Polynomial entropy of Morse-Smale diffeomorphisms on surfaces

A classical problem in dynamical systems is to measure the complexity of a map in terms of its orbits. One of the main tools we have to achieve this goal is topological entropy. The topological entropy of a map studies the exponential growth rate at which orbits are separated. Although it is a crucial tool for classifying highly chaotic dynamical systems, in many interesting families of dynamical systems, every system has vanishing entropy, and therefore, another tool is needed. J. P. Marco in [15] introduced the concept of polynomial entropy, and recently, the first author and E. Pujals introduced the notion of generalized topological entropy in [5]. The later definition, instead of quantifying the complexity of a system with a single number, it works directly in the space of the orders of growth $\mathbb{O}$.

J. P. Marco, C. Labrousse and P. Blanchard were the first to study dynamical systems with vanishing entropy from the perspective of polynomial entropy. First, J. P. Marco introduced the concept of polynomial entropy in [15], and this topic was subsequently studied in [12], [11], [14] and [2] in the context of Hamiltonian dynamical systems and geodesic flows with zero-entropy. For maps on the interval, we have the works of C. Labrousse [13], M. J. D. Carneiro and J. B. Gomes [3] and S. Roth, Z. Roth and L. Snoha [16] to refer to. Lastly, for general dynamical systems, we have the work of A. Artigue, D. Carrasco-Olivera and I. Monteverde [1].

We would like to coment on two aspects of the work of L. Hauseux and F. Le Roux in [9]. First, in [5], the authors observed that in the study of dynamical system with vanishing entropy, there are examples where the complexity of the overall system is greater than the complexity of the system restricted to its recurrent part. This phenomenon does not occur in the presence of chaos due to the variational principle. It is our opinion that the definition of wandering entropy given in [9] should give account of said jump. Second is with respect to the technique they developed to compute the polynomial entropy of a system whose non-wandering set consists of only one fixed point. In this article, we translate this technique to the context of generalized entropy and extend it to maps with a finite non-wandering set. We would like to mention that J. Katić and M. Perić in [10] use said approach from [9] to study Morse gradient systems with the singularities of a specific index.

Let us consider $S$ a compact surface and $f:S\to S$ a Morse-Smale diffeomorphism. We recall that these maps are the ones whose non-wandering set is finite and, therefore, consists only of periodic points. These periodic points have to be hyperbolic; moreover, the intersection between any stable and any unstable manifolds is transverse. Let us suppose that $\Omega(f)=\cup_{i=1}^{k}\theta(p_{i})$ where $\theta(p_{i})$ denotes the orbit of point $p_{i}$. We can construct a finite graph $G=(V,E)$, where the vertices $V$ are the orbits $\theta(p_{i})$ and the edges represent the existence of an orbit whose past is $\theta(p_{i})$ and future is $\theta(p_{j})$. Thus, we define

If the graph $G$ had any cycle, then there would be a non-trivial homoclinic class, and therefore, the non-wandering set would not be finite. Now, let us define $L(f)$ as the length of the longest possible path in $G$. We represent the generalized entropy of $f$ by $o(f)$, and the polynomial entropy of $f$ by $h_{pol}(f)$.

As an immediate corollary, we have a rigidity result.

We can apply theorem 1 to understand the dispersion of orbits in the border of zero-entropy. Consider $Emb^{r}(\mathbb{D}^{2})$ the space of $C^{r}$ embeddings of the disk $\mathbb{D}^{2}$ into itself. In [8], J. M. Gambaudo, S. van Strein and C. Tresser constructed a system $f\in Emb^{r}(\mathbb{D}^{2})$ that is Kupka-Smale and infinitely re-normalizable and happens to be in the border of chaos. This means that it can be perturbed into a system with positive classical entropy. The said example is also the limit of Morse-Smale diffeomorphisms and has a geometry that allow us to apply theorem 1. We would like to highlight that maps with said properties are far from being an isolated example. In [6], S. Crovisier and E. Pujals introduced the concept of mildly dissipative maps in $Emb^{r}(\mathbb{D}^{2})$, and with C. Tresser in [7], they proved that any map with mild dissipation in the border of chaos is infinitely re-normalizable. In order for our following result to be a direct application of theorem 1, we need an additional hypothesis that we shall call standard geometry and explain in detail in subsection 2.2. This property holds in the Gambaudo - van Strein - Tresser example as well as the Hénon-like maps with small Jacobian. The latter is proved by A. de Carvalho, M. Lyubich and M. Martens in [4].

Let $\mathbb{P}$ stand for the family of polynomial orders of growth $[n^{t}]$ with $0<t<\infty$.

Regarding the generalized entropy in the border of chaos, the standard geometry property is probably removable. However, a deeper understanding of the dispersion of wandering orbits near recurrent sets is needed. The Kupka-Smale property allows us to apply theorem 1 directly, yet it may not be necessary. It seems intuitive for us that $o(f)=\sup(\mathbb{P})$ should hold for every map in said set.

We can also deduce that the map $f\mapsto o(f)$ is continuous in some sense.

In retrospect, in the naturality of this result lies an indication of why theorem 1 is also true. We know that the polynomial entropy of Morse-Smale maps is positive because it has wandering points. When we consider a map in the border of chaos, we naturally want that $h_{pol}(f)=\infty$ and if it is approximated by Morse-Smale, the polynomial entropy of them should grow toward infinity. It is expected that maps in the border of chaos show a reminiscent phenomenon of the period doubling cascade of bifurcations for interval maps. In this context, the only dynamical quantity that is growing and has a global understanding of the map is $L(f)$.

Interestingly enough, in their recent work [16], S. Roth, Z. Roth and L. Snoha proved the counterpart of our results in the one-dimensional context. They showed that for endomorphisms of the interval with zero entropy, their polynomial entropy is equal to a quantity equivalent to our $L(f)$, and they obtained analogous results to our corollaries 2 and 3. In particular, for the logistic map, they showed that in each bifurcation of the period doubling cascade, the polynomial entropy increases by $1$, as well as that the infinitely re-normalizable map has infinite polynomial entropy.

The proof of theorem 1 has two steps: understanding the geometrical configuration of a Morse-Smale and computing the generalized entropy through a nice codification of orbits. In the study of the separation of the orbits of wandering dynamics, Hauseux and Le Roux introduced the concept of wandering mutually singular sets in [9]. We would like to point out that for dynamical systems with finite non-wandering set, this concept is in some sense the equivalent of the Markov partitions for uniformly hyperbolic systems.

We denote $\Omega(f)$ as the non-wandering set of $f$. We say that the subsets $Y_{1},\cdots,Y_{L}$ of $M\setminus\{\Omega(f)\}$ are mutually singular if for every $N>0$, there exists a point $x$ and times $n_{1},\cdots,n_{L}$ such that $f^{n_{i}}(x)\in Y_{i}$ for every $i=1,\cdots,L$, and $|n_{i}-n_{j}|>N$ for every $i\neq j$.

A more precise statement of this theorem is given in subsection 2.3. We want to now discuss the interplay of theorems 1, 2 with the main theorem in [9]. A consequence of theorem 2 is that the generalized entropy of a homeomorphism with a finite non-wandering set always verifies $[n]\leq o(f)\leq\sup(\mathbb{P})$. Let us consider $\mathcal{H}$ as a family of homeomorphism on a surface such that its non-wandering set is finite. Let us also consider $\mathcal{H}_{1}$ as the subset of $\mathcal{H}$ such that the surface is the sphere and its non-wandering set is only one fixed point. Part of the translation of the result in [9] in our context is that for every $t\geq 2$, there exists $f\in\mathcal{H}_{1}$ such that $o(f)=[n^{t}]$. This result in conjunction with our previous observation can be summarized as follows:

For every $f\in\mathcal{H}$, $[n]\leq o(f)\leq\sup(\mathbb{P})$. Moreover, for every $t\geq 2$, there exists $f\in\mathcal{H}$ such that $o(f)=[n^{t}]$.

Theorem 1 tells us that for those maps in $\mathcal{H}$ that are Morse-Smale diffeomorphisms, the set $\{o(f)\in\overline{\mathbb{O}}\}$ becomes discrete. It is not clear to us whether the differentiability plays a key role in this. We wonder if there is an obstruction for the examples built in [9] to be differentiable and whether this contrast is another case of pathological differences between the continuous world and the differentiable one.

A final comment in this topic is made toward the jump from $[n]$ to $[n^{2}]$ for the homeomorphisms in the sphere. In [9], the authors quote a result that seems to apply only for homeomorphisms in $\mathcal{H}_{1}$, and we would like to know if it possible to obtain a map in $\mathcal{H}$ for every $t\in(1,2)$ such that $o(f)=[n^{t}]$.

We finish this introduction with a small discussion regarding the higher dimension. From theorem 2 and an intermediary lemma for theorem 1, we deduce the following result.

Regarding the other inequality, it is unclear to us if our technique holds. The main obstruction we see is the possible interaction between the periodic saddles of different indices.

We would like to thank Enrique Pujals for the many discussions and insightful comments.

This work is structured as follows:

• •

  In section 2, we provide a quick summary of the preliminaries on the topic of generalized entropy, mildly dissipative maps and the coding of orbits developed in [9]. We also make a precise statement for theorem 2.

• •

  In section 3, we prove theorem 1, assuming theorem 2, and we prove corollaries 1, 2, 3 and 4.

• •

  In section 4, we prove theorem 2.

We use theorem 2 to prove theorem 1. However, the proof of theorem 2 is more technical and since the proof of theorem 1 has more geometrical components, we choose to prove 1 first and show the proof of theorem 2 later. At the end of this section, we provide a proof of corollaries 1, 2, 3 and 4.

Let us consider $S$ a compact surface and $f:S\to S$ a Morse-Smale diffeomorphism. As we are in dimension two, we have three possibilities for the hyperbolic periodic points. They are either sinks, sources or saddles of index 1. Using theorem 2, we need to first compute $[c_{f,\mathcal{F}}(n)]$ for finite families $\mathcal{F}$ of disjoint, mutually singular compact neighborhoods, and then take the supremum over such families. Our following lemma tells the location of the sets of any $\mathcal{F}\in\Sigma$.

The following figure represents the above-mentioned lemma.

Given $\mathcal{F}\in\Sigma$, take $L=\#\mathcal{F}$ and consider the periodic orbits $\theta(p_{1}),\cdots,\theta(p_{L})$ from lemma 3.1. Note that any point in the intersection between $Y_{1}$ and the stable manifold of $\theta(p_{1})$ must also belong to the unstable manifold of some periodic orbit $\theta(p_{0})$. Therefore, $\theta(p_{0}),\theta(p_{1}),\cdots,\theta(p_{L})$ induces a path of length $L$ in the graph constructed in the introduction, particularly $L\leq L(f)$.

By applying the above lemma to our previous reasoning, we infer that $[c_{f,\mathcal{F}}(n)]\leq[n^{L}]\leq[n^{L(f)}]$, and by theorem 2, we conclude $o(f)\leq[n^{L(f)}]$. Thus, it only remains for us to prove the other inequality, for which we only need to find a family $\mathcal{F}\in\Sigma$ with $L(f)$ elements such that $[c_{f,\mathcal{F}}(n)]=[n^{L(f)}]$. Before moving on, we prove lemma 3.2.

In the context of a homeomorphism such that its non-wandering set is finite, it is easier to work with the assumption that every periodic point is indeed a fixed point. However, to make such an assumption, we need to develop two notions: the bounded jump property and $N$-combinatorially complete.

We say that a class of orders of growth $[a(n)]$ verifies the bounded jump property if there exists a constant $C>0$ such that $a(n+1)\leq Ca(n)$. Note that this definition does not depend on the choice of the class representative. We recall that a set $B\subset\mathbb{N}$ is syndetic if there exists $N\in\mathbb{N}$ such that for all $n$, the interval $[n,n+N]$ contains at least one point of $B$.

The first lemma tell us that $[c_{f,\mathcal{F}}(n)]$ always verifies the bounded jump property. The second lemma tell us that if we know an order of growth in a syndetic set and said order of growth verifies the bounded jump property, then we understand the order of growth in $\mathbb{N}$. For example, if we prove that $c_{1}n^{L}\leq c_{f,\mathcal{F}}(n)\leq c_{2}n^{L}$ for every $n\in B$, then $[c_{f,\mathcal{F}}(n)]=[n^{L}]$.

Given $\mathcal{F}\in\Sigma$, we say that it is $N$-combinatorially complete if there exist $k_{1},\cdots,k_{L-1}$ such that for every $n_{1}\geq k_{1},\cdots,n_{L-1}\geq k_{L-1}$ with $n_{i}-k_{i}$ divisible by $N$, there are points $x_{1}\in Y_{1},\cdots,x_{L}\in Y_{L}$ with the property $x_{i+1}=f^{n_{i}}(x_{i})$. If $\mathcal{F}$ is 1-combinatorially complete, we say that it is combinatorially complete.

Let us consider $N$ such that every periodic point of $f^{N}$ is a fixed one. Note that if $\mathcal{F}\in\Sigma(f^{N})$, then $\mathcal{F}$ also belongs to $\Sigma(f)$. Our following two lemmas explain why we can work with a map $f$ that has only fixed points.

We proceed now to prove both lemmas.

We are now in a condition to prove theorem 1