THE GRADUATE STUDENT SECTION WHAT IS… a Blender? Ch. Bonatti, S. Crovisier, L. J. Díaz, A. Wilkinson Communicated by Cesar E. Silva A blender is a compact, invariant set on which a diffeo- morphism has a certain behavior. This behavior forces topologically “thin” sets to intersect in a robust way, pro- ducing rich dynamics. The term “blender” describes its function: to blend together stable and unstable manifolds. Blenders have been used to construct diffeomorphisms with surprising properties and have played an important role in the classification of smooth dynamical systems. One of the original applications of blenders is also one of the more striking. A diffeomorphism 푔 of a compact manifold is robustly transitive if there exists a point 푥 whose orbit {푔푛(푥) ∶ 푛 ≥ 0} is dense in the manifold, and moreover this property persists when 푔 is slightly perturbed. Until the 1990s there were no known robustly transitive diffeomorphisms in the isotopy class of the identity map on any manifold. Bonatti and Díaz (Ann. of Math., 1996)1 used blenders to construct robustly transitive diffeomorphisms as perturbations of the identity map on certain 3-manifolds. To construct a blender one typically starts with a proto- blender; an example is the map 푓 pictured in Figure 1. The function 푓 maps each of the two rectangles, 푅1 and 푅2, affinely onto the square 푆 and has the property that the vertical projections of 푅1 and 푅2 onto the horizontal Figure 1. An example of a proto-blender. The map 푓 is direction overlap. Each rectangle contains a unique fixed defined on the union of the two rectangles, 푅1 and 푅2, point for 푓. in the square 푆; 푓 sends each 푅푖 onto the entire square 푆 affinely, respecting the horizontal and Ch. Bonatti is CNRS researcher in mathematics at the Université vertical directions, with the horizontal expansion de Bourgogne, Dijon, France. His email address is bonatti@u- factor less than 2. Note that 푓 fixes a unique point in bourgogne.fr. each rectangle 푅푖. S. Crovisier is CNRS researcher in mathematics at the Université Paris-Sud. His email address is [email protected] psud.fr. −푛 L. J. Díaz is professor of mathematics at the Pontifícia Universi- The compact set Ω = ⋂푛≥0 푓 푆 is 푓-invariant, meaning dade Católica in Rio de Janeiro. His email address is lodiaz@mat. 푓(Ω) = Ω, and is characterized as the set of points in 푆 on puc-rio.br. which 푓 can be iterated infinitely many times: 푥 ∈ Ω if and A. Wilkinson is professor of mathematics at the University of only if 푓푛(푥) ∈ 푆 for all 푛 ≥ 0. Ω is a Cantor set, obtained Chicago. Her email address is [email protected]. by intersecting all preimages 푓−푖(푆) of the square, which 1This is also where the term “blender” was coined. nest in a regular pattern, as in Figure 2. For permission to reprint this article, please contact: Any vertical line ℓ between the fixed points in 푅1 and in [email protected]. 푅2 will meet Ω. To prove this, it is enough to see that for DOI: http://dx.doi.org/10.1090/noti1438 every 푖 the vertical projection of the set 푓−푖(푆) (consisting November 2016 Notices of the AMS 1175 THE GRADUATE STUDENT SECTION metric point of view, the fractal set Ω, when viewed along nearly vertical directions, appears to be 1-dimensional, allowing Ω to intersect a vertical line robustly. If the rectangles 푅1 and 푅2 had disjoint projections, the proto- blender property would be destroyed. This type of picture is embedded in a variety of smooth dynamical systems, where it is a robust mechanism for chaos. The search for robust mechanisms for chaotic dy- namics has a long history, tracing back to Henri Poincaré’s discovery of chaotic motion in the three-body problem of celestial mechanics. Figure 3(a) depicts the mechanism behind Poincaré’s discovery, a local diffeomorphism of the plane with a saddle fixed point 푝 and another point 푥 whose orbit converges to 푝 both under forward and backward iterations (that is, under both the map and Figure 2. The invariant Cantor set Ω produced by the its inverse). Meeting at 푝 are two smooth curves 푊푠(푝) proto-blender 푓 is the nested intersection of and 푊푢(푝), the stable and unstable manifolds at 푝, respec- preimages of 푆 under 푓. Any vertical line segment ℓ tively. 푊푠(푝) is the set of points whose forward orbit close to the center of the square intersects Ω in at converges to 푝, and 푊푢(푝) is the set of points whose least one point. The line segment can be replaced by backward orbit converges to 푝. a segment with nearly vertical slope or even a In Figure 3(a), the intersection of 푊푠(푝) and 푊푢(푝) is smooth curve nearly tangent to the vertical direction. transverse at 푥: the tangent directions to 푊푠(푝) and 푊푢(푝) at 푥 span the set of all directions emanating from 푥—the tangent space at 푥 to the ambient manifold, in this case of 2푖 horizontal rectangles) onto the horizontal is an the plane. The point 푥 is called a transverse homoclinic interval. This can be checked inductively, observing that point for 푝. In Figure 3(b) a slight variation is depicted: the projection of 푓−푖−1(푆) is the union of two rescaled here there are two periodic saddles, 푝 and 푞, such that copies of the projection of 푓−푖(푆), which overlap. 푊푠(푝) and 푊푢(푞) intersect transversely at a point 푥, and A more careful inspection of this proof reveals that 푊푢(푝) and 푊푠(푞) intersect transversely at another point the intersection is robust in two senses: First, the line ℓ 푦. The points 푥 and 푦 are called transverse heteroclinic can be replaced by a line whose slope is close to vertical points, and they are arranged in a transverse heteroclinic or even by a 퐶1 curve whose tangent vectors are close cycle. to vertical; second, the map 푓 can be replaced by any 퐶1 In the classification of the so-called Axiom A diffeo- map 푓̂ whose derivative is close to that of 푓. Such an 푓̂ is morphisms, carried out by Stephen Smale in the 1960s, called a perturbation of 푓. transverse homoclinic and heteroclinic points play a cen- The (topological) dimension of the Cantor set Ω is 0, tral role. Any transverse homoclinic point or heteroclinic the dimension of ℓ is 1, the dimension of the square is 2, cycle for a diffeomorphism is contained in a special Can- and 0+1 < 2. From a topological point of view, one would tor set Λ called a horseshoe, an invariant compact set with not expect these sets to intersect each other. But from a strongly chaotic (or unpredictable) dynamical properties (see [2] for a discussion). Two notable properties of a horseshoe Λ are: (1) Every point in Λ can be approximated arbitrarily well by a periodic point in Λ. (2) There is a point in Λ whose orbit is dense in Λ. Horseshoes and periodic saddles are both examples of hyperbolic sets: a compact invariant set Λ for a diffeo- morphism 푔 is hyperbolic if at every point in Λ there are transverse stable and unstable manifolds 푊푠(푥) and 푊푢(푥) with 푔(푊푠(푥)) = 푊푠(푔(푥)) and 푔(푊푢(푥)) = 푊푢(푔(푥)). For a large class of diffeomorphisms known as Axiom A sys- Figure 3. (a) A transverse homoclinic intersection of tems, Smale proved that the set of recurrent points can stable and unstable manifolds, first discovered by be decomposed into a disjoint union of finitely many Poincaré in his study of the 3-body problem. (b) A hyperbolic sets on which (1) and (2) above hold. This horseshoe Λ produced by a pair of transverse hetero- theory relies on the most basic property of transverse in- clinic points 푥 and 푦. Every point in the Cantor set Λ tersections, first investigated by René Thom: robustness. can be approximated arbitrarily well both by a peri- A transverse intersection of submanifolds cannot be de- odic point and by a point whose orbit is dense in Λ. stroyed by a small perturbation of the manifolds; in the dynamical setting, a transverse intersection of stable and 1176 Notices of the AMS Volume 63, Number 10 THE GRADUATE STUDENT SECTION the cube 푄 is stretched and folded across itself by a local diffeomorphism 푔. The horseshoe Λ in Figure 5 is precisely the set of points whose orbits remain in the future and in the past in 푄. The set 푊푢(Λ) of points in the cube that accumulate on Λ in the past is the cartesian product of the Cantor set Ω with segments parallel to the third, expanded direction. 푊푢(Λ) is the analogue of the unstable manifold of a saddle, but it is a fractal object rather than a smooth submanifold. The set Λ is an example of a blender, and its main geometric property is that any vertical curve crossing 푄 Figure 4. (a) Transverse cycle. (b) Nontransverse close enough to the center intersects 푊푢(Λ). In other cycle. words, this blender is a horseshoe whose unstable set behaves like a surface even though its topological dimen- sion is one. This property is robust. While the definition unstable manifolds of two saddles cannot be destroyed of blender is still evolving as new constructions arise, a by perturbing the diffeomorphism. working definition is: A blender is a compact hyperbolic Classifying Axiom A systems was just the beginning.