How do hyperbolic homoclinic classes collide at heterodimensional cycles

We present a model illustrating heterodimensional cycles (i.e., 
cycles associated to saddles having different indices) as a 
mechanism leading to the collision of hyperbolic homoclinic 
classes (of points of different indices) and thereafter to the 
persistence of (heterodimensional) cycles. The collisions are 
associated to secondary (saddle-node) bifurcations appearing in the 
unfolding of the initial cycle.


Introduction.
1.1. Context. A diffeomorphism f has a heterodimensional cycle if there are hyperbolic sets Λ and Σ with different indices (dimension of the unstable bundle) such that the stable manifold of Λ meets the unstable one of Σ and vice-versa (i.e., W s (Λ) ∩ W u (Σ) = ∅ and W u (Λ) ∩ W s (Σ) = ∅). The simplest case of heterodimensional cycle occurs when Λ and Σ are both periodic saddles. We analyze how the phenomena of collision of non-trivial hyperbolic sets, intermingled homoclinic classes, and persistence of heterodimensional cycles arise at simple heterodimensional cycles.
We consider one-parameter families of diffeomorphisms (f t ) t∈[−1,1] starting in a hyperbolic region (Morse-Smale systems) and crossing the boundary of hyperbolicity at a first bifurcation (say t = 0) corresponding to a heterodimensional cycle associated to a pair of saddles, say P and Q. We study the dynamics in a neighbourhood of the cycle set (i.e., the f -invariant set defined as closure of the orbits of the saddles P and Q in the cycle and the intersections W s (P ) ∩ W u (Q) and W u (P ) ∩ W s (Q) between their invariant manifolds). We fix a small neighbourhood W of the cycle set and, for small t > 0, study the part of the non-wandering set Ω(f t ) of f t in W , the so-called resulting non-wandering set for the parameter t, denoted by Ω(f t ) ′ .
The goal (see [36,Chapter 7]) is to describe the dynamics of the resulting nonwandering set Ω(f t ) ′ having some persistence or prevalence (in terms of the parameter t) after the bifurcation (i.e., for small t > 0). Such a dynamic depends essentially on the choice of the parameter and on the restriction of the bifurcating diffeomorphism to the cycle set.
We adopt the following strategy for studying bifurcations. The type of bifurcation of the diffeomorphism f (saddle-node, flip, Hopf, homoclinic tangency, heterodimensional cycle) defines a (local) bifurcation submanifold Σ (i.e., any g ∈ Σ has a bifurcation as the one of f ). We assume that Σ is in the boundary of the hyperbolic systems and has codimension one. More precisely, we consider arcs (f t ) t∈ [−1,1] intersecting transversely Σ for (say) t = 0 and such that, for every t < 0, the diffeomorphism f t is hyperbolic. Then we say that (f t ) t∈ [−1,1] crosses the boundary of hyperbolicity at t = 0 and unfolds the bifurcation for positive t. [32] conjectured that the generic way of crossing such a boundary is either by the loss of hyperbolicity of some periodic orbit or by the loss of transversality of the intersection between the invariant manifolds of hyperbolic periodic points (creation of cycles). This conjecture holds for surface diffeomorphisms in the boundary of the Morse-Smale systems and it remains open in other cases.
The first sort of bifurcation above (loss of hyperbolicity) is local (the dynamics on a finite orbit) while the second one (creation of cycles) is semi-global involving the orbit of a non-periodic point (a non-transverse intersection between invariant manifolds). Interesting dynamical features arise when there is some interplay between these local and semi-global bifurcations. For instance, bifurcations homoclinic bifurcation (tangencies) generate infinitely many saddle-node, period doubling and Hopf bifurcations, see [41] and [36,Chapter 3]. For the converse, loss of hyperbolicity bifurcations generate cycles, some semi-global hypotheses on the dynamics are necessary. Such an interaction occurs, for instance, in the so-called saddle-node cycles introduced in [33]: there is a saddle-node whose unstable manifold intersects its stable manifold (or vice-versa). Especially interesting saddle-node cycles are the saddle-node horseshoes introduced in [42], where the saddle-node belongs to a (topological) horseshoe (the archetypal saddle-node horseshoe is depicted in Figure 1).
In some cases, saddle-node horseshoes lead to a string of bifurcations and phenomena as Hénon-like attractors and persistence of tangencies, see [40,21,11,8,10,23,15], about saddle-node cycles and saddle-node horseshoes of surface diffeomorphisms, and [17], for partially hyperbolic saddle-node cycles in higher dimensions. For a survey on this subject see [20]. We study the occurrence of saddle-node horseshoes at heterodimensional cycles. In fact, saddle-node cycles are a simple type of the collision bifurcations we will discuss in the next paragraphs.

Collision bifurcations and heterodimensional cycles.
We study partially hyperbolic heterodimensional cycles: the cycle set has a partially hyperbolic splitting E s ⊕ E c ⊕ E u , where E s and E u are uniformly hyperbolic (non-trivial) and E c is one-dimensional and is not hyperbolic. Thus the dimension of the ambient is at least three (heterodimensional cycles only occur in dimensions strictly greater than two). We describe how the unfolding of these cycles generate twin saddle-node horseshoes with a common saddle-node, corresponding to the collision of horseshoes This paper illustrates the interplay between local (saddle-node) and semi-global (heterodimensional cycles) bifurcations: the unfolding of a heterodimensional cycle accomplishes a sequence (t n ), t n → 0 + , of colliding twin saddle-node horseshoes. Each saddle-node bifurcation generates persistence of cycles and intermingled homoclinic classes (see the definitions below). In rough terms, the each cycle generates two horseshoes having different indices. For each t n , the two horseshoes become saddle-node horseshoes sharing a saddle-node (following [22], this is a collision bifurcation of hyperbolic sets). Finally, these saddle-nodes generate persistent of heterodimensional cycles.
Let us state our main theorem informally. First, recall that the homoclinic class of a (hyperbolic) saddle P of a diffeomorphism f , denoted by H(P, f ), is the closure of the transverse intersections of the stable and unstable manifolds of the orbit of P . Important properties of a homoclinic classes H(P, f ) are: (i) f -invariance, (ii)transitivity (i.e., existence of a dense orbit), and (iii) the subset of periodic points with the same index as P is dense in H(P, f ).
Concerning (iii), note that a homoclinic class may have saddles of different indices, thus being non-hyperbolic. This phenomenon occurs at heterodimensional cycles, see [12,13,18]. This paper gives examples of such a situation. Finally, a homoclinic class is trivial if it consists of a single orbit, and two homoclinic classes are intermingled if they have non-empty intersection.
Theorem 1. There are arcs (f t ) t∈ [−1,1] unfolding at t = 0 (t ≥) a partially hyperbolic heterodimensional cycle associated to saddles P and Q having sequences of scaled parameter intervals C n = [a n , t n ) = λ n [1 + µ − , 1 + µ ⋆ ) and H n = (t n , b n ] = λ n (1 + µ ⋆ , 1 + µ + ], such that: : (A) Hyperbolicity: for every n and for all t ∈ H n , the resulting nonwandering set of f t is hyperbolic and equal to the disjoint union of the (nontrivial) homoclinic classes of P and Q. : (B) Secondary saddle-node collision bifurcations: for every n, f tn has a saddle-node S n such that the intersection of the homoclinic classes of P and Q is exactly the orbit of S n . : (C) Persistence of cycles at collision bifurcations: for every n, C n = (a n , t n ) is an interval of persistence of heterodimensional cycles: for every parameter t ∈ C n , the diffeomorphism f t has a heterodimensional cycle associated to hyperbolic sets Υ t of index one contained in the homoclinic class of P and Ξ t of index two contained in the homoclinic class of Q.
We observe that, for every saddle-node parameter t n , every compact invariant set contained in either H(P, f tn ) or in H(Q, f tn ) and disjoint from the saddle-node is hyperbolic.
Theorem 1 shows how homoclinic classes may change from the metric point of view. One aims to understand the map (P f , f ) → H(P f , f ) that associates to a saddle P f (depending continuously on f ) its homoclinic class. The map H(·, ·) is lower semi-continuous. This follows from the continuous dependence on the dynamics of the invariant manifolds of saddles and of their transverse intersections (on compact parts). A metric bifurcation of a homoclinic class H(P f , f ) is discontinuity of H(·, ·). A well-known metric bifurcation is the creation of cycles. The theorem describes bifurcations of homoclinic classes via collision. Let us shortly discuss this notion.
Consider an arc of (f t ) t∈ [a,b] such that, for all t ∈ [a, b], there are saddles P t and Q t (depending continuously on t). The homoclinic classes of P t and Q t collide at In the collisions in the theorem the following holds: (i) the colliding homoclinic classes are both non-trivial and hyperbolic; (ii) at the collision parameter, there are twin saddle-node horseshoes intersecting along the orbit of a saddle-node; and (iii) after the collision, the homoclinic classes simultaneously explode. See Figure 3. In our definition of collision the saddles P c and Q c are defined, thus their homoclinic classes are defined at and after the collision. Thus it is stronger than the following definition: the homoclinic classes of P t and Q t have a weak collision at t = c if H(P t , f t ) ∩ H(Q t , f t ) = ∅ for all t < c and the distance between H(P t , f t ) and H(Q t , f t ) goes to zero as t → c. A priori, the saddles P c and Q c to be defined. The simplest case of weak collision is the collision of two isolated saddles via a saddle-node bifurcation. A more interesting case happens when the saddle-node is non-isolated. In dimension two, the main examples are the Derived from Anosov bifurcations, [39], where a repeller and a non-trivial hyperbolic attractor collide, and the saddle-node horseshoes above.
In the weak collisions above, at least a homoclinic class is trivial (a sink or a repeller) while in our construction both colliding classes are hyperbolic and nontrivial. Collisions of two non-trivial and non-hyperbolic homoclinic classes were studied in [22]. In [22] the collision leads to intermingled homoclinic classes: after the collision, the two classes explode and coincide.
1.3. Dynamics at partially hyperbolic heterodimensional cycles. We now focus on the dynamics generated by partially hyperbolic heterodimensional cycles (with one-dimensional central direction) of codimension one (the bifurcating submanifold has codimension one). From now on, we assume these two conditions, which imply that the cycle is related to saddles P and Q of indices s and s + 1. In such a case, the unfolding does not generate homoclinic tangencies and all the cycles in the unfolding are heterodimensional ones. In fact, the unfolding accomplishes infinitely many cycles. We next discuss the transitions from hyperbolicity to intermingled homoclinic as the parameter evolves. We review some results and see how they are used in this paper.
In this setting, for all small t > 0 the homoclinic classes of P and Q explode and are both non-trivial (the dependence of the saddles on t is omitted). Thus, by the Smale's homoclinic theorem, there are horseshoes Λ P and Λ Q such that P ∈ Λ P ⊂ H(P, f t ) and Q ∈ Λ Q ⊂ H(Q, f t ). These horseshoes have different indices and, in some cases, the homoclinic classes of P and Q are equal to Λ P and Λ Q , thus the homoclinic classes are disjoint. But there are some cases where the homoclinic classes of P and Q coincide and properly contain the horseshoes Λ P and Λ Q . A goal of this paper is to study how these hyperbolic homoclinic classes can bifurcate and describe the dynamics following these secondary bifurcations. In this direction, three crucial steps are: : (i) to determine whether the homoclinic classes of P and Q are hyperbolic; : (ii) to describe the resulting non-wandering set of f t after the bifurcation; : (iii) to study secondary bifurcations generated by the unfolding of the first cycle.
We now recall some results related to (i)-(iii) in the simplest case of bifurcations from Morse-Smale systems. We first fix some notations. Consider (f t ) t∈[−1,1] unfolding a cycle, say at t = 0 and for positive t. Given a dynamical property P, consider the sets of parameters Following [36,Chapter 7], we say that the property P is robust (after the bifurcation) if there is some t > 0 such that P(t) = (0, t]. Similarly, property P is prevalent if lim inf t→0 + |P(t)|/t > 0 and totally prevalent lim inf t→0 + |P(t)|/t = 1 (| · | stands for the Lebesgue measure).
We are interested in three dynamical features: hyperbolic dynamics of the resulting non-wandering set (property H), intermingled homoclinic classes (prop. I), and existence of heterodimensional cycles associated to hyperbolic sets (prop. C).

As above, let
Note that H ∩ I = ∅ and H ∩ C = ∅.
Concerning question (i) above, [12,13] construct open set of arcs (f t ) t∈[−1,1] such that there is t 0 > 0 with I(t 0 ) = (0, t 0 ]. Moreover, [18] states that the phenomenon I of intermingled homoclinic classes is always prevalent at the bifurcation. On the other hand and in the opposite direction, [16] gives open sets of arcs (f t ) t∈[−1,1] which are prevalently hyperbolic at the bifurcation. Since I ∩ H = ∅, [18] above implies that lim sup t→0 + (|H(t)|/t) < 1. Thus hyperbolicity is not totally prevalent at the bifurcation. But [16] assures that one can construct (f t ) t∈ [−1,1] with frequency of hyperbolicity close to one: fixed any ε > 0, there are arcs with We now discuss questions (ii) and (iii) above. In [16] above, for every t > 0, the parameter interval [0, t] contains infinitely many intervals corresponding alternately to hyperbolic and non-hyperbolic diffeomorphisms. Thus a natural problem is to understand the transition from hyperbolic to non-hyperbolic dynamics (secondary bifurcations). For the hyperbolic parameters t in [16] the non-wandering set of f t is the (disjoint) union of the homoclinic classes of P and Q. Thus, in this case, there are two natural possibilities for secondary bifurcations: new heterodimensional cycles and loss of hyperbolicity of some saddle. These two possibilities correspond to the semi-global and local bifurcations discussed in Section 1.1. We now focus on local bifurcations.
Typically, the loss of hyperbolicity occurs via saddle-node bifurcations (in this context flip and Hopf bifurcations are forbidden). The saddle-node may correspond either to an emerging saddle-node or to the collision of a pair of old saddles in the homoclinic classes of P and Q. In the emerging case, the saddle-node generates a pair of saddles A and B of different indices. In some cases, these new saddles may be independent of P and Q (i.e., the saddles do not belong to the homoclinic classes of P and Q). Moreover, throughout the unfolding, cycles related to A and B can appear. Then this new cycle generates homoclinic points of A and B. Therefore non-trivial homoclinic classes independent of the ones of P and Q arise. Hence the hyperbolicity of the homoclinic classes of the saddles P and Q does not imply the global hyperbolicity of the non-wandering set (the homoclinic classes of A and B may fail to be hyperbolic). This sort of bifurcations and dynamics are described in [19].
In this paper, we study the non-emerging case: the saddle-node corresponds to a collision of saddles in the hyperbolic homoclinic classes of P and Q (thus to a collision of homoclinic classes). We describe how the phenomena of intermingled homoclinic classes and of persistence of heterodimensional cycles arise from these bifurcations.
1.4. Ingredients of the proof. We next describe some ideas behind the results above we use in this paper. The hypotheses (partial hyperbolicity and bifurcation from Morse-Smale) imply that the intersection W s (P, f 0 )∩W u (Q, f 0 ) is the orbit of a periodic curve γ, named connection, tangent to the central direction E c and whose extremes are the saddles P and Q. The heuristic principle in [12,13,16,19] is that the restriction of the diffeomorphism f 0 to the connection γ (shortly, the central dynamics) determines the dynamics after the bifurcation. The different kinds of dynamics after the bifurcation we described before correspond to different choices of parameters and of dynamics of f 0 along the connection. In rough terms: (a) small distortion along the connection γ generates robustly intermingled homoclinic classes (of P and Q); (b) to get hyperbolic parameters one needs appropriate big distortion along γ.
It is interesting to compare the previous results, suggesting that (for heterodimensional cycles) the central dynamics determines the dynamics following the bifurcation, to the papers [28,34,35,26,38,27] showing that, in the setting of homoclinic tangencies, the dynamics after the bifurcation is determined by the fractal dimensions of the bifurcating hyperbolic set. For heterodimensional cycles, the role of fractal dimensions is unknown. Here we describe the creation of thick Cantor sets at heterodimensional cycles and derive consequences from this fact.
More precisely, the restriction of f 0 to the connection γ induces a one-parameter family of maps (L µ ) µ∈[−δ,δ] , we call limit central one-dimensional dynamics, as follows. There are sequences t ± n of parameters, t ± n → 0, and of reparametriza- , such that the dynamics of f ̺n(µ) in the resulting nonwandering set is the product of L µ (the central part) by a hyperbolic saddle dynamics. The central part L µ is associated to a system of iterated functions generated by: : (i) the linearizations of f 0 in the central directions of Q and P ; : (ii) (assume that index(Q) = index(P ) + 1) a transition map given by the iterates from a (scaled) fundamental domain of Q in γ to a (scaled) fundamental domain of P in γ; : (iii) a translation corresponding to the rescaling of the parameter t unfolding the cycle.
The persistent (heterodimensional) cycles are associated to the creation of thick Cantor sets (see Definition 6.1 and the Gap Lemma 7.1). Thus the mechanism is somewhat similar to the one of persistence of homoclinic tangencies. However, the dynamics generating these thick sets are different. In the homoclinic setting, the thick Cantor sets are related to the critical dynamics and obtained using the quadratic family, see [36,Chapter 6]. Here the limit dynamics is non-critical, but we prove that the family (L µ ) µ∈[−δ,δ] also generates thick Cantor sets. We see that every L µ has two hyperbolic sets Υ ′ µ and Ξ ′ µ with non-empty intersection. Such an intersection is assured by a condition on the product of their thickness (see the Gap Lemma 7.1). The sets Υ ′ µ and Ξ ′ µ are the projections along the stable and unstable directions of two hyperbolic sets Υ ̺n(µ) and Ξ ̺n(µ) of f ̺n(µ) having different indices. Thus intersections of Υ ′ µ and Ξ ′ µ correspond to (heterodimensional) cycles associated to Υ ̺n(µ) and Ξ ̺n(µ) . Some comments about our constructions arise naturally. In rough terms, our construction is a skew-product, where we fix a one-dimensional dynamics and construct a cycle whose central limit dynamics is the fixed one. We do not know the flexibility of our construction. For instance, whether our result holds for arcs of diffeomorphisms in a neighbourhood of the arcs in this paper. The skew-product structure allows us to estimate the thickness of Cantor sets created in the unfolding of the cycle, concluding the persistence of cycles thereafter. But, in general, estimates of the thickness in dimensions strictly bigger than two are tough and subtle, depending on the regularity of the invariant foliations, see [37]. Probably, the ideas sketched in [27] for homoclinic tangencies in higher dimensions could be an ingredient for extending our results to a wider setting.
Finally, related to our results, in the context of vector fields, [3] gives a vector field with two homoclinic classes whose intersection is the closure of the unstable manifold of a hyperbolic singularity. See also [24], where it is proved that a singular hyperbolic set (see [25] for the definition) containing a unique singularity is either transitive or the union of two homoclinic classes. This paper is organized as follows. In Section 2, we describe a model arc unfolding a heterodimensional cycle. In Section 3, we introduce a one-parameter system of iterated functions describing the central dynamics after the bifurcation (limit dynamics). After introducing the basic terminology, we sketch the proof of the theorem in Section 3.3. In Section 4, we choose the central dynamics by considering saddle-node arcs. In Section 5, we construct two hyperbolic sets (of different indices) and describe their collision via a saddle-node bifurcation. In Section 6, we construct two one-dimensional hyperbolic Cantor sets (contracting and expanding), associated to the system of iterated functions, the expanding one with large thickness. Finally, we see that these Cantor sets correspond to two hyperbolic sets of diffeomorphisms of the arc (see Sections 6.1 and 6.3) and prove, in Section 7, that these hyperbolic sets have heterodimensional cycles.
2. The model one-parameter family. For notational simplicity, we will carry our constructions in dimension 3, the extension to higher dimensions is straightforward. We consider a one-parameter family of (local) diffeomorphisms (f t ) t∈[−τ,τ ] defined on the cube R = [−1, 1] × [−1, 5] × [−1, 1] of R 3 . Using local coordinates, this construction can be carried to any manifold of dimension three.
The definition of the arc (f t ) t∈[−τ,τ ] is done in three steps: (a) semi-local product partially hyperbolic dynamics, (b) existence and unfolding of the heterodimensional cycle, and (c) semi-global hypotheses (existence of a filtration). The filtration enables us to control the dynamics after the bifurcation. The product partial hyperbolicity allow us to reduce the study of the dynamics of the resulting non-wandering set to a one-dimensional family of iterated functions.
(a) Partially hyperbolic product dynamics.
where F is a strictly increasing function to be fixed in Section 4 such that: HOW DO HYPERBOLIC HOMOCLINIC CLASSES COLLIDE?

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: local affinity: The domination condition implies that λ s < λ < β < λ u and that the restriction of f to R has a partially hyperbolic splitting E s ⊕ E c ⊕ E u with three non-trivial bundles given by the coordinate directions. Observe also that, by definition of the diffeomorphism f , and and Thus W s (P ) and W u (Q) meet transversely along the connection ∈ R for all 0 < j < k 0 . So A, B ∈ W u (P ) ∩ W s (Q) = ∅ and there is a heterodimensional cycle associated to P and Q.
Take a small compact neighbourhood : . Thus W u (P ) and W s (Q) meet quasi-transversely throughout the orbit of the heteroclinic point A, Thus, as W s (P ) and W u (Q) have non-empty transverse intersection, f has a (codimension one) heterodimensional cycle related to P and Q.
We next define the arc arc (f t ) t∈[−τ,τ ] unfolding the cycle above at t = 0. We let f t (x, y, z) = f (x, y, z) for all (x, y, z) ∈ R and define the restriction of f k0 t to U A the composition of f k0 and a t-translation parallel to the Y direction: Note that [−1, 1] × (0, 5) × {0} ⊂ W s (P, f t ) and, by construction, We now assume some properties of the bifurcating diffeomorphism f that allow us to control the global dynamics of f . First, a pair ( : (c) Semi-global dynamics: filtrations and relative Morse-Smale dynamics. The neighbourhood of the cycle It follows that, for every small |t|, the set W is also a filtrating neighbourhood of f t . Hence, for all small |t|, The semi-local dynamics of f t implies that, for t < 0, the restriction of f t to W is Morse-Smale (in particular, the homoclinic classes of P and Q are both trivial). Thus at t = 0 there is a Ω-explosion: for positive t, the classes of H(P, f t ) and H(Q, f t ) are both non-trivial.
We finish this section with a remark about invariant foliations. For X = (x 0 , y 0 , z 0 ) ∈ R, define its strong stable, strong unstable and central leaves, denoted by F s (X), F u (X), and F c (X), by This defines three foliations (strong stable, strong unstable, and central) on R. For each t, we extend them, via f t , to the whole W . These foliations (denoted by . Note that if X ∈ R then Df t (X) uniformly contracts in the X-direction and uniformly expands in the Z-direction, and if X ∈ U A then Df k0 t (X) is the identity. These remarks and the partial hyperbolicity of f t in R imply that Λ t is partially hyperbolic: the X and Z-directions are hyperbolic and dominate the (central) Y-direction. Thus the dynamics of f t is mainly determined by its central dynamics. We study the central dynamics in the next section.
3. Normalized central dynamics, selection of parameters, and outline of the construction. In this section, we introduce the main ideas in the proof of Theorem 1 and sketch its proof. We first present the main terminology in our constructions: itineraries (Section 3.1) and normalized central dynamics (Section 3.2). Finally, we outline the proof of Theorem 1 in Section 3.3.

Itineraries. Take fundamental domains
Fix small t > 0 and X ∈ Λ t ∩ ∆ + . There are two possibilities for positive iterates of X: either there is n > 1 with f n t (X) ∈ ∆ + or not. In the first case, the smallest n > 1 with f n t (X) ∈ ∆ + is the forward return time of X, denoted by r + (X). In the second case, X ∈ W s (P ) ∪ W s (Q). Similarly, for negative iterates of X, either there is n > 1 with f −n t (X) ∈ ∆ + or not. In the first case, the backward return time of X, r − (X), is defined as above. In the second case, X ∈ W u (P ) ∪ W u (Q). These properties follow using the filtration and the geometry of the cycle. See [16,Lemma 4.1]. Since, by the definition of f t , the invariant manifolds of P and Q meet transversely or quasi-transversely, according to the case, the previous results may be read as follows: Lemma 3.1. Given small t > 0 and any point X ∈ Λ t ∩ ∆ + there are the following possibilities: • X has infinitely many forward or backward returns to ∆ + , • X has finitely many forward and finitely many backward returns to ∆ + , then there are three possibilities: W s (Q, f t ) meet quasi-transversely at X, and 3. X ∈ W s (P ) ∩ W u (Q) (heteroclinic connection point) and W s (P, f t ) and W u (Q, f t ) meet transversely at X.
Assume that X has a first forward return r + (X) =q + k 0 +p + N to ∆ + ,q =q(X) andp =p(X), where k 0 is as in the definition of the cycle, F N (D + ) = D − , and • If X ∈ ∆ + has a first forward return r + (X) =q Proof: Fix t = t n (1 + µ), µ ∈ (−δ, δ), and X = (x, y, z). By definition of f t and since F is affine in [3,5], Since X returns to ∆ + , the unfolding condition (1) implies that Finally, the estimate forq follows similarly noting that y k0+p+N < t.
and we say that the first return r + (X) of X is of type (q(X), p(X)). Since β n = t −1 n and λ n = t n , the y-coordinate of X r + (X) is Equation (2) leads to the following one-parameter family of maps: where D q,p µ is the maximal subset of D + consisting of points y with Φ q,p µ (y) ∈ D + (see Figure 6). The Remark 3.3 justifies this name and implies that Φ q,p µ gives the y-coordinate of the (q, p)-returns of f tn (1+µ) .
• By the monotonicity of F , D q,p µ is either empty or a closed subinterval of D + . • Suppose that X = (x 0 , y 0 , z 0 ) ∈ ∆ + has a (q, p)-return. Then y 0 ∈ D q,p µ and the y-coordinate of f q+n+k0+p+n+N t (X) is Φ q,p µ (y 0 ). Conversely, given any y 0 ∈ D q,p µ , there is at least one point of the form X = (x, y 0 , z) ∈ ∆ + with a (q, p)-return.
• (consecutive returns) Assume that X 0 = (x 0 , y 0 , z 0 ) ∈ ∆ + has a (q 0 , p 0 )return X 1 and X 1 has a (q 1 , p 1 )-return X 2 . Figure 6. The maps Φ q,p µ By Remark 3.3, the maps Φ p,q µ give the Y-coordinate (central dynamics) of returns of f t for appropriate t. Now Df t is uniformly contracting in the X-direction and uniformly expanding in the Z-direction. These facts together with the product structure imply that some dynamical properties of subsets of Λ t , such as hyperbolicity, existence of periodic points, and bifurcations, have a translation to dynamical properties of the maps Φ p,q µ , for appropriate µ, p, and q. For example, to a fixed point of Φ p,q µ corresponds a periodic point of f t , where p and q determines the central coordinate of the iterates of the point as well as its index. The dynamics of the maps Φ p,q µ depend only on F N , λ, and µ. In Section 4, we choose F and λ accordingly.

3.3.
Outline of the constructions. Sketch of the proof. In this section, we present the main steps of our constructions and sketch the proof of Theorem 1.

Selection of the central dynamics.
To define the map F , corresponding to the central dynamics, we first consider the auxiliary saddle-node arc g α with a saddle-node at the point 1/2 for the parameter α = 0 (here ε > 0 is small). The saddle-node is unfolded for negative α: for α > 0, the map g α has two hyperbolic fixed points close to 1/2, collapsing to 1/2 for α = 0, and disappearing thereafter. We select an interval L containing the point 1/2 in the reference fundamental domain D + = [β −1 , 1]. To each parameter α (close to 0), we consider a map F α defining a possible central dynamics of f = f α in the cube R as follows, This is a key property of F α , see condition (F3) in Section 4.1. We will choose α ⋆ > 0 close to 0 and let F = F α ⋆ . Then, by equation (2), for parameters t n = λ n the map g α ⋆ gives the y-coordinate of the points having a return of type (0, 0). In very rough and purposely vague terms, the choice of α ⋆ is done for guaranteeing the creation of one dimensional (central) Cantor sets with large thickness after unfolding the saddle-node bifurcations that appear throughout the unfolding of the initial heterodimensional cycle (for details, see the next paragraphs). The creation of these thick Cantor sets is a key step for obtaining the persistence of heterodimensional cycles in Theorem 1.
Dynamics of the returns. The selection of α in the definition of the central dynamics is done as follows. We fix small µ ⋆ < 0 and consider the auxiliary saddlenode family ϕ α (y) = g α (y) + µ ⋆ . This family has a saddle-node bifurcation for some We consider a new saddle-node arc (L µ ) µ∈(−δ,δ) , L µ = g α ⋆ + µ, with a saddlenode for µ ⋆ < 0. By equations (2) and (4) and Remark 3.3, for t = t n (1 + µ) ∈ B n and X ∈ [−1, 1] × L × [−1, 1] with a return of type (0, 0), the map L µ defines the y-coordinate of such returns, Secondary (rescaled) saddle-node bifurcations. Take the saddle-node arc (L µ ) µ∈(−δ,δ) and let s µ ⋆ ∈ L (close to 1/2) be the corresponding saddle-node. The construction of (f t ) t∈[−τ,τ ] and the comments above imply that, for any t ⋆ n = t n (1 + µ ⋆ ), the diffeomorphism f t ⋆ n has a saddle-node at some point S t ⋆ n in ∆ + whose ycoordinate is s µ ⋆ . For every t ⋆ n , the saddle-node is essentially the same, the only difference being that the periods π n of S t ⋆ n increase as n, but the local central dynamics of the f πn t ⋆ n at the saddle-nodes S t ⋆ n are always the same (given by h µ ⋆ ). Hyperbolicity before the unfolding of the saddle-node. By Remark 3.3, for every t = t n (1 + µ) with small µ, the system (Φ q,p µ ) gives the central returns to the reference cube ∆ + . We prove that there is µ + , µ + > µ ⋆ , such that the system (Φ q,p µ ) is hyperbolic for all µ ∈ (µ ⋆ , µ + ]. For the precise meaning of this assertion see Section 5.1. The relevant fact is that the hyperbolicity of (Φ q,p µ ) implies the hyperbolicity of f t for t = t n (1 + µ). Let us explain this point.
For µ > µ ⋆ , the map L µ = Φ 0,0 µ has two fixed points s − µ (repelling) and s + µ (contracting) collapsing to the saddle-node s µ ⋆ at µ ⋆ . As above, for t = t n (1 + µ), µ > µ ⋆ , the points s − µ and s + µ correspond to saddles S − t and S + t of f t , of indices 2 and 1, whose y-coordinate are s − µ and s + µ . Using Remark 3.3, the product structure of the dynamics, and noting that every f t uniformly contracts in the X-direction and uniformly expands in the Z-direction, we translate the hyperbolicity of (Φ q,p µ ) to f t for every t = t n (1+µ) with µ ∈ (µ ⋆ , µ + ] (any n). Finally, using the family (Φ q,p µ ), we prove that for such parameters one has that • H(P, f t ) = H(S + t , f t ) and H(Q, f t ) = H(S − t , f t ); • the homoclinic classes H(P, f t ) and H(Q, f t ) are both hyperbolic (thus disjoint); and • the resulting non-wandering set of f t is the union of H(P, f t ) and H(Q, f t ).
Finally, we see that these hyperbolic homoclinic classes collide for t ⋆ n . Persistence of cycles at the collisions. To get the intervals of persistence of cycles (associated to the saddle-nodes t ⋆ n ) we choose carefully the parameter α ⋆ (of the arc (g α )) in the definition of F . In fact, in the previous steps about hyperbolicity the choice of α ⋆ is essentially irrelevant.
Recall that, roughly, the thickness of a Cantor set is the quotient length(bridge)/length(gap), where in each step of the construction of the Cantor set a gap is a removed open interval and its two bridges are the adjacent resulting connected components, see Definition 6.1.
gaps Figure 8. Naive gaps Remark 3.3 now implies that Σ µ is contained in the the projection in the central direction of a hyperbolic set Ξ t , t = t n (1 + µ), of index two.
We also consider a hyperbolic set Υ t of index one in the homoclinic class of P whose projection in the central direction is a Cantor set with thickness uniformly bounded from 0, see Section 6.1.
The Gap Lemma (Lemma 7.1) claims that two Cantor sets with non-disjoint convex hulls are and whose product of their thickness is larger than one have nonempty intersection. The previous constructions (for small |α ⋆ |) imply that product of the thickness of the projections of Ξ t and Υ t is bigger than one. One checks that for small |α ⋆ | the convex hulls of these sets are non-disjoint. Thus they verify the Gap Lemma. Hence the projections of Ξ t and Υ t are non-disjoint. The product structure now gives the persistence of cycles (associated to Ξ t and Υ t ) in Theorem 1.

4.
Definition of the map F . Saddle-node arcs. In this section, we define the map F giving the central dynamics of the diffeomorphisms f t .

4.1.
A saddle-node family. For small positive ε (further restrictions on the size of ε will appear throughout the text) consider the saddle-node arc (g α ), small |α|, The arc g α unfolds, at α = 0, a saddle-node at 1/2 as follows. For α > 0, the map g α has two hyperbolic points. These points collapse to the saddle-node at α = 0 and disappear for negative α.
. We will take the map F giving the central dynamics, see Section 2, equal to some F α , for a convenient α, to be chosen in Proposition 6.6.
For the existence of a map F α satisfying the properties above, note first that (F1) and (F3) are compatible. On one hand, by definition, Since g 0 is strictly increasing in [0, 1], c 0 > 0, and 2 β −1 < 1, this condition is equivalent to Since β −1 = ε/4, HOW DO HYPERBOLIC HOMOCLINIC CLASSES COLLIDE?

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Thus, g 0 (2 β −1 ) > β −1 is equivalent to which is automatically verified for small positive ε. Finally, to check condition (F5), observe that by (F3), Note that for α < 0, given any w ∈ . This proposition will be used to estimate the thickness of the dynamically defined Cantor sets to be constructed in Section 6.
The main step in the proof of Proposition 4.1 is the next estimate of the derivative of g nα(w) α : Lemma 4.2. There is α ǫ > 0 such that and anyw ∈ I α = [g α (1), 1].
Proof of the lemma: To prove the second inequality it is enough to check it for α = 0. The general case follows from the continuous dependence of d m (α), d M (α), |J α |, and |I α | on the parameter α. Since g ′ 0 is decreasing, I 0 and J 0 are contained in (−∞, d 0 ), and c 0 > β −1 , By definition, |I 0 | = ε/4. It remains to estimate |J 0 |.
5. Hyperbolic dynamics. In this section, we prove the parts of Theorem 1 about hyperbolicity and collisions of homoclinic classes. First, in Section 5.1, we select the map F and state a hyperbolicity result for the family (Φ q,p µ ), for parameters µ ∈ (µ ⋆ , µ + ], where µ ⋆ is the saddle-node parameter of (L µ ) in Section 3.3 (see equation (7)). In Section 5.2, using Remark 3.3 and the product structure, we translate the hyperbolicity of (Φ p,q µ ) to f t , for t ∈ t n (1 + µ ⋆ , 1 + µ + ]. Finally, in Section 5.3, we state the collision of homoclinic classes, related to the saddlenode bifurcations µ ⋆ of L µ = Φ 0,0 µ . Since we will follow closely [16, Section 4] and [19, Sections 3,7, and 8] we will omit some details of the proofs, just giving the corresponding precise references.
• All the returns of points of L s (µ) to D + are of the form (0, p) with p ≥ 0, i.e., there are no returns of type (q, p) with q > 0.
is uniformly expanding for all q ≥ 0.

Prevalent hyperbolicity for the arc (f t ) t∈
To each point X 0 = (x 0 , y 0 , z 0 ) ∈ ∆ + ∩Λ t (recall that Λ t is the maximal invariant set of f t in the neighbourhood W of the cycle) we associate a sequence i(X 0 ) = (i k (X 0 )) k∈E(X0) of symbols s, c and u, as follows. We let i 0 (X 0 ) = j if y 0 ∈ L j (µ). Let X 1 = (x 1 , y 1 , z 1 ) and X −1 = (x −1 , y −1 , z −1 ) be the first forward and backward returns of X 0 to ∆ + by f t (if they are defined); write i ±1 (X 0 ) = j if y ±1 ∈ L j (µ). If possible, we inductively define first forward and backward returns X ±(n+1) of X ±n and let i n+1 The set E(X 0 ) ⊂ Z is the maximal interval (in Z) where these indices are defined. The sequence i(X 0 ) = (i k (X 0 )) k∈E(X0) is the itinerary of X 0 . We also write r i (X) = r i = (q, p) if the i-th return of X is of type (q, p).
Using the cycle configuration, it is easy to see that every X ∈ Λ t , X = P, Q, has some (backward or forward) iterate in ∆ + . So we can associate to each X ∈ (Λ t \ {P, Q}) a sequence i(Y ), where Y is some iterate of X in ∆ + . These sequences are well defined and coincide (up to a shift) for points whose returns do not have central coordinates equal to β −1 or 1. In fact, for the parameters that we select, we will see that the central coordinates β −1 and 1 correspond to wandering points. Thus, from now on, we assume that X ∈ ∆ + , otherwise we replace X by some iterate of it.
Define now the sets: Fix t = t n (1 + µ) ∈ H n and take X = (x, y, z) ∈ ∆ + with a first return X 1 = (x 1 , y 1 , z 1 ) to ∆ + . Assume that this return is of type (q, p). Then, by Remark 3.3, y 1 = Φ q,p µ (y). This fact and Lemmas 5.2, 5.3, and 5.4 imply that the itinerary of any non-wandering point X of Λ t is constant (equal to s, c or u). These lemmas also imply the following: • If i k (X) = s for all k ∈ E(X) then all the returns are of type (0, p), p ≥ 0. Thus, if µ = µ ⋆ , the returns are uniformly contracting in the Y-direction. • If i k (X) = u for all k ∈ E(X) then all the returns are of type (q, 0), q ≥ 0.
Thus, if µ = µ ⋆ , the returns are uniformly expanding in the Y-direction. • If i k (X) = c for all k ∈ E(X) then all the returns are of type (0, 0). Actually, there is no non-wandering points X with i k (X) = c, for all k ∈ E(X). This follows from the first item in Lemma 5.3: the restriction of Φ 0,0 µ to L c (µ) is monotone without fixed points, see [19,Section 8 and Lemma 8.3]. Similarly, any point having a return with central coordinate equal to β −1 or 1 is wandering. These facts are the ingredients of the proof of the next lemma (see [16,Section 4]). Note that the hyperbolicity result in Theorem 1 follows from Lemma 5.5. • Ω(f t ) ∩ W = Ω s (t) ∪ Ω u (t). In particular, Ω c (t) = ∅.
• There are no cycles associated to the homoclinic classes of P and Q.

5.3.
Sequence of saddle-node bifurcations. Let t = λ n (1 + µ), µ ∈ (µ ⋆ , µ + ], and consider the corresponding hyperbolic fixed points of Φ 0,0 µ , s + µ (attracting) and s − µ (repelling). Having in mind Remark 3.3 and borrowing Lemma 8.1 of [19], we obtain that associated to these points there are hyperbolic periodic points S ± t of f t in ∆ + whose central coordinates are s ± µ . Moreover, the returns of the points S + n is hyperbolic. This lemma follows as Lemma 5.5 using Remark 3.3 and Lemmas 5.2, 5.3, and 5.4. First, note that L c (µ ⋆ ) = ∅. For p ≥ 0, the map Φ 0,p µ ⋆ is uniformly contracting in any compact subset of L s (µ ⋆ ) that does not contain s µ ⋆ . For q ≥ 0, the map Φ q,0 µ ⋆ is uniformly expanding in any compact subset of L u (µ ⋆ ) disjoint from s µ ⋆ . By the product structure, these properties can be translated to f t ⋆ n . Finally, the fact that , Lemmas 5.2, 5.3, and 5.4, and that s µ ⋆ is a saddle-node of Φ 0,0 µ ⋆ .
6. Dynamically defined Cantor sets. In this section, we construct dynamically defined Cantor sets for the family (Φ q,p µ ) having arbitrarily big thickness. In this step, the selection of the parameter α ⋆ in the definition of F (so in the definition of (Φ q,p µ )) is crucial. Note that in the previous sections we just need to consider small α. Therefore, for the sake of clearness, we write explicitly the two parameters µ and α involved in our constructions. In this way, we consider a two parameter family of maps Φ q,p µ,α defined as Φ q,p µ just taking the central map F equal to F α . Similarly, we also have a two-parameter family of diffeomorphisms f t,α .
In what follows, we use dynamically defined Cantor sets and their thickness. For further information on this topic we refer to [36,Chapter 4]. We begin by recalling some definitions. Definition 6.1. A presentation of a Cantor set Υ ⊂ R is an enumeration G = {G n } of the bounded gaps of Υ (i.e., the bounded components of R \ Υ). The thickness of the presentation G, τ (Υ, G), is defined as follows. Let I be the convex hull of Υ. For each n and each point g in the boundary of a gap G n , let B g be the connected component of I \ (G 1 ∪ · · · ∪ G n ) containing g. The interval B g is the left or right bridge of G n , according to its relative position. Then Finally, the thickness of Υ is where the supremum is taken over all the presentations of Υ. Thus, for any presentation G of Υ, τ (Υ, G) is a lower bound of τ (Υ).
We define the central localization of an f t,α -invariant set Λ in the neighbourhood W by Λ c = {y 0 ∈ D + : there is X ∈ Λ of the form (x, y 0 , z)}.
(10) In this section, for appropriate α, we get a sequence (C n ) n of t-intervals such that for every t ∈ C n there are two (transitive) hyperbolic sets Υ t,α and Ξ t,α of f t,α such that • Υ t,α has index one and is contained in the homoclinic class of P ; • Ξ t,α has index two and is contained in the homoclinic class of Q; and • the product of the thickness of the central localizations of Υ t,α and Ξ t,α is bigger than one. These facts (and Lemma 7.1) are the key for getting a t-parameter interval with persistence of cycles, (item (C) in Theorem 1).
6.1. Construction of the hyperbolic set Υ t,α of index one. Lemma 6.2. There are parameters µ 0 and α 0 > 0 and strictly positive constants d c and τ c such that, for every t ∈ B n = [t n (1 − µ 0 ), t n (1 + µ 0 )], t n = λ n , and α ∈ [−α 0 , α 0 ], there exists a (transitive) hyperbolic set Υ t,α of f t,α of index one, contained in the homoclinic class of P , such that the diameter and the thickness of (Υ t,α ) c are lower bounded by d c and τ c , respectively.
Proof: The proof of this lemma has two parts. We first construct a dynamically defined Cantor set for the system Φ 0,p µ,α , for (µ, α) close to (0, 0). Next, using Remark 3.3, we see that such a set is the central localization of a hyperbolic set of f tn (1+µ),α contained in the homoclinic class of P .
It remains to prove that Υ t(µ),α is a transitive subset of the homoclinic class of P . As a consequence of the contraction in the X-direction and the expansion in the Z-direction and the f t,α -invariance of the planes XZ (product structure), the set Υ t(µ),α intersects every plane parallel to the XZ-plane in at most one point. Thus there is a one-to-one correspondence between Υ t(µ),α ∩ ∆ + and (Υ t(µ),α ) c . Hence the dynamics (via f t(µ),α ) of the returns of the points of Υ t(µ),α ∩ ∆ + to ∆ + and the dynamics of (Υ t(µ),α ) c = K p1,p1+1 µ,α by the system Φ 0,p1 µ,α and Φ 0,p1+1 µ,α are conjugate. Hence, the periodic points are dense in Υ t(µ),α and this set is transitive.
Finally, to see that any periodic point A ∈ Υ t(µ),α is homoclinically related to P note that, by construction, the orbit of the unstable manifold of A meets the rec- To see that W u (P, f t(µ),α ) and W s (A, f t(µ),α ) have some transverse intersection note that, by construction and considering negative iterations by f t(µ),α , the stable manifold of A contains a disk of the form [−1, 1] × D + × {z 0 } and that W u (P, f t(µ),α ) intersects any disk of this form (any z 0 ∈ [−1, 1]). This completes the proof of the lemma. 6.2. Construction of the thick hyperbolic set Ξ t,α of index two. As in Section 6.1, we first construct a dynamically defined Cantor set Σ µ,α associated to the family Φ q,0 µ,α , q ≥ 0, with large thickness (Proposition 6.6). Next we see that this set corresponds to the central localization of a transitive hyperbolic set of index two (the announced set Ξ t,α ).
We take small µ < 0 with where d c and τ c are as in Lemma 6.2.
We are now ready to construct a dynamically defined Cantor set Σ µ,α , associated to Γ µ,α , with large thickness and small gaps. We see that, for t = λ n (1 + µ), the set Σ µ,α is the central localization is a hyperbolic set Ω t,α of index 2. The key step in this construction is Proposition 6.6. This proposition together with Lemma 6.2 and (Gap) Lemma 7.1 are the ingredients of the persistence of cycles in Theorem 1, see Section 7.
Archetypally, the set Σ µ,α consists of the points in D + whose orbits by Γ µ,α do not meet G 0 (µ, α). In general, this set fails to be dynamically defined (and thus we do not know how to calculate its thickness). To bypass this difficulty, we consider a dynamically defined subset of it. This is done by constructing a Markov partition for Γ µ,α and defining Σ µ,α as the maximal invariant set associated to such a partition.
Notational remark: From now on, the parameter α is fixed and equal to α(µ ⋆ ), the saddle-node parameter of the arc (g α + µ ⋆ ) α , for some small negative µ ⋆ . Thus for notational simplicity, the dependence on α = α(µ ⋆ ) of the maps and the associated points and sets will be omitted.
For µ < µ ⋆ < 0 and close to µ, to construct the set Σ µ we consider a Markov partition. There are two cases, according to the position of G k0 (µ).
Proof: The inequalities in item (1) follow from Proposition 4.1, the first one, and equation (11), the second one. To prove item (2), let G k0 (µ) = (1 − v µ , 1). By construction, considering the branch of Γ µ in . Thus, since v µ = |G k0 (µ)|, using the first part of the lemma, one has that ending the proof of the inequalities in the second item of the lemma.
We now define the intervals corresponding to the initial gaps and bridges of the Cantor set Σ µ : • For j ∈ {1, . . . , (k 0 − 1)}, A j (µ) = (a j , b j ) is an interval containing G j (µ) such that Γ µ (a j ) = q µ and Γ µ (b j ) = p µ . More precisely, a j is the biggest point in I j−1 (µ) with Γ µ (a j ) = q µ and b j ∈ I j (µ) is chosen close to the right extreme of G j (µ). A priori, there are infinitely many possibilities for the choice of b j , we will explicit this choice below. The gaps A 0 (µ) and A 1 (µ) are depicted in Figure 13. • For j = 0, . . . , k 0 − 1, B j (µ) = [b j , a j+1 ], where a k0 = q µ and b 0 = p µ . Remark 6.9. By construction, (Φ 0,0 µ ) k (B k (µ)) = B 0 (µ). In [p µ , c µ ] ⊂ I 0 (µ), the map Ψ µ = Γ µ has finitely many discontinuities, say r. Note that the number r increases as p µ approaches to ℓ µ , so r can be taken arbitrarily large after an appropriate choice of p µ . We choose the points b j above such that the restriction of Γ µ to each B j (µ) also has r discontinuity points d 1 j < d 2 j < · · · < d r j , where Γ µ is bi-valuated and takes the values β −1 and 1. Thus the restriction of Γ µ to each B j (µ) has (r + 1) (injective) branches, where To get a Markov partition of Γ µ , we refine the bridges B j (µ) by adding new gaps (corresponding to the discontinuities d i j ), see Figure 13. • For each j ∈ {0, . . . , k 0 − 1} and i ∈ {1, . . . , r}, A i j (µ) is the smallest interval (a i j , b i j ) containing the discontinuity d i j such that Γ µ (a i j ) = q µ and Γ µ (b i j ) = p µ . • The intervals B 0 j (µ), B 1 j (µ), . . . , B r j (µ) are the connected components of B j (µ)\ ∪ r i=1 A i j (µ), ordered in such a way B i j (µ) is at the left of B i+1 j (µ). By construction, Γ µ (B i j (µ)) = [p µ , q µ ]. Note that B i j (µ) = [b i j , a i+1 j ] and that b 0 0 = p µ . The previous constructions and Lemma 6.5 imply the following: Figure 13. Initial gaps and bridges , and Σ µ be the maximal invariant set of Γ µ in B µ . The set Σ µ is a (expanding) dynamically defined Cantor set associated to the Markov partition B µ .
Throughout this section, in our estimates of the thickness, we use repeatedly the next result: Lemma 6.11. (Bounded distortion property, [36, Theorem 1, page 58]) Let Σ be a dynamically defined Cantor set of a C 1+ε -expanding map Γ. Given any δ > 0, there is κ(δ) > 0 such that: for any pair of points x and y and n ≥ 1 such that |Γ n (x) − Γ n (y)| ≤ δ and the interval of extremes Γ i (x) and Γ i (y) is contained in the domain of Γ, for all i = 0, . . . , (n − 1), it holds Moreover, the constant κ(δ) can be taken such that κ(δ) → 0 as δ → 0.
We apply this lemma to the dynamically defined Cantor set Σ µ of the expanding map Γ µ (restricted to the Markov partition B µ ). In our context, it is enough to fix δ = (1 − β −1 ), which is an upper bound for the length of any interval of the partition, and let κ µ = κ µ (δ), the constant given by the lemma. Using the definition of Γ µ , see (15) and (16), and that it is C 2 , one gets where ρ > 1 is as in Lemma 6.5. Thus we can take κ instead of κ µ , so the distortion constant can be taken independent of small µ. This inequality (whose proof we omit here) is obtained adapting straightforwardly the proof of Lemma 6.11 to the C 2 -case (see [36, page 59]).
To estimate the thickness of Σ µ we choose the following presentation A µ of it: The presentation A µ of the set Σ µ : The generation (0, 1) of gaps is {A 1 (µ), A 2 (µ), . . . , A k0 (µ)}. The generation (0, 2) of gaps is defined by . . , A r k0 (µ)}. Inductively, for each i = 1 or 2, we define the (k + 1, i)-generation of gaps as the pre-images by Γ µ of the gaps of the (k, i)-generation, enumerated according to the ordering in R. Moreover, any gap of generation (k + 1, 2) is posterior to any gap of generation (k + 1, 1).
The heuristic principle in our construction is that to estimate the thickness of the Cantor set Σ µ it is enough to calculate the quotients bridge/gap for the first generation of gaps. These gaps are the generating ones: gaps of higher generations are pre-images of them. This principle uses that the maps defining set Σ µ have small distortion. This follows from Remark 6.12 below.
generation (1, 1) generation (1, 2) Figure 14. New generation of gaps Take a gap G of (n, i)-generation and a bridge B of G. Then the number n and any pair of points x ∈ G and y ∈ B verify the hypotheses of Lemma 6.11. Moreover, by construction, Γ n µ (G) is a gap of (0, i)-generation and Γ n µ (B) is a bridge of it. Using Lemma 6.11 one gets the following (which is a standard argument): Remark 6.12. Take τ > 0 such that |B|/|G| > τ for any gap G of generation (0, 1) or (0, 2) and any bridge B of G. Equation (18) implies that the thickness of Σ µ is bigger than τ e −κ .
By Remark 6.12, to get a lower bound for τ (Σ µ ) (i.e. to prove item (a) of Proposition 6.6 in Case 1) it suffices to estimate the ratios between bridges and gaps of generations (0, 1) and (0, 2). This is done in the next lemma, which immediately implies item (a) of Proposition 6.6 in Case 1. Lemma 6.13. There is K 0 , independent of small µ ⋆ < 0, such that, for every gap G of generation (0, 1) or (0, 2) and any bridge B of G, it holds |B|/|G| > K 0 |µ| , for every µ < µ ⋆ close to µ ⋆ .
As in the case of the map Γ µ , we consider two cases according to the position of 1.
Case (a). In this case, we argue as in Case 1 (Section 6.2.2) with the following modifications.
Lemma 6.2 and Proposition 6.6 imply that each Cantor set is not contained in a gap of the other one (recall the conditions on the size of the gaps of Σ µ and on the diameters of these two sets). Thus, by Lemma 7.1, Σ µ ∩ Υ µ = ∅.