The source paper (C. G. Moreira, Acta Mathematica 2011) proved that there are no C1C^1-stable intersections of regular Cantor sets — any pair of dynamically defined Cantor sets can be approximated in the C1C^1 topology by a pair with empty intersection — and that a generic C1C^1 surface diffeomorphism has no tangencies between the stable foliation of one horseshoe and the unstable foliation of another. The author presented this as evidence toward the classical question of whether hyperbolicity is dense in the C1C^1 topology for surface diffeomorphisms. The weaker Palis dichotomy (approximation by a hyperbolic map or one with a homoclinic tangency) was proved for C1C^1 surfaces by Pujals–Sambarino; Crovisier–Pujals reduced the problem to the gap between essential hyperbolicity and genuine Axiom A; the toy-model family of Benedicks–Carleson (a one-parameter family of Hénon-like planar maps) satisfies C1C^1 density of Axiom A; and in dimension at least three Axiom A is provably not C1C^1-dense, so surfaces are the one remaining case. The problem of C1C^1 density of Axiom A on compact surfaces remains open, with a precise, nonempty open core.

Background

Let MM be a compact surface and let Diff1(M)\mathrm{Diff}^1(M) be the space of C1C^1 diffeomorphisms of MM endowed with the C1C^1 topology. A diffeomorphism ff satisfies Smale's Axiom A if its non-wandering set Ω(f)\Omega(f) — the set of points every neighbourhood of which returns to intersect itself under some positive iterate of ff — is uniformly hyperbolic, meaning the tangent bundle over Ω(f)\Omega(f) splits into two invariant directions, one uniformly contracted and one uniformly expanded by the derivative, and periodic points are dense in Ω(f)\Omega(f). Together with the no-cycle condition (no cyclic intersections between the invariant manifolds of distinct basic pieces), Axiom A characterizes Ω\Omega-stable dynamics by the Palis–Smale theorem, so the question of whether such systems are dense is a question about how far one can push arbitrary dynamics toward the best-understood class. The density question goes back to Smale, who introduced Axiom A in his 1967 survey, and he later posed the general problem 'In which class of systems is hyperbolicity dense?' explicitly in his 1998 list of mathematical problems for the next century; it became a central open problem of smooth dynamics. A homoclinic tangency — a non-transverse intersection between the stable and unstable manifolds of a hyperbolic periodic point — is the archetypal local obstruction: a diffeomorphism with a homoclinic tangency cannot be Axiom A, and tangencies generate rich non-hyperbolic behavior. A heterodimensional cycle is the analogous obstruction built from periodic points whose stable manifolds have different dimensions; it can only occur in ambient dimension at least three, since all saddle periodic points of a surface diffeomorphism have one-dimensional stable manifolds.

For surface diffeomorphisms the answer depends dramatically on the regularity. In the C2C^2 topology Axiom A is not dense: Newhouse constructed persistent ('wild') non-hyperbolic C2C^2 surface diffeomorphisms, and related Newhouse phenomena give open sets of C2C^2 surface diffeomorphisms with infinitely many coexisting sinks. The mechanism is fractal. A horseshoe — a hyperbolic invariant Cantor set of a surface map — meets a transverse section in regular Cantor sets (compact sets dynamically defined by a CrC^r expanding map of finitely many intervals). Newhouse's thickness is a fractal invariant of such Cantor sets, continuous and positive in the C2C^2 (indeed C1+αC^{1+\alpha}) topology, and when the product of the thicknesses of two regular Cantor sets exceeds one and their convex hulls overlap, the two sets intersect stably, i.e. the intersection persists under perturbation; stably intersecting stable and unstable Cantor sets of horseshoes produce persistent tangencies. Moreira and Yoccoz later proved that generic pairs of C2C^2-regular Cantor sets whose Hausdorff dimensions sum to more than one even have translations with stable intersection.

The C1C^1 world is different. Ures showed that thickness is 'terribly discontinuous' in the C1C^1 topology — generic C1C^1-regular Cantor sets have zero thickness — and Moreira proved that there are no C1C^1-stable intersections of regular Cantor sets at all: any pair (K,K′)(K,K') of regular Cantor sets can be approximated arbitrarily well in the C1C^1 topology by pairs (K~,K~′)(\tilde K,\tilde K') with empty intersection, and generic pairs of C1C^1-regular Cantor sets have an arithmetic difference K−K′={x−y:x∈K,y∈K′}K-K'=\{x-y:x\in K,y\in K'\} with empty interior. The same paper shows that a residual set of C1C^1 surface diffeomorphisms has no tangencies between the stable foliation of one horseshoe and the unstable foliation of another. Since stable Cantor-set intersections are the main known obstruction to hyperbolicity on surfaces, Moreira's result gives, in the author's words, 'some hope of proving density of hyperbolicity in the C1C^1 topology for diffeomorphisms of surfaces' — a question he attributes to Smale.

The question sits inside the Palis density conjecture, which predicts that every diffeomorphism can be approximated by one that is hyperbolic or exhibits a homoclinic tangency or a heterodimensional cycle. On surfaces, where heterodimensional cycles between saddles cannot occur, Pujals and Sambarino proved this dichotomy affirmatively in the C1C^1 topology: every C1C^1 surface diffeomorphism can be C1C^1-approximated by one that is hyperbolic or displays a homoclinic tangency. Crovisier and Pujals proved the analogue in arbitrary dimension: every diffeomorphism is C1C^1-approximated by one exhibiting a tangency or a heterodimensional cycle, or by an essentially hyperbolic one (finitely many hyperbolic attractors and repellers whose basins form an open dense subset of MM). In ambient dimension at least three the strong density statement fails outright: Bonatti and Díaz constructed C1C^1-robust homoclinic tangencies using blenders (hyperbolic sets whose unstable set behaves as if it had larger dimension than its unstable leaves, creating robust intersections with curves and surfaces), so there are C1C^1-open sets of diffeomorphisms containing no Axiom A element; subsequent work extended these robust-tangency mechanisms, including C1C^1-stable intersections of higher-dimensional regular Cantor sets (Asaoka) and uncountably many C1C^1-robust tangencies unfolding from coindex-one heterodimensional cycles (Li). These results delimit the problem to dimension two, where Moreira's theorem kills the classical obstruction but does not settle density: the surface case of the density question remains open.

Problem Statement

Are Axiom A diffeomorphisms dense in Diff1(M)\mathrm{Diff}^1(M) equipped with the C1C^1 topology, for every compact surface MM? Equivalently: given any C1C^1 diffeomorphism ff of a compact surface MM and any ε>0\varepsilon>0, does there exist a diffeomorphism gg with dC1(f,g)<εd_{C^1}(f,g)<\varepsilon whose non-wandering set is uniformly hyperbolic and has periodic points dense (Axiom A; one may equivalently demand Axiom A together with the no-cycle condition, i.e. Ω\Omega-stability, since the no-cycle condition is accessible within the Axiom A class)? A negative answer amounts to exhibiting, for some compact surface MM, a non-empty C1C^1-open subset of Diff1(M)\mathrm{Diff}^1(M) containing no Axiom A diffeomorphism. This is often called Smale's problem for surface diffeomorphisms; it is strictly stronger than the surface Palis dichotomy proved by Pujals–Sambarino, which allows a hyperbolic approximation or one with a homoclinic tangency. Moreira's theorem that stable intersections of regular Cantor sets — the classical Newhouse obstruction — do not exist in the C1C^1 topology removes the main known mechanism obstructing density on surfaces, but it does not by itself settle the question.

References

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