1.1 Existence of robust NUH endomorphisms

Any map $f:\mathbb {T}^2\to \mathbb {T}^2$ is homotopic to a linear endomorphism $E$ induced by an integer matrix that we denote by the same letter. In this work, we are interested in non-invertible local diffeomorphisms and will therefore only consider the case when $|\!\det E | \geq 2$ . Considering the projective action ${\mathbb {P}} E: {\mathbb {P}} {\mathbb {R}^2} \to {\mathbb {P}} {\mathbb {R}^2}$ that induces $E$ , we have four possibilities:

1. – ${\mathbb {P}} E=\mathrm {Id}$ ;

2. – there exists a parabolic fix point $p\in {\mathbb {P}} {\mathbb {R}^2}$ that is repelling on one side and attracting on the other;

3. – there exist two hyperbolic fix points $p,q\in {\mathbb {P}} {\mathbb {R}^2}$ , and the dynamics is of the north pole–south pole type; or

4. – ${\mathbb {P}} E$ is conjugated to a rotation.

Note that the first corresponds to the case when $E$ is a homothety.

If $f:\mathbb {T}^2\to \mathbb {T}^2$ is of class $\mathcal C^1$ we consider the class of $\mathcal C^1$ smoothly homotopic maps to $f$ ,

\begin{equation*} [f]=\{g:\mathbb {T}^2\to \mathbb {T}^2: \text {there exists a }\mathcal C^1\text { homotopy between $f$ and $g$}\}. \end{equation*}

We denote by $\mathrm {End}_{\mu }^r(\mathbb {T}^2)$ the set of $\mathcal C^r$ local diffeomorphisms of $\mathbb {T}^2$ preserving the Lebesgue measure $\mu$ that are not invertible. For $f\in \mathrm {End}^1_{\mu }(\mathbb {T}^2), (x,v)\in T^1\mathbb {T}^2$ we define

(1) \begin{equation} I(x,v;f^n)=\sum _{y\in f^{-n}(x)}\frac {\log \|(D_yf^n)^{-1}v\|}{|\!\det (D_yf^n)|}, \end{equation}

and let

(2) \begin{equation} C_{\chi }(f)=\sup _{n\in \mathbb N}\frac 1n\inf _{(x,v)\in T^1\mathbb {T}^2}I(x,v;f^n). \end{equation}

The set

(3) \begin{equation} \mathcal U:=\{f\in \mathrm {End}^1_{\mu }(\mathbb {T}^2):\ C_{\chi }(f)\gt 0\} \end{equation}

is open in the $\mathcal C^1$ topology: indeed, if $f\in \mathcal U$ then for some $n$ it follows that $(1/n)\inf _{(x,v)\in T^1\mathbb {T}^2}I(x,v;f^n)\gt {C_{\chi }(f)}/{2}$ , and hence taking a sufficiently small $\mathcal C^1$ neighborhood $\mathcal V$ of $f$ we can guarantee that any $g\in \mathcal V$ is an endomorphism and satisfies $(1/n)\inf _{(x,v)\in T^1\mathbb {T}^2}I(x,v;g^n)\gt {C_{\chi }(f)}/{4}$ , thus implying that $g\in \mathcal U$ .

Our first main result is the following.

This, in particular, establishes the existence of $\mathcal C^1$ open sets of NUH endomorphisms. The above is a flexibility theorem: unlike the diffeomorphism case, there are no topological obstructions to the existence of robust NUH endomorphisms in $\mathbb {T}^2$ . We also point out that this is the first robustly NUH (area-preserving) example of an endomorphism, that it is not expanding, and that it does not admit an invariant distribution.

Remark 1.1. If $\tau _1(E)=\gcd {e_{ij}}, \tau _2(E)=|\!\det E|/\tau _1(E)$ , then the Smith Normal Form of $E$ is the diagonal matrix with entries $\tau _1(E)$ and $\tau _2(E)$ (see § 2.4). The condition on $\tau _2(E)$ from Theorem A is valid for all pairs $(\tau _1(E),\tau _2(E))$ with the exception of those in

\begin{equation*} \{(1,2), (1,3), (1,4), (2, 2), (2,4), (3,3), (4,4)\}. \end{equation*}

This requirement on $\tau _2(E)$ and the non-homothety assumption are made to avoid some technical difficulties, and we expect that with additional work this restriction can be lifted to any (non-invertible) endomorphism; see § 1.5. Let us also point out that our methods give effective bounds for the $\mathcal {C}^1$ distance between constructed elements $f\in [E]\cap \mathcal U$ and $E$ .

Observe the following curious consequence.

For any such expanding map, both its exponents are necessarily positive (in fact, this is the case with respect to any of its invariant measures). Nevertheless, it can be obtained as a quotient of a system which has contracting direction at Lebesgue almost every point of its phase space.

1.2 Continuity of the exponents

A natural question which arises is how the Lyapunov exponents depend on the map $f$ . Since we do not know a priori that $f$ is ergodic, it makes sense to ask about the regularity of the maps $f\mapsto \int _{\mathbb {T}^2}\chi ^-(f)\,d\mu$ , $f\mapsto \int _{\mathbb {T}^2}\chi ^+(f)\,d\mu$ , for $f\in \mathrm {End}^r_{\mu }(\mathbb {T}^2)$ , $r\geq 1$ .

The classical results of Mañe, Bochi and Viana show that one cannot expect continuity in the $\mathcal C^1$ topology for diffeomorphisms without dominated (Oseledets) splitting. Continuity in finer topologies is known for certain restricted classes of diffeomorphism, but the general panorama is still not clear. The reader is referred to [Reference Liang, Marin and YangLMY18] and references therein for a discussion in the diffeomorphism (partially hyperbolic) setting. For continuity of Lyapunov exponents for cocycles over homeomorphisms, the reader can check the survey in [Reference VianaVia18].

On the other hand, the results in [Reference Viana and YangVY19] for cocycles over endomorphisms suggest that we may be able to obtain better regularity than in the invertible counterpart. Indeed, we obtain the following result.

Define

\begin{equation*} C_{\det }(f)=\sup _{n\in \mathbb N}\frac 1n\inf _{x\in M}\log |\!\det (Df_x^n)|\gt 0 \end{equation*}

and

(4) \begin{equation} \mathcal U_1=\left \{ f\in \mathrm {End}^1_{\mu }(\mathbb {T}^2): C_{\chi }(f)\gt -\tfrac 12C_{\det }(f)\right \}, \end{equation}

where $C_{\chi }(f)$ is defined in (2). Clearly $\mathcal U_1$ is open and contains the set $\mathcal U$ of NUH endomorphisms constructed in Theorem A.

An immediate consequence of the Pesin formula for endomorphisms [Reference Qian and ZhuQZ02] is the result below.

1.3 Stable ergodicity

Some of the examples constructed can be shown to be stably ergodic.

The study of stably ergodic diffeomorphisms (that are not uniformly hyperbolic) was initiated by the results of Grayson, Pugh and Shub [Reference Grayson, Pugh and ShubGPS94], and was promoted by Pugh and Shub to a theory which is currently attracting a large amount of research. Citing all this literature would lead us astray, so we will content ourselves by indicating some significant recent articles and referring the reader to them for additional information: [Reference Burns and WilkinsonBW10, Reference Hertz, Hertz and UresHHU08, Reference Avila, Crovisier and WilkinsonACW21].

Despite its popularity in the context of invertible dynamics, stable ergodicity for endomorphisms has never been given serious attention. In fact, we are unaware of any earlier attempt to give examples outside the realm of uniformly expanding systems. On the other hand, there are quite a few studies of physical measures for endomorphisms, and some of these can be translated into results about stable ergodicity. Most notably, for maps that are non-uniformly expanding in the sense of [Reference Alves, Bonatti and VianaABV00], the basins of physical measures are open. In a conservative context, this translates into the statement that ergodic components of Lebesgue measure are open (modulo sets of zero Lebesgue measure). Therefore, robust almost everywhere non-uniform expansion, together with robust transitivity, implies stable ergodicity.

For partially hyperbolic endomorphisms we have a very complete picture in the dissipative setting thanks to the seminal work of Tsujii [Reference TsujiiTsu05]: $\mathcal C^2$ -generic partially hyperbolic local diffeomorphisms on $\mathbb {T}^2$ have a finite number of physical measures whose basins cover a set of full Lebesgue measure of the manifold. One of the curious facts is that these measures are absolutely continuous with respect to Lebesgue, even if the central Lyapunov exponent is slightly negative. In their recent paper [Reference Tsujii and ZhangTZ24] Tsujii and Zhang construct smooth families of endomorphisms that are non-uniformly hyperbolic with respect to the physical measure. These works seem to require regularity higher than $\mathcal C^2$ . In the conservative setting it is likely that one can generalize the concept of accessibility to partially hyperbolic endomorphisms (by looking at the inverse limit and considering $\pi _{\mathrm {ext}}(x)$ as the strong stable manifold of $x$ ) and then adapt the Grayson–Pugh–Shub program to obtain stable ergodicity.

For partially hyperbolic endomorphisms having only admissible measures (the analog of Gibbs $u$ -states for non-invertible maps) and negative central Lyapunov exponents, the first author proved in [Reference AnderssonAnd10] that having a single physical measure is a $\mathcal C^2$ open property. For conservative maps, this translates into the statement that ergodicity implies ( $\mathcal C^2$ ) stable ergodicity. We find it very likely that such maps are indeed $\mathcal C^1$ stably ergodic, meaning that the map is ergodic and that every $\mathcal C^2$ map $\mathcal C^1$ close to it is also ergodic. This was proved by Yang [Reference YangYan21] in the analogous setting for diffeomorphisms, and we see no reason why his proof should not work for endomorphisms as well.

In this work we produce examples of stably ergodic endomorphisms of a rather exotic kind: robustly non-uniformly ergodic maps without any dominated splitting. This stands in sharp contrast to the invertible case. Indeed, it was shown by Bochi, Fayad and Pujals [Reference Bochi, Fayad and PujalsBFP06] that any stably ergodic diffeomorphism can be approximated by one having a dominated splitting.

Our contribution to the theory is the following.

Thus, we obtain examples of stably ergodic non-uniformly hyperbolic endomorphisms that do not admit any invariant distribution, and therefore are not partially hyperbolic.

Theorem C above will be obtained as a consequence of the following general result, which is of independent interest. For the discussion of global stable manifolds and Pesin theory see § 5.

1.4 Comments on the proofs

We will make some comments on the ideas of the proofs involved in this article.

A fundamental tool is the natural extension of the endomorphism $f$ : this is the space $\mathbf {L}_{f}$ of the backward orbits, with a natural projection $\pi _{\mathrm{ext}}$ to the torus. The endomorphism $f$ lifts to an invertible map $\hat {f}$ and the area $\mu$ lifts to an invariant measure $\hat {\mu }$ . The Lyapunov exponents of $f$ coincide with the Lyapunov exponents of the $Df$ cocycle lifted to $\mathbf {L}_{f}$ . An important observation is that $\lim _{n\rightarrow \infty } ({1}/{n})I(x,v;f^n)$ is exactly the integral of the Lyapunov exponents of the lifted cocycle at $(\hat {x},v)$ on the fiber $\pi _{\mathrm{ext}}^{-1}(x)$ with respect to the disintegration of $\hat {\mu }$ on $\pi _{\mathrm{ext}}^{-1}(x)$ . This will ultimately imply that any map in $\mathcal U$ must be non-uniformly hyperbolic.

Let us mention that the property defining $\mathcal U$ has a similar counterpart in the setting of random maps, namely the ‘uniform expansion property’ of Dolgopyat and Krikorian [Reference Dolgopyat and KrikorianDK07]. This property has several nice consequences for the properties of stationary measures, and was investigated in a series of papers including [Reference Dolgopyat and KrikorianDK07, Reference Brown and Rodriguez HertzBRH17, Reference ChungChu20, Reference LiuLiu16].

The basic ideas in the construction of the specific examples in the homotopy class of $E$ are the following. We modify $E$ such that we make the map (strongly) hyperbolic in some ‘good’ region $\mathcal G$ , which is chosen to be large enough that most of the backward orbits spend a substantial time inside it. In general, without some further control this may not be enough to obtain hyperbolicity, because even if one obtains a relatively good expansion on some large piece of orbit, it is possible that, once we loosen the control, the next entry in the good region is such that the expanding direction goes to the contracting direction, and thus all the hyperbolicity which we gained is lost. In order to avoid this we have to make some kind of non-degeneracy assumption; in particular, we construct the map with slightly different hyperbolicity in different parts of the good region $\mathcal G$ . Thus we may lose the hyperbolicity for some particular orbits, but we will maintain it for other ones, and this turns out to be enough to obtain the conclusion.

The proof of the continuity of the exponents is based on methods developed for the study of the Lyapunov exponents for cocycles with holonomies. The definition of $\mathcal U_1$ guarantees that the Lyapunov spectrum is simple at almost every point. We consider the projectivization of the cocycle over $\hat {f}$ , and the invariant measure $\hat {\mu }$ lifts to convex combinations of Dirac measures supported at the Oseledets subspaces at almost every point. Assume that a sequence $f_n$ converges to $f$ in the $\mathcal C^1$ topology. The only way the continuity of the exponents can fail is if a sequence of stable lifts of $\hat {\mu }$ for $f_n$ (lifts supported at the subspaces corresponding to the smallest Lyapunov exponent) does not converge to the stable lift of $\hat {\mu }$ for $f$ , or, in other words, the limit contains some parts of the unstable lift of $\hat {\mu }$ for $f$ . The stable lifts are invariant under the holonomy given by the projection $\pi _{\mathrm{ext}}$ , and this property can be passed to the limit. This would imply that there is a nontrivial part of the unstable lift of $\hat {\mu }$ for $f$ which is invariant under the $\pi _{\mathrm{ext}}$ holonomy. It turns out that this is impossible because of the definition of $\mathcal U_1$ , which already implies that the (strong) unstable Oseledets subspaces cannot be independent of the past, so they cannot be constant on $\pi _{\mathrm {ext}}$ fibers. This condition is reminiscent of the ‘non-random stable bundle’ case appearing in the classification theorem of Brown and Rodriguez Hertz [Reference Brown and Rodriguez HertzBRH17] of hyperbolic stationary measures for random diffeomorphisms of surfaces.

For the proof of the stable ergodicity we rely on the classical Hopf argument. In order to obtain the intersections between the stable and the unstable manifolds we introduce a new method: we first prove that the stable manifolds at almost every point have large diameter. This is different from the classical notion of ‘large Pesin manifolds’, because we do not have control on how the manifolds turn inside the torus. However, we can use ideas from Rodriguez Hertz, Rodriguez Hertz, Tahzibi and Ures [Reference Hertz, Hertz, Tahzibi and UresHHTU11] to show that in the case of endomorphisms on surfaces, large stable manifolds in our sense imply that the ergodic components are open modulo zero sets. In order to conclude the proof we use a result of Andersson [Reference AnderssonAnd16] on the transitivity of area-preserving endomorphisms on $\mathbb {T}^2$ .

1.5 Further questions

Here we make some comments and we ask some further questions related to this work.

1. (1) Non-uniformly hyperbolic endomorphisms in all homotopy classes: We proved that the $\mathcal {C}^1$ open set $\mathcal U$ of non-uniformly hyperbolic endomorphisms intersects most of the homotopy classes of endomorphisms.

   Question 1. Is it true that $\mathcal U$ intersects all the homotopy classes of endomorphisms on $\mathbb {T}^2$ ?

   It is reasonable to expect that this is true. However, if the endomorphism $E$ is a homothety then one has to construct a more complicated good hyperbolic region $\mathcal G$ , and if the degree of $E$ is very small then one should take into consideration the position of several pre-images of every point, in order that most of them fall into the good region.

2. (2) Stable ergodicity in a more general setting: We can obtain the stable ergodicity of the non-uniformly hyperbolic endomorphisms under various restrictions. One of them is a restriction of the homotopy class (the eigenvalues must have absolute value different from one) in order to apply Andersson’s result on the transitivity of endomorphisms.

   Question 2. Are there stably ergodic non-uniformly hyperbolic endomorphisms in every homotopy class of endomorphisms on $\mathbb {T}^2$ ?

   Another restriction comes from the proof of the existence of large stable manifolds: we need the good region to be large enough, with good hyperbolicity, and the critical region to be small. On the other hand, because of the area expansion, the unstable manifolds of a non-uniformly hyperbolic endomorphism are always large.

   Question 3. Is it true that every transitive non-uniformly hyperbolic endomorphism on a surface is stably ergodic?

   Our methods cannot be applied directly to large unstable manifolds, because they do not form a lamination, so one cannot immediately obtain intersections of stable and unstable manifolds. One does, however, obtain intersections between different unstable manifolds, and, with some further work, this may be enough to obtain the ergodicity. If the Jacobian of the endomorphism is constant, the proof of the ergodicity is straightforward from the absolute continuity of the $\pi _{\mathrm{ext}}$ holonomy, so the difficult question is when the Jacobian is not constant.

3. (3) More continuity of the exponents: In general the continuity of the exponents may fail if parts of the stable lift of the measure to the projectivization of the cocycle move in the limit to parts of the unstable lift. However, this would imply that the strong unstable subspaces are invariant under the $\pi _{\mathrm{ext}}$ holonomy, and this seems to be a rigid situation.

   Question 4. Is it true that the average Lyapunov exponents are continuous in (an open dense subset of) the space of $\mathcal C^1$ area-preserving endomorphisms of surfaces?

   One may also ask how the Lyapunov exponents depend on the invariant measure, not necessarily the Lebesgue measure.

   Question 5. Can we say something about the continuity of the Lyapunov exponents of maps in $\mathcal U_1$ with respect to the invariant measure, in a neighborhood of the Lebesgue measure?

4. (4) Flexibility of Lyapunov exponents for endomorphisms on $\mathbb {T}^2$ : Bochi, Katok and Rodriguez Hertz proposed in [Reference Bochi, Katok and HertzBKH22] a program regarding the flexibility of Lyapunov exponents for diffeomorphisms. The same questions can be addressed in the case of endomorphisms:

   Question 6 (Weak flexibility on $\mathbb {T}^2$ ). Given any two real numbers $\lambda _1,\lambda _2$ with $\lambda _1+\lambda _2\gt 0$ , does there exist an ergodic measure-preserving (smooth) endomorphism of $\mathbb {T}^2$ with Lyapunov exponents $\lambda _1$ and $\lambda _2$ ?

   Question 7 (Strong flexibility on $\mathbb {T}^2$ ). Given a linear endomorphism $E$ of $\mathbb {T}^2$ and $\lambda _1,\lambda _2\in {\mathbb {R}^2}$ with $0\lt \lambda _1+\lambda _2$ $\leq \log (|\!\det (E)|)$ , does there exist an ergodic measure-preserving (smooth) endomorphism of $\mathbb {T}^2$ , isotopic to $E$ , with Lyapunov exponents $\lambda _1$ and $\lambda _2$ ?

   Our results answer the second question for many isotopy classes and for $\lambda _1\ll 0\ll \lambda _2$ , $\lambda _1+\lambda _2$ close to $\log (|\!\det (E)|)$ . Any progress on Questions 1–5 above would also contribute to the flexibility program for endomorphisms on $\mathbb {T}^2$ .

5. (5) Dissipative examples: In our paper we consider endomorphisms which preserve the Lebesgue measure on the torus. We could similarly consider other invariant measures, once we have some control on the disintegrations of the lift to the natural extension along the fibers of $\pi _{\mathrm{ext}}$ . A natural question is the following.

   Question 8. Can we deduce non-uniform hyperbolicity and ergodicity for other meaningful invariant measures of our examples and dissipative perturbations of them and, in particular, non-uniform hyperbolicity and uniqueness of equilibrium states for some specific potentials?

   Of special interest is the existence and uniqueness of SRB measures.

   Question 9. Can we construct (robust) examples of dissipative endomorphisms on surfaces having (unique) non-uniformly hyperbolic SRB measures and no dominated splitting?

6. (6) Weak partially hyperbolic diffeomorphisms with a two-dimensional center-unstable bundle: Our examples can be seen (in the natural extension) as weak partially hyperbolic attractors of diffeomorphisms having a (super-)strong stable bundle and a center-unstable bundle of dimension two. It is therefore natural to ask how far our results can be transferred to the setting of weak partially hyperbolic diffeomorphisms with a two-dimensional area-expanding center-unstable bundle (or with a two-dimensional area-contracting center-stable bundle, taking the inverse map).

   Question 10. Can one construct examples of $\mathcal C^2$ stably ergodic non-uniformly hyperbolic diffeomorphisms with a stable bundle and a two-dimensional center-unstable bundle, such that the center-unstable bundle has a positive and a negative exponent and no dominated (sub) splitting? How about dissipative examples with SRB measures?

   Somewhat similar examples of non-uniformly hyperbolic diffeomorphisms with two-dimensional center-stable and center-unstable bundles were constructed in [Reference Berger and CarrascoBC14], and ergodicity was proved in [Reference ObataOba20]. The stable holonomy for the natural extension of endomorphisms is given by $\pi _{\mathrm{ext}}$ which is very nice, and we make strong use of this fact in our proofs. In the case of weak partially hyperbolic diffeomorphisms one would have to assume better regularity of the map and maybe a bunching condition, in order to obtain some good properties of the stable holonomy.

7. (7) Higher-dimensional examples: Another natural question is whether similar examples can be constructed for higher-dimensional endomorphisms with a given number of positive and negative exponents. Furthermore, it would be interesting to see if such examples could be adapted to weak partially hyperbolic diffeomorphisms.

1.6 Organization of the article

The rest of the article is organized as follows. In the next section we introduce some notation and discuss some prerequisites for the rest of the work; in particular, we review the concept of inverse limit and prove that every endomorphism in $\mathcal U$ must be non-uniformly hyperbolic (Proposition 2.2). The third section is devoted to the proof of the second part of Theorem A, and we construct examples of maps in $[E]\cap \mathcal U$ for most homotopy classes $[E]$ . At the end of this section we include a specific computation of an example to convince the reader of the controllability of all constants. This example is a so-called ‘expanding’ standard map, which has additional interest and has recently received some attention. The fourth section contains the proof of Theorem B, and includes a discussion about measures with product disintegrations. In the fifth section we review some facts of Pesin theory for our context, particularly the notion of stable and unstable manifolds for endomorphisms. This is used in the last part to establish the stable ergodicity and the Bernoulli property of the examples. In Appendix A we discuss the inverse limit space, and how the area lifts to an invariant measure on it.