\Phi$ be the flow of a Morse-Smale vector field on a manifold $M$ . Then $h_{\mathrm{pol}}(\Phi)=0$ .

This model statement (which does not seem to have been noticed before) is a good illustration of the type of questions we would like to investigate for more general systems. On the one hand, the polynomial entropy of systems which are totally wandering is quite well understood, mainly after the study of [5] and the recent preprint [12]. On the other hand, the Conley decomposition theorem makes the wandering set of a system appear (with a grain of salt) as “fibered” over the nonwandering set. It is therefore natural to try to apply the previous remark in this context, and one obviously has first to set out a more local version of the polynomial entropy. This is the aim of this section.

3.1 The Polynomial Torsion

Let $(X,f,\pi,B,g)$ be a fibered system.

Definition 2

Given $b\in B$, we define the polynomial torsion of $f$ on the fiber $\pi^{-1}(b)$ by

$${\tau_{\mathrm{pol}}}\big{(}f,\pi^{-1}(b)\big{)}=\lim_{\delta\rightarrow 0}h_{\mathrm{pol}}\big{(}f,\pi^{-1}(B(b,\delta))\big{)},$$

where $B(b,\delta)$ stands for the ball centered at $b$ and of radius $\delta$ for the metric $d_{B}$. We also say that ${\tau_{\mathrm{pol}}}\big{(}f,\pi^{-1}(b)\big{)}$ is the torsion of the fiber $\pi^{-1}(b)$.

Note that the limit w.r.t. $\delta$ exists since $h_{\mathrm{pol}}\big{(}f,\pi^{-1}(B(b,\delta))\big{)}$ is nonincreasing w.r.t. $\delta$. We now state some immediate properties of the polynomial torsion. Since the proofs are similar to those of Proposition 2, we only give the first one.

$\bullet$ Independence of the metrics. Let us first prove that the polynomial torsion of $f$ on $\pi^{-1}(b)$ is invariant by the choice of topologically equivalent metrics on $X$ and $B$.

Proof

Let $d_{X}$, $d^{\prime}_{X}$ (resp $d_{B}$, $d^{\prime}_{B}$) be topologically equivalent metrics on $X$ (resp. $B$). Given $\delta>0$, there is a $\delta^{\prime}$ such that (with obvious notation) $B^{\prime}(b,\delta^{\prime})\subset B(b,\delta)$, so that

$$h_{\mathrm{pol}}\big{(}f,\pi^{-1}(B^{\prime}(b,\delta^{\prime})\big{)}\leqslant h_{\mathrm{pol}}\big{(}f,\pi^{-1}(B(b,\delta)\big{)},$$

where $h_{\mathrm{pol}}$ is relative to $d_{X}$. Now since $d_{X}$ is equivalent to $d^{\prime}_{X}$,

$$h_{\mathrm{pol}}^{\prime}\big{(}f,\pi^{-1}(B^{\prime}(b,\delta^{\prime})\big{)}=h_{\mathrm{pol}}\big{(}f,\pi^{-1}(B(b,\delta^{\prime})\big{)},$$

where $h_{\mathrm{pol}}^{\prime}$ is relative to $d^{\prime}_{X}$. Hence, ${\tau_{\mathrm{pol}}}^{\prime}\big{(}f,\pi^{-1}(b)\big{)}\leqslant{\tau_{\mathrm{pol}}}\big{(}f,\pi^{-1}(b)\big{)}$, and one gets the equality by symmetry.     $\square$

$\bullet$ Invariance. We pass to conjugacies that we have to set out in the context of fibered systems. Consider another fibration $(X^{\prime},\pi^{\prime},B^{\prime})$ and a fibered system $(X^{\prime},f^{\prime},\pi^{\prime},B^{\prime},g^{\prime})$. We say that the systems $(X,f)$ and $(X^{\prime},f^{\prime})$ are fiber-conjugate if there exist homeomorphisms $\varphi:X\rightarrow X^{\prime}$ and $\psi:B\rightarrow B^{\prime}$ satisfying $\psi\circ\pi=\pi^{\prime}\circ\varphi$ such that

$$\varphi\circ f=f^{\prime}\circ\varphi,\qquad\psi\circ g=g^{\prime}\circ\psi.$$

Then, transporting the metrics on $X$ and $B$ by $\varphi$ and $\psi$ and using the previous proposition yield the equality

$${\tau_{\mathrm{pol}}}\big{(}f^{\prime},(\pi^{\prime})^{-1}(\psi(b))\big{)}={\tau_{\mathrm{pol}}}\big{(}f,\pi^{-1}(b)\big{)}$$

for all $b\in B$.

$\bullet$ Factors. We say that a system $(X^{\prime},f^{\prime})$ fibered over $(B^{\prime},g^{\prime})$ relative to a surjective continous map $\pi^{\prime}:X^{\prime}\to B^{\prime}$ is a factor of $(X,f)$ if there are two continuous surjective maps $\varphi:X\rightarrow X^{\prime}$ and $\psi:B\to B^{\prime}$ satisfying $\psi\circ\pi=\pi^{\prime}\circ\varphi$ such that

$$\varphi\circ f=f^{\prime}\circ\varphi,\qquad\psi\circ g=g^{\prime}\circ\psi.$$

Then if $(X^{\prime},f^{\prime})$ is a factor of $(X,f)$, for all $b\in B$:

$${\tau_{\mathrm{pol}}}\big{(}f^{\prime},(\pi^{\prime})^{-1}(\psi(b))\big{)}\leqslant{\tau_{\mathrm{pol}}}\big{(}f,\pi^{-1}(b)\big{)}.$$

$\bullet$ Restrictions. For any $Y\subset X$ and for all $b\in\pi(Y)$,

$${\tau_{\mathrm{pol}}}\big{(}f_{|Y},\pi^{-1}(b)\cap Y\big{)}\leqslant{\tau_{\mathrm{pol}}}\big{(}f,\pi^{-1}(b)\big{)}.$$

$\bullet$ Iterates. For all $m\in\mathbb{N}$ and $b\in B$,

$${\tau_{\mathrm{pol}}}(f^{m},\pi^{-1}(b))={\tau_{\mathrm{pol}}}\big{(}f,\pi^{-1}(b)\big{)}.$$

$\bullet$ Products. Let $(X^{\prime},f^{\prime})$ be fibered over $(B^{\prime},g^{\prime})$ relative to $\pi^{\prime}$. Then, for all $(b,b^{\prime})\in B\times B^{\prime}$,

$${\tau_{\mathrm{pol}}}\big{(}f\times f^{\prime},\big{(}\pi\times\pi^{\prime})^{-1}(b,b^{\prime})\big{)}={\tau_{\mathrm{pol}}}\big{(}f,\pi^{-1}(b)\big{)}+{\tau_{\mathrm{pol}}}\big{(}f^{\prime},\pi^{-1}(b^{\prime})\big{)}.$$

Again, one gets similar definitions and properties for continuous flows.

3.2 The Polynomial Entropy of Fibered Systems

The first main result of this paper is the following.

Theorem 1

Let $(X,f,\pi,B,g)$ be a fibered system. Then

$$h_{\mathrm{pol}}(f)=\max_{b\in B}{\tau_{\mathrm{pol}}}\big{(}f,\pi^{-1}(b)\big{)}.$$

(3.1)

Proof

By definition of the polynomial torsion, for all $b\in B$, ${\tau_{\mathrm{pol}}}\big{(}f,\pi^{-1}(b)\big{)}\leqslant h_{\mathrm{pol}}(f)$, so it remains to prove the converse inequality. Fix $\delta>0$. There is a covering $\{B(b_{i},\delta)$, $1\leqslant i\leqslant N(\varepsilon)\}$ of $B$ by $d_{B}$-balls of radius $\delta$. Hence,

$$X=\pi^{-1}(B)=\bigcup_{i=1}^{N(\varepsilon)}\pi^{-1}\big{(}B(b_{i},\delta)\big{)}.$$

Therefore, there is an $i(\delta)\in\{1,\ldots,N(\varepsilon)\}$ such that $h_{\mathrm{pol}}(f)=h_{\mathrm{pol}}\big{(}f,\pi^{-1}(B(b_{i(\delta)},\delta))\big{)}$. Up to extraction, the sequence $b_{i(1/m)}$ converges as $m\to\infty$ to a point $b\in B$. Hence, for any $\alpha>0$, there is an $m$ large enough such that $B\big{(}b_{i(1/m)},1/m\big{)}\subset B(b,\alpha)$, so that

$$h_{\mathrm{pol}}\big{(}f,\pi^{-1}(B(b,\alpha))\big{)}\geqslant h_{\mathrm{pol}}\Big{(}f,\pi^{-1}\big{(}B\big{(}b_{i(1/m)},1/m\big{)}\big{)}\Big{)}=h_{\mathrm{pol}}(f).$$

Therefore,

$${\tau_{\mathrm{pol}}}\big{(}f,\pi^{-1}(b)\big{)}\geqslant h_{\mathrm{pol}}(f)$$

which completes the proof.     $\square$

3.3 The Torsion Function

To simplify the notation, we now set ${\tau_{\mathrm{pol}}}\big{(}f,\pi^{-1}(b)\big{)}=\tau_{f}(b)$, so that $\tau_{f}:B\to[0,+\infty]$ is a well-defined function, which we call the torsion function of $f$. Our second main result is the following.

Theorem 2

The torsion function $\tau_{f}$ is upper semicontinuous.

Proof

Fix $b\in B$. Given $\alpha>0$, there is a $\delta_{0}>0$ such that for $0<\delta\leqslant\delta_{0}$

$$\tau_{f}(b)+\alpha\geqslant h_{\mathrm{pol}}\big{(}f,\pi^{-1}(B(b,\delta))\big{)}.$$

Let $b^{\prime}\in B(b,\delta_{0}/2)$. Then $B(b^{\prime},\delta_{0}/2)\subset B(b,\delta_{0})$ and

$$h_{\mathrm{pol}}\big{(}f,\pi^{-1}(B(b,\delta_{0})\big{)}\geqslant h_{\mathrm{pol}}\big{(}f,\pi^{-1}(B(b^{\prime},\delta_{0}/2))\big{)}\geqslant\lim_{\delta\to 0}h_{\mathrm{pol}}\big{(}f,\pi^{-1}(B(b^{\prime},\delta))\big{)}=\tau_{f}(b^{\prime}),$$

so that $\tau_{f}(b)+\alpha\geqslant\tau_{f}(b^{\prime})$ for $b^{\prime}\in B(b,\delta_{0}/2)$, which proves the upper semicontinuity of $\tau_{f}$.     $\square$

We obtain a finiteness criterion as an immediate consequence of the previous two theorems.

Corollary 2

The polynomial entropy $h_{\mathrm{pol}}(f)$ is finite if and only if the function $\tau_{f}$ is everywhere finite.

Proof

If $h_{\mathrm{pol}}(f)<+\infty$, then ${\tau_{\mathrm{pol}}}(b)\leqslant h_{\mathrm{pol}}(f)<+\infty$ for every $b\in B$. Conversely, an upper semicontinuous function defined on a compact space with values in $\mathbb{R}$ is bounded from above. Hence, if $\tau_{f}$ is everywhere finite, $h_{\mathrm{pol}}(f)$ is finite by (3.1).     $\square$

3.4 The Torsion Function as a Conjugacy Invariant

Observe finally that the torsion function $\tau_{f}$ is equivariant under fibered conjugacy, that is, with the notation of Section 3.1:

$$\tau_{f}=\tau_{f^{\prime}}\circ\psi.$$

In particular, we will be interested in systems for which the torsion function $\tau$ admits a finite set of discontinuities $\{b_{1},\ldots,b_{m}\}$ at which it experiences a “jump”, that is,

$$\tau(b_{i})=\lim_{x\to b_{i},\ x\neq b_{i}}\tau(x)+\ell_{i},$$

where the jump $\ell_{i}$ is negative⁴ . When this is the case, the number of discontinuities together with the values of their jumps are new examples of invariants under fibered conjugacy.

4 INTEGRABLE HAMILTONIAN SYSTEMS AND GEODESIC FLOWS

Let $(M,\Omega)$ be a $2n$-dimensional $C^{\infty}$ symplectic manifold and fix a $C^{2}$ Hamiltonian $H:M\to\mathbb{R}$. Then $H$ is said to be completely integrable when there exist $C^{2}$ functions $\pi_{1},\ldots,\pi_{n}$ such that $\{\pi_{i},\pi_{j}\}=0$ for all $i,j$ (where $\{,\}$ is the Poisson bracket associated with $\Omega$), $\{\pi_{i},H\}=0$ for all $i$ and the map $\pi=(\pi_{1},\ldots,\pi_{n}):M\to\mathbb{R}^{n}$ is a submersion on an open dense subset of $M$.

Many completely integrable systems which occur from geometry or physics are actually much simpler: the singular points of the map $f$ are localized on submanifolds of $M$ and the behavior of $f$ transverse to those submanifolds is nondegenerate in the Morse sense. This is in particular the case for the so-called Bott integrable systems introduced by Fomenko and, more generally, for the systems which satisfy a Williamson condition (see [13]), which we call Williamson integrable.

Completely integrable systems have a natural fibered structure: their time-one flow $\Phi$ (which is well-defined, for instance, when the Hamiltonian is a proper map) and the map $\pi:M\to\pi(M)\subset\mathbb{R}^{n}$ satisfy the relation

$$\pi\circ\Phi=\pi.$$

Let $R$ be the subset of regular values of $\pi$. Then, under suitable compactness conditions, the subset $\pi^{-1}(R)\subset M$ admits a covering by “angle-action” domains: this is the content of the Liouville-Mineur-Arnold theorem. More precisely, for any compact connected component $T$ of a preimage $\pi^{-1}(b)$, where $b$ is a regular value of $\pi$, the previous theorem asserts that $T$ is diffeomorphic to a torus $\mathbb{T}^{n}$ and admits a tubular neighborhood endowed with symplectic angle-action coordinates $(\theta,r)\in\mathbb{T}^{n}\times B$, where $B$ is a ball in $\mathbb{R}^{n}$, relative to which the initial Hamiltonian depends only on the action $r$.

It turns out that the torsion function induced by $\Phi$ can be computed in (the image by $\pi$ of) angle-action domains, which will be proved in Section 4.1. This makes Williamson integrable systems (under natural nondegeneracy conditions) good examples of systems for which the torsion function admits a finite number of jumps.

In Section 4.2, we examine the case of integrable Hamiltonian systems or diffeomorphisms in a weaker sense, relaxing the $C^{2}$ assumption on the first integrals (see [1]) or assuming only the existence of suitable $C^{0}$ Lagrangian foliations of their phase space (see [2]). We will give a sufficient condition for these systems to have finite polynomial entropy (and thus everywhere finite torsion).

Finally, a Hamiltonian system on $(M,\Omega)$ defined by a proper Hamiltonian function $H$ has a natural fibered structure over any compact interval $[a,b]$ of $\mathbb{R}$: with our standard notation, set $X=H^{-1}([a,b])$, $f=\Phi$, $B=[a,b]$, $g={\rm Id}$ and $\pi=H$ (we say that $X$ is a slice). It is in general difficult to compute the torsion function of such a system, however, this is possible for (co)-geodesic systems once the polynomial entropy of the restriction to the unit tangent bundle is known. In Section 4.3 we use this remark to estimate the polynomial entropy of geodesic flows on slices.

4.1 The Torsion of Angle-Action Systems and Some Consequences

The annulus $T^{*}\mathbb{T}^{d}=\mathbb{T}^{d}\times\mathbb{R}^{d}$ will be equipped with the angle-action coordinates $(\theta,r)\in\mathbb{T}^{d}\times\mathbb{R}^{d}$ and the canonical symplectic form. The time-one Hamiltonian flow generated by a $C^{2}$ Hamiltonian function $h$ which depends only on the action variables takes the simple form

$$f(\theta,r)=\big{(}\theta+\omega(r),r\big{)},$$

(4.1)

where $\omega(r)$ is the (projection on $\mathbb{T}^{n}$ of) the gradient vector $\nabla h(r)$, and is therefore $C^{1}$. The following result was proved in [7].

Theorem. Assume that $h:\mathbb{R}^{n}\to\mathbb{R}$ is $C^{2}$ . Then, given $r^{0}\in\mathbb{R}^{n}$ and $R>0$ ,

$$h_{\mathrm{pol}}\big{(}f,\mathbb{T}^{n}\times B(r^{0},R]\big{)}=\max_{r\in B(r^{0},R]}{\rm rank}({\rm Hessh})(r),$$

where $B(r^{0},R]$ stands for the closed Euclidean ball centered at $0$ with radius $R$ .

Fix now $R>0$. Then, setting $X=\mathbb{T}^{n}\times B(0,R]$, the restriction of $f$ to $X$ (still denoted by $f$) has a fibered structure over $B(0,R]$ relative to the second projection $\pi:\mathbb{T}^{n}\times B(0,R]\to B(0,R]$. The previous theorem immediately yields the following result.

Corollary 3

Under the previous assumptions, the torsion function $\tau_{f}:B(0,R]\to[0,+\infty]$ associated with $f$ reads

$$\tau_{f}(r^{0})=\lim_{\delta\to 0}\max_{r\in B(r^{0},\delta)}{\rm rank}({\rm Hessh})(r).$$

Consequently, $\tau_{f}(r^{0})={\rm rank}({\rm Hessh})(r^{0})+d(r^{0})$, where $d(r^{0})\in\big{\{}0,\ldots,n-{\rm rank}({\rm Hessh})(r^{0})\big{\}}$ takes positive values at the discontinuities of $\tau$. Since any closed subset of $[-R,R]$ is the set of zeroes of a $C^{1}$ (and even $C^{\infty}$) function, one can produce examples of angle-action systems on $T^{*}\mathbb{T}$ with arbitrary discontinuities of their torsion (with natural generalizations to higher dimensional systems). However, in most known examples, the situation is much simpler.

Example 1

The simple pendulum on $T^{*}\mathbb{T}$, governed by the Hamiltonian $P(\theta,r)={\tfrac{1}{2}}r^{2}-\cos(2\pi\theta)$, admits three maximal domains in which there exist conjugacies to angle-action systems: one inside the separatrices (without the elliptic point) and two outside the separatrices. It turns out that the angle-action systems associated to the domains outside the separatrices have nondegenerate rotation number $\omega$ and, therefore, the torsion function is everywhere equal to $1$. As for the inner domain, the corresponding $\omega$ has isolated singularities, hence, the torsion function is everywhere constant too. Therefore, the simple pendulum admits a single jump fiber $P^{-1}(1)$, which corresponds to the union of the separatrices and the hyperbolic point. One deduces from the study of [7] that the torsion of this fiber is equal to 2.

Example 2

A more general example was studied in [7] from the global point of view of the polynomial entropy. Let $S$ be a compact orientable surface without boundary endowed with a symplectic form $\Omega$. Let $H:S\to\mathbb{R}$ be a $C^{2}$ Morse function, in the sense that all its critical points are nondegenerate. Its saddle points can take the same values, which yields the existence of fibers $H^{-1}(c)$ containing several saddle points (these fibers are called polycycles), as shown in Fig. 1.

Fig. 1

From [7]: a critical fiber with 6 saddle points. A single component ${\mathscr{C}}(\rho_{0})$ is represented in gray, together with the boundary curve $C_{\rho_{0}}\subset H^{-1}(c\pm\rho_{0})$.

Given such a fiber and a $\rho_{0}>0$ small enough, the connected components of a slice of the form $H^{-1}\big{(}]c,c+\rho_{0}]\big{)}$ or $H^{-1}\big{(}[c-\rho_{0},c[\big{)}$ are annuli, their number is finite (namely, 8 in Fig. 1) and their union, together with the polycycle, form a neighborhood of the polycycle. It has been proved in [7] that, for such a component ${\mathscr{C}}(\rho_{0})$, the flow $\Phi$ of the Hamiltonian vector field generated by $H$ satisfies

$$h_{\mathrm{pol}}\big{(}\Phi,{\mathscr{C}}(\rho_{0})\big{)}=2.$$

Moreover, one readily checks that $h_{\mathrm{pol}}\big{(}\Phi,H^{-1}(c)\big{)}=1$, and this proves that ${\tau_{\mathrm{pol}}}\big{(}H^{-1}(c)\big{)}=2$.

The critical fibers of $H$ are either polycycles or isolated elliptic points. The complement in $S$ of the union of the critical fibers is covered by angle-action domains on which the system is conjugated to the system (4.1) (with $n=1$). Adding the (generic on $H$) assumption that the singularities of the rotation number $\omega$ (relative to such a conjugacy domain) are isolated, the upper semicontinuity property of $\tau_{\Phi}$ proves that it is equal to $1$ in the complement of the critical values. By the same token, one sees that the torsion at the critical values corresponding to elliptic points is also equal to $1$.

We therefore proved that, for a generic family of Morse Hamiltonians on $S$, the only jump fibers are those which contain saddle points, and that the jump of the torsion at these fibers is equal to $1$.

Some extensions of this result can be proved in the context of Bott integrable systems following the global study of [6], and we conjecture that the torsion of Williamson integrable systems takes only integer values: this is an example of a rigid behavior in the terminology of [12]. We will go back to this question in a subsequent work.

4.2 Weakly Integrable Systems

In this section we are interested in generalizations of the notion of integrability under weaker regularity conditions on the first integrals. In this context, in contrast with the previous $C^{2}$ case, the entropic behavior is flexible. Let us begin with a “weak angle-action model”.

Proposition

Fix $n\geqslant 1$ and let $f:\mathbb{T}^{n}\times B(0,R]\to\mathbb{T}^{n}\times B(0,R]$ be defined by

$$f(\theta,r)=\big{(}\theta+\omega(r),r\big{)},$$

(4.2)

where $\omega:B(0,R]\to\mathbb{R}^{n}$ is $(c,\alpha)$ -locally Hölderian, that is, there is a $\rho>0$ such that

$$\|\omega(r)-\omega(r^{0})\|\leqslant c\|r-r^{0}\|^{\alpha}$$

for any pair $(r,r^{0})$ such that $\|r-r^{0}\|<\rho$ . Then $h_{\mathrm{pol}}(f)\leqslant 1/\alpha$ .

Proof

Fix $m>0$, $r^{0}\in B(0,R)$ and ${\varepsilon}>0$. Then, given $\theta^{0}\in\mathbb{T}^{n}$, the set of all $(\theta,r)$ such that

$$\|\theta-\theta^{0}\|<{\varepsilon}/2,\qquad\|\omega(r)-\omega(r^{0})\|<{\varepsilon}/2m,$$

is contained in the $(m,{\varepsilon})$-ball $B_{m}\big{(}(\theta^{0},r^{0}),{\varepsilon}\big{)}$. Therefore, for ${\varepsilon}<\rho$, the set of all $(\theta,r)$ such that

$$\|\theta-\theta^{0}\|<{\varepsilon}/2,\qquad\|r-r^{0}\|<\Big{(}\frac{1}{c}\frac{{\varepsilon}}{2m}\Big{)}^{1/\alpha}:=\delta({\varepsilon})$$

is also contained in $B_{m}\big{(}(\theta^{0},r^{0}),{\varepsilon}\big{)}$. Varying the center $\theta^{0}$, this yields a covering of $\mathbb{T}^{n}\times B\big{(}r^{0},\delta({\varepsilon})\big{)}$ by at most $\overline{c}/{\varepsilon}^{n}$ $({\varepsilon},m)$-balls, where $\overline{c}$ is a uniform constant. Therefore, the set $\mathbb{T}^{n}\times B(0,R)$ admits a covering by at most $\delta({\varepsilon})\overline{c}/{\varepsilon}^{n}$ $({\varepsilon},m)$-balls and

$$h_{\mathrm{pol}}(f)\leqslant\limsup_{m\to\infty}\frac{1}{\log m}\log\left(\frac{\delta({\varepsilon})\overline{c}}{{\varepsilon}^{n}}\Big{(}\frac{{\varepsilon}}{2\overline{c}m}\Big{)}^{1/\alpha}\right)=1/\alpha,$$

which proves the claim.     $\square$

The flexibility of the entropic behavior at the polynomial scale is already seen when $n=1$.

Proposition

Assume that $n=1$ and that the function $\omega$ in (4.2) satisfies

$$\omega(r)=\left|r\right|^{\alpha},$$

where $0<\alpha<1$ . Then $\tau_{f}(r)=1$ for $r\in[-R,R]\setminus\{0\}$ and $\tau_{f}(0)=\frac{1}{\alpha}$ . Consequently, $h_{\mathrm{pol}}(f)=1/\alpha$ .

Proof

The proof follows the same lines as the previous one (using the estimate from below (2.1)), so we omit it.     $\square$

Observe finally that, if $\omega$ has a super-Hölderian growth at $0$, such as $\omega(r)=-1/\log\left|r\right|$, then $h_{\mathrm{pol}}(f)={\tau_{\mathrm{pol}}}_{f}(0)=+\infty$.

In [1], the authors introduce the notion of $GC^{1}$ integrability for $C^{2}$ Hamiltonian systems: a $C^{2}$ Hamiltonian $H:T^{*}\mathbb{T}^{n}\to\mathbb{R}$ is said to be $GC^{1}$-integrable when there are $C^{1}$ functions $\pi_{1},\ldots,\pi_{n}$ on $T^{*}\mathbb{T}^{n}$ such that $\{\pi_{i},\pi_{j}\}=0$ for all $i,j$ and $\{\pi_{i},H\}=0$ for all $i$, and, setting $\pi=(\pi_{1},\ldots,\pi_{n})$, such that the levels $\pi^{-1}(a)$ are the graphs of $C^{1}$ maps from $\mathbb{T}^{n}\to\mathbb{R}^{n}$. They then prove that on subsets of the form $\pi^{-1}\big{(}B(0,R)\big{)}$, the Hamiltonian system defined by $H$ is $C^{0}$ conjugate to a system of the form (4.2). A natural question would be to determine additional conditions under which the function $\omega$ can be proved to be Hölderian. This question seems to be open in the general case.

However, in [2], the authors define the notion of Lipschitz integrability for twist maps of the annulus. They then prove that any Lipschitz integrable twist map is conjugate to a system (4.2) with bi-Lipschitz $\omega$.

Corollary 4

The polynomial entropy of a Lipschitz integrable twist map is $\leqslant 1$ .

4.3 Geodesic Flows

In this section we consider a smooth compact Riemannian manifold $(M,g)$ and we denote by $\Phi$ the associated geodesic flow on $TM$. Then, thanks to the homogeneity of the metric, given $(x,v)\in TM$, for any $\lambda>0$:

$$\Phi\big{(}t,(x,\lambda v)\big{)}=\Phi\big{(}\lambda t,(x,v)\big{)}.$$

Given $\nu>0$, we denote by $T^{(\nu)}M$ the subset of $TM$ formed by the tangent vectors of norm $\nu$ and by $\Phi^{(\nu)}$ the restriction of $\Phi$ to $T^{(\nu)}M$. Then the previous reparameterization, together with a generalization of our result on the polynomial entropy of the iterates of a continuous map⁵ , yields the following.

Proposition

For any $\nu>0$ :

$$h_{\mathrm{pol}}\big{(}\Phi,T^{(\nu)}M\big{)}=h_{\mathrm{pol}}\big{(}\Phi,T^{(1)}M\big{)}\leqslant+\infty.$$

The geodesic flow has the obvious fibered structure whose fibers are the levels $T^{(\nu)}M$, so that the notion of polynomial torsion makes sense.

Theorem 3

For any $\nu>0$ :

$${\tau_{\mathrm{pol}}}\big{(}\Phi,T^{(\nu)}M\big{)}=h_{\mathrm{pol}}\big{(}\Phi,T^{(\nu)}M\big{)}+1\leqslant+\infty.$$

Proof

We endow $TM$ with the Sasaki metric and the usual Riemannian measure. Given $\nu_{0}>0$ and $0<\lambda<1$, the slice $TM\big{(}[\lambda^{-1}\nu_{0},\lambda\nu_{0}]\big{)}$ is diffeomorphic to the direct product $T^{(\nu_{0})}M\times[\lambda^{-1},\lambda]$ by the map $(x,v)\mapsto\big{(}(x,v),\|v\|/\nu_{0}\big{)}$, and the Sasaki distance is equivalent to the product $d$ of its restriction to $T^{(\nu_{0})}M$ with the length on $[\lambda^{-1},\lambda]$. In the same way, we endow the slice with the product of the Liouville measure on $T^{(\nu_{0})}M$ with the length on $[\lambda^{-1},\lambda]$.

Let us first prove that ${\tau_{\mathrm{pol}}}\big{(}\Phi,T^{(\nu_{0})}M\big{)}\leqslant h_{\mathrm{pol}}\big{(}\Phi,T^{(\nu_{0})}M\big{)}+1$. Let $\kappa$ be a Lipschitz constant for the $C^{1}$ flow $\Phi$ restricted to $[-1,1]\times T^{(\nu)}M$. Fix ${\varepsilon}>0$ and $T>0$. Fix an $a\in T^{(\nu_{0})}M$ and a $(T,{\varepsilon}/2)$ ball $B:=B_{T}(a,{\varepsilon}/2)\subset T^{(\nu_{0})}M$ for the restriction of $\Phi$ to $T^{(\nu_{0})}M$. Fix $(x,v)\in B$. Then for $\lambda>0$ and for $0\leqslant t\leqslant T$:

$$\Phi\big{(}t,(x,\lambda v)\big{)}=\Phi\big{(}\lambda t,(x,v)\big{)}=\Phi\big{(}t(\lambda-1),\Phi\big{(}t,(x,v)\big{)}\big{)},$$

(4.3)

hence,

$$d\Big{(}\Phi\big{(}t,(x,\lambda v)\big{)},\Phi\big{(}t,(x,v)\big{)}\Big{)}\leqslant\kappa\left|t(\lambda-1)\right|.$$

Therefore, if $\kappa\left|T(\lambda-1)\right|<{\varepsilon}/2$, the point $(x,\lambda v)$ is $(T,{\varepsilon}/2)$-close to $(x,v)$ and therefore $(T,{\varepsilon})$-close to $a$. Consequently, the ball $B_{T}(a,{\varepsilon})\subset TM$ contains the subset

$$\bigcup_{(x,v)\in B}\{x\}\times\Big{[}\big{(}1-{\varepsilon}/(2T\kappa)\big{)}v,\big{(}1+{\varepsilon}/(2T\kappa)\big{)}v\Big{]}.$$

Hence, a minimal covering of $T^{(\nu_{0})}M$ by a number $G(T,{\varepsilon})$ of $(T,{\varepsilon})$-balls yields a covering of the slice $TM\Big{(}\Big{[}\big{(}1-{\varepsilon}/(2T\kappa)\big{)}\nu,\big{(}1+{\varepsilon}/(2T\kappa)\big{)}\nu\Big{]}\Big{)}$ with the same cardinality. Taking the homogeneity under account, one sees that there is a fixed $c>0$ such that the slice $TM\big{(}[\lambda^{-1}\nu_{0},\lambda\nu_{0}]\big{)}$ admits a covering by $(T,{\varepsilon})$-balls with cardinality $cT\cdot G(T,{\varepsilon})$. This proves (taking the $\lambda\to 1$ limit) that

$${\tau_{\mathrm{pol}}}\big{(}\Phi,T^{(\nu_{0})}M\big{)}\leqslant h_{\mathrm{pol}}\big{(}\Phi,T^{(\nu_{0})}M\big{)}+1.$$

Conversely, there exists $\overline{\kappa}>0$, $\overline{t}>0$ such that for $\left|t\right|\leqslant\overline{t}$ and any point $(x,v)\in TM\big{(}[\lambda^{-1}\nu_{0},\lambda\nu_{0}]\big{)}$

$$d\big{(}\Phi\big{(}t,(x,v)\big{)},(x,v)\big{)}>\overline{\kappa}\left|t\right|.$$

Fix $a\in T^{(\nu_{0})}M$ and ${\varepsilon}<\overline{t}$, and set $B=B_{T}^{\Phi}(a,{\varepsilon})$ and $B^{(\nu_{0})}=B\cap T^{(\nu_{0})}M$. Then, for any $(x,v)\in B$, the projection $(x,\nu_{0}v/\|v\|)$ on $T^{(\nu_{0})}M$ belongs to $B^{(\nu_{0})}$ by definition of $d$ (and so of $d_{T}^{\Phi}$).

Fix $(x,v)\in B$ and set $\overline{v}=\sigma v$ with $\sigma=\nu_{0}/\|v\|$. Then

$$d\Big{(}\Phi\big{(}T,(x,\overline{v})\big{)},\Phi\big{(}T,(x,v)\big{)}\Big{)}=d\Big{(}\Phi\big{(}T(\sigma-1),\Phi\big{(}T,(x,v)\big{)}\big{)},\Phi\big{(}T,(x,v)\big{)}\Big{)}\geqslant\overline{\kappa}\left|T(\sigma-1)\right|.$$

Therefore, if $\overline{\kappa}\left|T(\sigma-1)\right|>2{\varepsilon}$,

$$d_{T}^{\Phi}\big{(}(x,\overline{v}),(x,v)\big{)}>2{\varepsilon},$$

and so $d_{T}^{\Phi}\big{(}(x,\overline{v}),a\big{)}>{\varepsilon}.$ Consequently,

$$B_{T}^{\Phi}(a,{\varepsilon})\subset\bigcup_{(x,v)\in B^{(\nu_{0})}}\{x\}\times\big{[}1-2{\varepsilon}/\overline{\kappa}T,1+2{\varepsilon}/\overline{\kappa}T\big{]}.$$

Let $\mu_{\rm max}^{(\nu_{0})}(T,{\varepsilon})$ be the maximal measure of a $(T,{\varepsilon})$-ball for the restriction of $\Phi$ to $T^{(\nu_{0})}$. Then the maximal measure $\mu_{\rm max}(T,{\varepsilon},\nu_{0})$ of a $(T,{\varepsilon})$-ball for $\Phi$ centered on $T^{(\nu_{0})}$ satisfies

$$\mu_{\rm max}(T,{\varepsilon},\nu_{0})\leqslant\frac{4{\varepsilon}}{\overline{\kappa}T}\mu_{\rm max}^{(\nu_{0})}(T,{\varepsilon}).$$

A similar reasoning using the homogeneity of $\Phi$ proves that there is a constant $c({\varepsilon})>0$ such that for any $\nu\in[\lambda^{-1}\nu_{0},\lambda\nu_{0}]$:

$$\mu_{\rm max}(T,{\varepsilon})\leqslant\frac{c({\varepsilon})}{T}\mu_{\rm max}^{(\nu_{0})}(T,{\varepsilon}).$$

Hence,

$$\lim_{\varepsilon\rightarrow 0}\limsup_{T\rightarrow\infty}\dfrac{-\log\big{(}\mu_{\rm max}(T,{\varepsilon})\big{)}}{\log(T)}\geqslant 1+\lim_{\varepsilon\rightarrow 0}\limsup_{T\rightarrow\infty}\dfrac{-\log\big{(}\mu^{(\nu)}_{\rm max}(T,{\varepsilon})\big{)}}{\log(T)}=1+h_{\mathrm{pol}}(\Phi,T^{(\nu_{0})}),$$

which proves our claim.     $\square$

Example 3

The flat torus $T^{(n)}$ is endowed with the Riemannian metric induced by the Euclidean norm. Up to the Legendre diffeomorphism, this corresponds to the Hamiltonian $h(r)={\tfrac{1}{2}}\|r\|^{2}$ on $T^{*}T^{(n)}$. The previous equality proves that the polynomial entropy of the restriction of the flow to an energy level is $n-1$, therefore, we recover a result of [7].

Example 4

It is proved in [6] that the geodesic flow $\Phi$ of the torus of revolution ${\mathscr{T}}$ in $\mathbb{R}^{3}$ (endowed with its natural Riemannian structure) satisfies $h_{\mathrm{pol}}(\Phi,T^{(1)}{\mathscr{T}})=2.$ Consequently, given $0<a<b$:

$$h_{\mathrm{pol}}\big{(}\Phi,T([a,b])\big{)}=3.$$

5 ON A THEOREM OF MISIUREWICZ – SZLENK ABOUT THE ENTROPY OF INTERVAL MAPS

In this section we apply the notion of contracting metric structure to the case of $\ell$-modal maps to get a lower bound for their polynomial entropy, and answer a question from [3].

5.1 A Theorem of Misiurewicz – Szlenk

Before stating the theorem, let us recall the notions of turning points and lap number for piecewise monotone continuous maps.

Definition 3

Let $I=[0,1]$ and $f:I\rightarrow I$ be a piecewise monotone continuous map, meaning that the subset of points of $I$ where $f$ has a local extremum is finite and contained in the interior of $I$; such points are called turning points. A map $f$ is said to be $\ell$-modal if $f$ has precisely $\ell$ turning points and $f(\partial I)\subset\partial I$ $($when $\ell=1$, $f$ is said to be unimodal$)$. The lap number of an $\ell$-modal map is the number of intervals limited by the boundaries of $I$ and the turning points, that is, ${\rm L}(f)=\ell+1$.

The iterates of a piecewise monotone continuous map are piecewise monotone. The following result relates the topological entropy of $f$ to the lap number of the iterates of $f$.

Theorem [10]. Let $f:I\rightarrow I$ be a piecewise monotone continuous map. Then:

$$h_{\mathrm{top}}(f)=\limsup_{n\rightarrow\infty}\dfrac{\log\big{(}{\rm L}(f^{n})\big{)}}{n}.$$

As for the polynomial growth rate, the following estimate for interval maps is proved in [3]:

$$h_{\mathrm{pol}}(f)\leqslant\limsup_{n\rightarrow\infty}\dfrac{\log\big{(}{\rm L}(f^{n})\big{)}}{\log(n)}+1.$$

In the same paper, the authors ask the following:

Question. Does the inequality $\displaystyle h_{\mathrm{pol}}(f)\geqslant\limsup_{n\rightarrow\infty}\dfrac{\log\big{(}{\rm L}(f^{n})\big{)}}{\log(n)}$ hold for piecewise monotone maps?

In the next section, we give a positive answer to this question using the polynomial torsion.

5.2 A Lower Bound to the Polynomial Entropy of Interval Maps

Theorem 4

Let $f:I\rightarrow I$ be an $\ell-$ modal map. Then,

$$h_{\mathrm{pol}}(f)\geqslant\limsup_{n\rightarrow\infty}\dfrac{\log\big{(}{\rm L}(f^{n})\big{)}}{\log(n)}.$$

Proof

The proof is an adaptation of the proof of Misiurewicz – Szlenk using symbolic dynamics given in [8]. We construct symbolic dynamics associated to $f$. Let $0<c_{1}<c_{2}<\cdots<c_{\ell}<1$ be the $\ell$ turning points of $f$ and consider the monotonicity intervals $I_{1}=[0,c_{1})$, $I_{2}=(c_{1},c_{2})$, $\ldots$, $I_{\ell+1}=(c_{\ell},1]$ of the function $f$.

Set

$$\Sigma_{0}(f)=\{\bar{x}=(x_{i})_{i\geqslant 0},\ x_{i}\in\{1,\cdots,\ell+1\}\mathrm{\ and\ }\displaystyle\cap_{i=0}^{n}f^{-1}(I_{x_{i}})\neq\varnothing,\ \forall n\in\mathbb{N}\}$$

equipped with the metric

$$d(\bar{x},\bar{y})=\sum_{i=0}^{\infty}\dfrac{\left|x_{i}-y_{i}\right|}{2^{i}},$$

for $\bar{x}=(x_{i})_{i\geqslant 0}$, $\bar{y}=(y_{i})_{i\geqslant 0}$. Then $(\Sigma_{0},d)$ is compact and invariant by the shift operator $\sigma$. Set

$$\Sigma_{I}(f)=\big{\{}(\bar{x},x)\in\Sigma_{0}(f)\times[0,1]:f^{i}(x)\in\overline{I_{x_{i}}},\ \forall i\in\mathbb{N}\big{\}},$$

so that $\Sigma_{I}(f)$ is compact for the natural product metric.

We define $\sigma_{I}:\Sigma_{I}(f)\rightarrow\Sigma_{I}(f)$ by

$$\sigma_{I}(\bar{x},x)=\big{(}\sigma(\bar{x}),f(x)\big{)}.$$

Let $\pi_{1}:\Sigma_{I}(f)\rightarrow\Sigma_{0}(f)$ and $\pi_{2}:\Sigma_{I}(f)\rightarrow[0,1]$ be the canonical projections. These maps are surjective and continuous and satisfy : $\pi_{1}\circ\sigma_{I}=\sigma\circ\pi_{1}$ and $\pi_{2}\circ\sigma_{I}=f\circ\pi_{2}$. Hence, by the factor property of the polynomial entropy $h_{\mathrm{pol}}(\sigma_{I})\geqslant h_{\mathrm{pol}}(\sigma)$.

Moreover, observe that $(\Sigma_{I},\sigma_{I})$ has a contracting fibered structure over $([0,1],f)$ (see the definition in Section 3). Indeed, one checks that $d\big{(}\sigma(\bar{x}),\sigma(\bar{y})\big{)}<d(\bar{x},\bar{y})$ for $(\bar{x},x)$ and $(\bar{y},y)$ in $\Sigma_{I}(f)$. Hence, $h_{\mathrm{pol}}(\sigma_{I})=h_{\mathrm{pol}}(f)$ (see the Theorem in [6] quoted in Section 3). We deduce that $h_{\mathrm{pol}}(\sigma)\leqslant h_{\mathrm{pol}}(f)$.

As in [8], we can show that

$$\limsup_{n\rightarrow\infty}\dfrac{\log\big{(}{\rm L}(f^{n})\big{)}}{\log(n)}\leqslant h_{\mathrm{pol}}(\sigma)\leqslant h_{\mathrm{pol}}(f),$$

which proves our claim.     $\square$

5.3 Application to the Logistic Map

This inequality allows us to get a lower bound of the polynomial entropy of some interval maps in terms of the polynomial growth of the lap number. Recall that the logistic map is the continuous map of the interval $[0,1]$ defined by $f_{\lambda}:x\mapsto\lambda x(1-x)$.

Using the kneading theory, it is proved in [9] that the growth rate of the lap number for the logistic map is faster than polynomial, but less than exponential when the parameter $\lambda$ is the Feigenbaum constant $\lambda_{\infty}$. The previous theorem shows that $h_{\mathrm{pol}}(f_{\lambda_{\infty}})=\infty$.

In [12] the authors compute the polynomial entropy of interval maps in terms of the Sharkovskii types of their periodic orbits. In particular, they obtained the same result for the polynomial entropy of $f_{\lambda_{\infty}}$ and showed that, for $1<\lambda<\lambda_{\infty}$, $h_{\mathrm{pol}}(f_{\lambda})=n+1$ if there is an attracting $2^{n}-$cycle.