The bifurcation set as a topological invariant for one-dimensional dynamics

For a continuous map on the unit interval or circle, we define the bifurcation set to be the collection of those interval holes whose surviving set is sensitive to arbitrarily small changes of their position. By assuming a global perspective and focusing on the geometric and topological properties of this collection rather than the surviving sets of individual holes, we obtain a novel topological invariant for one-dimensional dynamics. We provide a detailed description of this invariant in the realm of transitive maps and observe that it carries fundamental dynamical information. In particular, for transitive non-minimal piecewise monotone maps, the bifurcation set encodes the topological entropy and strongly depends on the behavior of the critical points.


Introduction
Given some dynamical system on a topological space and an open subset (called hole in the following), it is natural to study the associated surviving set, that is, the collection of all points which never enter this subset. In this framework, the theory of open dynamical systems is, for instance, concerned with escape rates, conditionally invariant measures and other closely related concepts, see for example [1][2][3][4][5][6][7][8] for more information and further references. Recently, there has been an increased interest in understanding families of suitably parametrized interval holes of one-dimensional maps whose surviving sets fulfill certain properties, see for instance [9][10][11][12][13][14][15]. As a matter of fact, this thread of research goes back to the classical work by Urbański [16].
In this spirit, we propose to study the family of all interval holes representing distinct surviving dynamics as a source of topological invariants. To be more precise, for a continuous map f on the interval [0, 1] or the circle T, we consider the bifurcation set B f which is given by all those intervals whose surviving set can change under arbitrarily small perturbations. To get a first impression of the bifurcation set, see Figure 1 below, where an approximation of B f for the doubling map on the circle is depicted.
Before we state our main results, let us introduce some basic definitions. Throughout this work, I refers to [0, 1] (in which case we set ∂I = {0, 1}) or T (in which case ∂I = ∅). If I = [0, 1], a hole is given by an open interval (a, b) with a, b ∈ I \ ∂I. 1 In this case, the collection of holes is naturally parametrized by ∆ := {(a, b) ∈ I × I : a < b, a, b ∈ ∂I}.
This project has received funding from the European Union's Horizon 2020 research and innovation program under the Marie Sklodowska-Curie grant agreement No 750865. Further, MG acknowledges support by the DFG grants JA 1721/2-1 and GR 4899/1-1. Moreover, this project has received funding from the CSIC project SD 618. Finally, the authors would like to thank Henk Bruin, Pablo Guarino, Sebastian van Strien and Björn Winckler for related discussions and many valuable comments. 1 The assumption that a and b avoid the boundary points {0, 1} simply reduces certain technicalities and is not of any further importance. For an explicit study of general continuous maps on [0, 1] with interval holes of the form [0, t) and (t, 1] where t ∈ [0, 1], see [13]. If I = T, then a hole is an open interval of positive orientation from a to b. In this case, the interval holes are naturally parametrized by the set ∆ := {(a, b) ∈ I × I : a = b}.
We denote the diagonal in I × I by ∆ 0 := {(a, a) : a ∈ I}. Observe that ∆ 0 is explicitly not included in ∆. If not stated otherwise, we consider ∆ equipped with the subspace topology of the product topology on I × I.
The geometric structure of B f is constituted by a configuration of vertical and horizontal segments. Let us introduce some notation in order to describe it.
Given a closed subset X ⊆ ∆, we define H(X) to be the family of non-trivial maximal horizontal line segments in X, and V(X) to be the family of non-trivial maximal vertical line segments in X. We define the set of double points D(X) to be the collection of points in X which are in the intersection of an element of H(X) and an element of V(X). The set of corner points C(X) ⊂ D(X) is given by those double points which are endpoints of an element of H(X) and of an element of V(X). Last, given x ∈ H∈H(X) H (x ∈ V ∈V(X) V ) we denote the element of H(X) (V(X)) containing x by H x (V x ).
Double points will play an important part in retrieving dynamical information from the bifurcation set. In particular, this holds for corner points x = (a 1 , a 2 ) ∈ X whose coordinates are links, that is, there is an element in H(X) whose second coordinate coincides with a 1 and an element in V(X) whose first coordinate equals a 2 . We refer to such an x as a step. Given a step x = (a 1 , a 2 ) ∈ X, we call the maximal collection of steps F x = {. . . , (a 1 , a 2 ), (a 2 , a 3 ), . . .} ⊆ C(X), where for each element y ∈ F x there is a finite sequence y = y 1 , . . . , y n = x ∈ F x such that y i shares a link with y i+1 (i = 1, . . . , n − 1), a stair. Note that F x is well defined and uniquely determined by x. Given F x = {(a 1 , a 2 ), . . . , (a p−1 , a p )} is finite and I = [0, 1], we also refer to a 1 and a p as terminal links. The length of a stair is the cardinality of its links. Let us point out that the above terminology originates from the situation described in Theorem A (2): for any step x ∈ D(B f ) the segments H x and V x accumulate at the diagonal, so that the set y∈Fx (H y ∪ V y ) resembles the shape of a stair (see also Figure 1). We can now state the first main assertion which is proven in Section 3.
Theorem A. Assume that f : I → I is continuous, transitive and not minimal. Then B f is closed and the following hold. ( is a sequence of continuous functions on I converging uniformly to f , then every accumulation point of B fn (w.r.t. the Hausdorff metric) is contained in B f . (7) Every stair of length p in B f corresponds to a unique periodic orbit of period p. Furthermore, all but finitely many periodic orbits correspond to a stair.
Observe that point (7) yields the important fact that periodic points and their periods can be identified in the bifurcation set. This in particular implies that the topological entropy for transitive non-minimal piecewise monotone maps can be deduced from the bifurcation set (see Section 2.1 for more details). Notice further that the second part of point (2) implies that the bifurcation set is a collection of horizontal and vertical segments, while the first part of (2) gives -together with point (4)that these segments can essentially be obtained by drawing a horizontal and vertical line from the double points to the diagonal. 2 This observation emphasizes the importance of double points which is even more prominent due to their close relation to nice points introduced in [17] (see also Remark 3.3).
As we will see, natural representatives of double points originate from the periodic points Per(f ) and the preperiodic points of f (see Proposition 3.8 below). It turns out that periodic and preperiodic orbits are of general importance also besides the associated double points. With Theorem B, our second main result, we obtain assumptions which guarantee that already by drawing vertical and horizontal lines from points in the bifurcation set with one periodic and preperiodic coordinate and taking the closure of the respective union of segments in ∆ recovers the bifurcation set.
In order to state Theorem B, we need to introduce some further notation. Let x = (a, b) be a corner point of B f . We say that x is isolated in B f whenever for some neighborhood U of x in ∆ it holds Moreover, we call x isolated from below whenever for some neighborhood U of x in ∆ it holds for every Otherwise we call x accumulated from below. 2 For I = T, this is true for all segments. For I = [0, 1], this is true for all but those lines in ∆ with arbitrarily small first or second coordinate, see also the previous footnote.
The next statement yields the sensitivity of B f on the dynamical behavior of the critical points Cri(f ) of f . We would like to remark that an essential ingredient of its proof are shadowing and stability properties of the surviving sets (see Section 4).
Theorem B. Suppose f : I → I is a continuous, transitive, not minimal and piecewise monotone map. Then the following hold.
(1) If Per(f ) ∩ Cri(f ) = ∅, then every step is isolated from below. Moreover, in case Cri(f ) is empty or contains only transitive points, we have that f is a continuity point 3 of the bifurcation set and that B f can be recovered from periodic and preperiodic points.
then there is at least one step accumulated from below or f is a discontinuity point for the bifurcation set.
Our last statement is an application of the above theorems and existing results concerning the family of restricted tent maps (see Section 5 for the details). The presentation here is a simplified version of Theorem 5.2. 2] be the family of restricted tent maps. Then there exist two disjoint and dense subsets of parameters, denoted by I and J where I has full measure, such that: (1) For s ∈ I every step is isolated from below and s is a continuity point of s → B Ts .
(2) For s ∈ J some step is accumulated from below and s is a discontinuity point of s → B Ts .
We close the introduction noting that although the bifurcation set itself is clearly not a dynamical invariant, we can easily introduce an induced invariant, see Section 2.1. By means of this idea, each topological property of the bifurcation set turns into a topological invariant. This aspect as well as the relation with periodic orbits, topological entropy, and some measure theoretic aspects are further explained in Section 2. From this discussion the following natural question arises: Which dynamical invariants of transitive non-minimal one-dimensional dynamics can be obtained from the bifurcation set?

Interpretation of B f and induced invariants
We start with a formal definition of the bifurcation set. Consider a continuous map f : I → I. The surviving set of f with respect to (a, b) ∈ ∆ is defined as Observe that surviving sets are f -invariant. Further, we define the bifurcation set of f as the following set of parameters, Note that if (a, b) ∈ B f , both a and b may belong to S f (a, b).
We omit the obvious proof of the next statement (which was formulated for transitive maps in Theorem A, already).  The proof of Proposition 2.2 is a consequence of the next lemma. In what follows we set and Proof. Note that for (a, b) as in the assumptions, there is Consider (a , b ) ∈ B ε (a, b) and suppose a ≤ a and b ≤ b (the other cases work similarly).
. This finishes the proof.
Observe that with ε and L as above, we hence have for all (a 0 , b 0 ) and (a 1 , b 1  Recall that given a probability measure µ on I, the (exponential) escape rate of a hole (a, b) with respect to µ is defined as )).
If the above limit does not exist, we may likewise consider the upper and lower escape rate by considering the lim sup and lim inf, respectively. For more information about escape rates and related concepts, see the references at the beginning of the introduction.
Another dynamical characterization of the bifurcation set is the following which is again a consequence of Lemma 2.3. Clearly, this result remains true when we consider non-exponential escape rates, too. 2.1. The bifurcation set as a strict invariant and deduced invariants. In the following, we discuss different dynamical invariants involved with the bifurcation set. First, let us assume that f and g are conjugate, i.e., π • f = g • π where π : I → I is a homeomorphism. Then B g = {(π(a), π(b)) ∈ ∆ : (a, b) ∈ B f } if π is order preserving and B g = {(π(b), π(a)) ∈ ∆ : (a, b) ∈ B f } otherwise. Hence, the bifurcation sets of conjugate maps are homeomorphic via a uniformly continuous self-homeomorphism on ∆. Now, for subsets X, Y ⊆ ∆ we can define an equivalence relation by setting X ∼ Y iff there is a uniformly continuous homeomorphism p : ∆ → ∆ with p(X) = Y . Then the equivalence class [B f ] defines a topological dynamical invariant for f . Furthermore, any topological property of B f , preserved under uniformly continuous homeomorphisms, is a dynamical invariant of f . In this spirit, if we start from [B f ], we can easily index the stairs and their lengths in B f . Accordingly, in the light of point (7) of Theorem A, we can index the periodic orbits (all but finitely many if I = [0, 1]) and their periods by an inspection of [B f ] for transitive maps.
In particular, we can deduce for a transitive non-minimal piecewise monotone map f : I → I that its topological entropy h(f ) can be recovered from B f . For this recall that a continuous map f : I → I is called piecewise monotone if there are finitely many intervals I 1 , . . . , I n in I with I ⊆ n =1 I such that f is monotone on each I . For this kind of maps we have that

see [18, Corollaries 3 and 3'
]. Moreover, in Remark 4.11 we explain that every transitive nonminimal piecewise monotone map is conjugate to a map with constant slope. This in turn implies that each monotone piece of f intersects the diagonal at most one time. Accordingly, we get h(f ) = lim sup n→∞ 1/n log #{x ∈ I : f n (x) = x} (see for example [19, p. 218] for more details) and we obtain the desired property of B f .
Another dynamical invariant visible in B f for a continuous self-map f on I is the group of automorphisms Aut(f ). These are all homeomorphisms π : I → I commuting with f , i.e., f • π = π • f . Each π ∈ Aut(f ) defines a mapπ : ∆ → ∆ mapping (a, b) to (π(a), π(b)) or (π(b), π(a)) depending on whether π is order preserving or reversing, respectively. Accordingly, we get that B f is invariant underπ and this means π represent a certain symmetry of the bifurcation set. For an example of this observation, see Figure 1, where the automorphism π = −Id of the doubling map is visible in the symmetry along the off-diagonal.
Finally, from an ergodic point of view, let us briefly come back to the so-called nice points from [17] which were introduced to study possible ergodic behavior of S-unimodal maps on the interval. In particular, it is known that every S-unimodal map without periodic attractors has the weak-Markov property, which implies the non-existence of positive Lebesgue measure attracting Cantor sets. Nice points are essential for proving this assertion and a simple inspection of their definition shows that they can be derived from the bifurcation set.

Proof of Theorem A
In this section, we study the topology of the bifurcation set in general and for transitive systems in particular. We will obtain Theorem A as a combination of several smaller propositions and lemmas proven in this part.
3.1. General properties of the bifurcation set. This section aims at a first understanding of basic topological properties of the bifurcation set.
For the sake of completeness, let us start by briefly recalling some standard notions from the theory of dynamical systems. For f : I → I and x ∈ I we refer to O(x) := {f n (x) : n ∈ N 0 } as the orbit of x. If O(x) is finite, we call x and likewise its orbit preperiodic. If f n (x) = x for some n ∈ N, then x as well as its orbit are referred to as periodic and we call n a period of x.
We denote the collection of all periodic and transitive points of f by Per(f ) and Tra(f ), respectively. If Tra(f ) = ∅, then we call f transitive and if Tra(f ) = I, we say f is minimal. It is well known and easy to see Observe that the next statement yields point (6) of Theorem A. In the following, we denote by d the standard metric on I and by d ∞ the supremum metric on the space of continuous self-maps on I.
. We may assume without loss of generality that δ < ε.
In the following, we say a set  )). In particular, each non-trivial maximal vertical or horizontal segment in B f accumulates at ∆ 0 .
Proof. For the first part, suppose x = (a, b) ∈ B f and assume without loss of generality that a ∈ S f (a, b). Clearly, a ∈ S f (a, b ) for every b ∈ (a, b] which proves that there is a vertical segment in B f which accumulates at ∆ 0 and contains x. For the second part, we may assume without loss of generality to be given an element V ∈ V(B f ). Denote by π 2 : ∆ → I the canonical projection to the second coordinate. Given (a, b) ∈ V , let us assume for a contradiction that a / , c) and which -as its orbit is dense and thus hits (a, c)-is not in S f (a, c). Therefore, (a, c) / ∈ B f contradicting the assumption that V ⊆ B f . This proves the statement.  Proof. Let (a n , b n ) n∈N be a sequence of points in V ∈V(B f ) V (the case of H∈H(B f ) H works similarly) converging to some (a, b) ∈ ∆. By Proposition 3.2, we know (a n , b n ) is contained in a vertical segment which accumulates at ∆ 0 . Hence, for each b ∈ (a, b] we have a sequence (a Proof. Given (a, b) ∈ B f , we find arbitrarily close (a , b ) such that a and b are transitive points and hence a , b / ∈ S f (a , b ).
Note that transitivity is not necessary in order to have int(B f ) = ∅. For example, on I = [0, 1], we may consider Here, [0, 1/2] and [1/2, 1] are transitive f -invariant subsets and we see, similarly as in the proof of Proposition 3.5, that int(B f ) = ∅.
Recall that the set of non-wandering points of f is defined by We straightforwardly obtain the following converse of Proposition 3.5.
Clearly, N W (f ) = I is not sufficient in order to have int(B f ) = ∅ as can be seen by considering the identity, for example.

3.2.
Transitive case. The statements of the previous section suggest that the additional assumption of transitivity allows for a substantially more detailed description of the bifurcation set. With this observation in mind, we are now taking a closer look at the transitive case.
intersects D(B f ) (note that we may assume without loss of generality that ε 0 < 1/ 2 · d(a, b)). Observe that if D(B f ) intersects one of the vertical sides of this boundary, this gives According to Corollary 3.4, the sets Recall the definition of steps, links and stairs from the introduction. Given a transitive self-map on I, it is easy to see that if {x 1 , x 2 , . . . , x p } is a periodic orbit, then each pair of adjacent points ( ∈ ∂I is a step and all elements in {x 1 , x 2 , . . . , x p } \ ∂I are links. In this way, each periodic orbit with at least two elements not contained in ∂I is naturally associated to a stair in B f . In fact, we have the following Proof. Assume we are given a stair of length p ∈ N ∪ {∞}. By definition, each element (x i , x i+1 ) of the stair is a corner point so that x i , x i+1 ∈ S f (x i , x i+1 ), due to Proposition 3.8. Further, as x i is a link, there is an element of H(B f ) which accumulates at (x i , x i ) ∈ ∆ 0 , so that Proposition 3.2 yields that x i ∈ S f (c, x i ) for some c < x i . Hence, the orbit of x i does not hit the set (c, x i ) ∪ (x i , x i+1 ) and can therefore not accumulate at x i . Likewise, we obtain that the orbit of x i+1 cannot accumulate at x i+1 . However, due to Proposition 3.8, the orbit of x i comes arbitrarily close to x i+1 and the orbit of x i+1 comes arbitrarily close to x i . This clearly yields that x i is an iterate of x i+1 and vice versa. Hence, x i and x i+1 are elements of a periodic orbit. We conclude that all links associated to a stair come from one and the same periodic orbit of period not bigger than p + 2. This proves the statement.
Corollary 3.10. Let f : I → I be continuous and transitive. Then, for all but finitely many p ≥ 2, there is a one-to-one correspondence between periodic orbits of minimal period p and stairs of length p.
Proof. By the above, there is a one-to-one correspondence between stairs and periodic orbits which contain at least two elements within I \ ∂I. Further, unless a given periodic orbit hits ∂I, its period obviously coincides with the length of the associated stair. As there are at most two periodic orbits which hit ∂I, the statement follows.
Remark 3.11. We would like to stress that in case of I = T, it is straightforward to see that the above one-to-one correspondence holds true for all periods p ≥ 2, in fact.
Slightly abusing notation, given a step x, we may also refer to the point-set S x = V x ∪H x ⊆ B f as a step. In a similar fashion, given a stair F x , we may also refer to the union of all maximal vertical and horizontal segments whose first and second coordinate, respectively, coincides with a link of F x as the stair F x . Notice that for I = [0, 1], this union not only includes all respective steps (considered as point-sets) but also the horizontal and vertical segments associated to terminal links. We may refer to these segments as terminal segments of F x . Observe that since each stair is realized by a periodic orbit, the terminal segments accumulate at {0}×I and I×{1}.
By a path in B f , we refer to a continuous map γ : Recall that B f is path-connected if for all x, y ∈ B f there is a path γ in B f from x to y, that is, γ(0) = x and γ(1) = y. In order to prove the path-connectedness of B f , we make use of the following observation whose proof is based on the classical fact that a continuous transitive and non-minimal self-map on I has a dense set of periodic points (for interval maps, see [20] and also [  Proof. We may assume without loss of generality that (a, b) and (a , b ) lie on neighboring steps, that is, (a, b) ∈ S (y 1 ,y 2 ) and (a , b ) ∈ S (y 2 ,y 3 ) for some (y 1 , y 2 ), (y 2 , y 3 ) ∈ F x (note that if (a, b) or (a , b ) lies on a terminal segment, the following proof works exactly the same). As f is transitive, there is a transitive point y ∈ (y 1 , y 2 ). By transitivity of y, there is n ∈ N such that f n (y) ∈ (y 2 , y 3 ). Clearly, for a small enough interval J ⊆ (y 1 , y 2 ) containing y, we have f n (J) ⊆ (y 2 , y 3 ). By denseness of periodic points, there is a periodic point z ∈ J. Let z 1 and z 2 be those points in the orbit of z which are the furthest to the right in O(z) ∩ (y 1 , y 2 ) and the furthest to the left in O(z) ∩ (y 2 , y 3 ), respectively. Clearly, (z 1 , z 2 ) is a step and S (z 1 ,z 2 ) intersects both S (y 1 ,y 2 ) and S (y 2 ,y 3 ) . Let γ 1 be some path in S (y 1 ,y 2 ) from (a, b) to the unique intersection point (c, d) of S (y 1 ,y 2 ) and S (z 1 ,z 2 ) ; let γ 2 be a path in S (z 1 ,z 2 ) from (c, d) to the unique intersection point (c , d ) of S (z 1 ,z 2 ) and S (y 2 ,y 3 ) ; let γ 3 be a path in S (y 2 ,y 3 ) from (c , d ) to (a , b ). Clearly, the concatenation of γ 1 , γ 2 and γ 3 is a path in B f from (a, b) to (a , b ). We next obtain point (5) of Theorem A. Lemma 3.13. If f : I → I is continuous and transitive, then B f is path-connected.
Proof. We first observe that given two points x and y on stairs F x and F y , respectively, there is a path in B f from x to y. To see this, it suffices -due to the previous statement-to show that there is a non-empty intersection between some segment associated to F x and some segment associated to F y . This, however, follows immediately from the fact that on I = T 1 , each stair wraps around ∆ 0 while on I = [0, 1], the horizontal and vertical terminal segment of each stair accumulates at {0} × I and I × {1}, respectively (see Figure 2). Now, suppose we are given arbitrary points x, y ∈ B f . Due to Proposition 3.2, we may assume without loss of generality that x = (a, b) lies on a non-trivial horizontal segment H. Due to the denseness of periodic points, we find a periodic point c ∈ I with c ∈ (a, b). Without loss of generality, we may assume that O(c) contains at least two points in I \ ∂I. Choose c to be the right-most point in O(c) ∩ (a, b). Then, the vertical segment (terminal or not) of the stair associated to O(c) which accumulates at (c , c ) clearly intersects H. Hence, there is a path γ 1 from x to a point z x on a stair F zx in B f . Likewise, we obtain a path γ 2 from y to a point z y on a stair F zy in B f whose inverse (from z y to y) we denote by γ 2 . By the above observation, there is a path γ 3 in B f from z x to z y . Altogether, the concatenation γ 2 · γ 3 · γ 1 is a path in B f from x to y which proves the statement.

Proof of Theorem B
In this section, we turn to the problem of identifying critical points and their dynamical behavior by means of the bifurcation set. For this recall that given a continuous map f : I → I, a point x ∈ I is referred to as critical if there is no neighborhood of x on which f is monotone. The collection of all critical points of f is denoted by Cri(f ).
Let us point out that Theorem B follows from Theorem 4.10 (the main result of this section), see Remark 4.11. 4.1. Implications of hyperbolicity. Besides transitivity, we will impose additional assumptions on the map f . In particular, we will assume certain forms of hyperbolicity. As we are dealing with results of a topological flavor, we consider the following definition of hyperbolicity: an f -invariant set A ⊆ I is referred to as hyperbolic for a continuous map f : I → I and a compatible metric d if there exist ε > 0 and λ > 1 and an open neighborhood U of A such that d(f (x), f (y)) > λ · d(x, y) for all x, y ∈ U with d(x, y) < ε. In this case, we may also say that f is ε-locally λ-expanding on U (with respect to d). Note that a smooth map f : I → I which is hyperbolic on an invariant set A in the classical sense is also hyperbolic in the above sense with respect to some metric d equivalent to the usual one (see for instance the proof of Theorem 2.3 in Chapter III of [24]). Henceforth, all metrics are considered to be equivalent to the standard metric on I and throughout denoted by d.
We call x ∈ I hyperbolic if O(x) is hyperbolic in the above sense. Notions like hyperbolic steps or hyperbolic double points are defined in the natural way. Suppose x ∈ I is a periodic point of f : I → I with minimal period p. We say that f preserves orientation at a ∈ O(x) whenever f p | J preserves orientation in some neighborhood J of a. Otherwise, we say that f reverses orientation at a. Given a, b ∈ O(x), we denote by n a,b the minimum time for going from a to b by iteration of f . We say that f preserves orientation from a to b whenever f n a,b | J preserves orientation in some neighborhood J of a. Otherwise, we say that f reverses orientation from a to b.
Concerning the next statement, recall that due to Proposition 3.9 every step is associated to a periodic point. We may hence refer to the period of this periodic point also as the period of the respective step.  Suppose (a, b) ∈ B f is a hyperbolic step of period p. The following holds.
(1) If f reverses orientation at a or b, then (a, b) is an isolated corner point of B f . f preserves orientation both at a and b, then (a, b) is isolated from below. Proof. We start by proving (1). For a contradiction, suppose there is (a , b ) ∈ B f \ (V (a,b) ∪H (a,b) ) arbitrarily close to (a, b). Without loss of generality, we may assume a ∈ S f (a , b ).
First, consider a = a. Then we necessarily have b ∈ [a, b] c (since otherwise we had (a , b ) ∈ V (a,b) ) and thus (a , b ) (a , b). As the orbit of a = a accumulates at b (see Proposition 3.8), this gives a / ∈ S f (a , b ). Now, consider a = a and assume without loss of generality that a ∈ (a, b) (the other case can be dealt with similarly) . Assuming that (a , b ) is sufficiently close to (a, b), we have f 2p (a ) ∈ (a , b ) since f is expanding in a neighborhood of O(a) (as it is hyperbolic on O(a)) and f 2p is order-preserving in a neighborhood of a. Hence, a / ∈ S f (a , b ). This contradicts the assumptions on a and finishes the proof of the first part.
Let us now turn to part (2). Assume for a contradiction that there is (a , b ) ∈ B f arbitrarily close to (a, b) with [a , b ] ⊆ (a, b). As f p is expanding and order preserving both in a and b, we have f p (a ), f p (b ) ∈ (a , b ) if a and b are sufficiently close to a and b. Hence, a , b / ∈ S f (a , b ) which contradicts the assumptions.
Remark 4.2. Assume the situation of the previous statement. It is not hard to see that if f preserves orientation at a and b and additionally preserves orientation from a to b, then (a, b) is actually isolated. In particular, if f is uniformly expanding, every step is isolated.
Recall that given x ∈ I, its ω-limit set ω f (x) is defined to be the collection of all accumulation points of O(x). It is well known and easy to see that ω f (x) is non-empty, compact and contains recurrent points, that is, there is y ∈ ω f (x) such that y ∈ ω f (y).
We call a double point (a, b) ∈ B f (pre)periodic, if both a and b are (pre)periodic. The proof of the next statement makes use of standard shadowing arguments. Moreover, if (a, b) ∈ B f is a double point which is not preperiodic and the orbits of a and b are hyperbolic, then (a, b) is accumulated by hyperbolic preperiodic double points.
Proof. Let a ∈ S f (a, b) (the other case is similar) and assume a is not preperiodic. Due to the assumptions, there is an open set U (and a compatible metric d) with O(a) ∪ ω f (a) ⊆ U such that f is δ-locally λ-expanding on U . Without loss of generality, we may assume that δ > 0 is such that B δ (x) ⊆ U for all x ∈ O(a) ∪ ω f (a).
Choose some ε < δ/2 and let c ∈ ω f (a) be a recurrent point. Pick n ∈ N with d(f n (c), c) < ε. Since f is δ-locally λ-expanding on U , we may assume without loss of generality that n is large enough to ensure that f n (B ε (c)) ⊇ B 2ε (f n (c)). Choose I to be the connected component of By the assumptions on n, we have f n (I) = B 2ε (f n (c)) ⊇ B ε (c) ⊇ I. Hence, there is a periodic point d ∈ I of period n whose orbit is 2ε-close to ω f (a) (by definition of I). Since f is δ-locally λ-expanding on U , there further is m ∈ N and a point a ∈ I such that f m (a ) = d and max =0,...,m−1 d(f (a ), f (a)) < 2ε. Setã to be the right-most point of O(a ) ∩ B 2ε (a). a, b). Then O(a) is at positive distance to b (otherwise a would not survive) and we may assume ε > 0 to be small enough to ensure that a does not come 2ε-close to b so that O(ã) ∩ (ã, b) = ∅, i.e.,ã ∈ S f (ã, b). As ε can be chosen arbitrarily small, the first part follows.  Proof. According to Proposition 3.1, given a sequence (f n ) n∈N of continuous maps f n : I → I with f n → f uniformly as n → ∞, it suffices to show that for each ε > 0 there is n 0 such that for all n ≥ n 0 we have B ε (B fn ) ⊇ B f . Pick ε > 0. Observe that due to Proposition 2.1, B f is precompact so that there are finitely many (a 1 , b 1 ((a j , b j )). As the elements of Cri(f ) are transitive, we further have that the orbits of the surviving endpoints of (a 1 , b 1 ), . . . , (a M , b M ) are at a positive distance to Cri(f ). By the assumptions, this yields that the orbits of the surviving endpoints are hyperbolic. Hence, by Lemma 4.3, there are (ã 1 ,b 1 and such that at least one of the surviving endpoints of each pair among (ã 1 ,b 1 ), . . . , (ã M ,b M ) is preperiodic and hyperbolic. We denote these surviving endpoints by y 1 , . . . , y N (where M ≤ N ≤ 2M ).
Let p be bigger than max =1,...,N #O(y ) and such that y ( = 1, . . . , N ) is eventually pperiodic. Observe that f p maps the points y 1 , . . . , y N to fixed points of f p . By possibly going over to multiples of p, we may assume without loss of generality that there is δ > 0 such that f p is (2δ)-locally 3-expanding in a neighborhood of N =1 O(y ). We may further assume δ to be small enough such that Now, f p (B δ (y )) ⊇ B 3δ (f p (y )) so that f p k (B δ (y )) ⊇ B 2δ (f p (y )) for = 1, . . . , N and k ≥ n 0 . Similarly, we have f p k (B δ (f p (y ))) ⊇ B 2δ (f p (y )). Altogether, this shows that for all k ≥ n 0 there is x k ∈ B δ (y ) with f p k (x k ) ∈ B δ (f p (y )) and f p k (f p k (x k )) = f p k (x k ). For j = 1, . . . , M , we define (a k j , b k j ) as follows: ifã j = y (for some ) is a surviving endpoint, we set a k j to be the right-most point in . Ifã j is not a surviving endpoint, thenb j is necessarily surviving and we proceed similarly. Note that (a k j , b k j ) ∈ B f k and d(a k j ,ã j ), d(b k j ,b j ) < ε 0 ≤ ε/2 (j = 1 . . . , N ) and hence B f ⊆ j=1 B 3ε ((a k j , b k j )). Since ε > 0 can be chosen arbitrarily small, this proves the desired statement.

4.2.
Critical steps. Throughout this section, we consider continuous self-maps f on I which are piecewise uniformly expanding (for the relation to piecewise monotone maps, see Remark 4.11). Recall that f is referred to as piecewise uniformly expanding, if there are finitely many intervals I 1 , . . . , I n with I ⊆ n =1 I such that f is uniformly expanding on each such interval, that is, there is λ > 1 such that d(f (x), f (y)) > λ · d(x, y) whenever there is with x, y ∈ I . Given these intervals are maximal, the corresponding boundary points which do not lie in ∂I coincide with the critical points of f . where I c = f n a,b (a) = b. In the complementary situation, we have that f either reverses orientation from a to b or we have that for an arbitrary small enough segment J containing a in its interior it holds f n a,b (J) = [c, b] for some c = f n a,b (a) = b in I. In either case we say that f is negative from a to b. The following statement shows that in several situations B f detects the periodicity of critical points explicitly. Proof. We first show that for every ε > 0 we have that there is n ∈ N and two distinct points x, y ∈ [a, a + ε] = I with f n (x), f n (y) ∈ O(a) (note that possibly f n (x) = f n (y) = a). To that end, we may assume without loss of generality that ε > 0 is small enough to guarantee that (a + ε, b ) is a non-empty subinterval of (a, b), where b is the left-most point of (a, b) ∩ (f n a,b (I) ∪ {b}). As f is transitive, there clearly exists a transitive point z ∈ I. In particular, there must be n ≥ 1 such that f n (z) ∈ (a + ε, b ). Note that this necessarily gives {a} ⊆ f n (I) ∩ O(a) or {b} ⊆ f n (I) ∩ O(a). In the first case, if n is not a multiple of the minimal period p of a, we are done since f n (a) obviously lies in O(a) which would hence give two points in O(a). If, however, n is a multiple of p, we must have another point besides a whose n-th image coincides with a as f is assumed to be negative at a. The second case can be dealt with similarly. Now, assume n ∈ N to be minimal with the discussed property and observe that the above argument also gives f j (I) ∩ (a, b) = ∅ for j = 1, . . . , n a,b − 1, n a,b + 1, . . . , n − 1.
By definition of n, we hence have x 0 ∈ I \{a} with f n (x 0 ) ∈ O(a) and such that f j (x 0 ) ∈ I\(a, b) for every j = 1, . . . , n a,b − 1, n a,b + 1, . . . , n − 1, due to (1). Clearly, given δ > 0 we can further guarantee that y 0 = f n a,b (x 0 ) is δ-close to b by choosing the above ε small enough. Then (x 0 , y 0 ) is a double point 2δ-close to (a, b) and below (a, b). Since δ > 0 was arbitrary, this proves the statement. Proof. The "if"-part is given by the previous statement. For the other direction consider a periodic corner point (a, b) ∈ B f accumulated from below. If it is hyperbolic, then it cannot be accumulated from below due to Lemma 4.1. Hence, it must be a critical periodic corner point. For a contradiction, suppose f is negative at a and negative at b (the other cases can be dealt with similarly) and assume there is (a , b ) ∈ B f with a < a < a + δ and b − δ < b < b for arbitrarily small δ > 0. We denote by p the minimal period of a and b. For small enough δ > 0, the negativity at b implies that f p (b ) ∈ (a , b ) and that there is ∈ N such that f n a,b + ·p (a ) ∈ (a , b ) since f is piecewise uniformly expanding. For such δ we have (a, a + δ) × (b − δ, b) ⊆ B c f which finishes the proof.
If I = T, we clearly have that if b is the second coordinate of a step, then it also is the first coordinate of the neighboring step of the associated stair. In this way, we obtain the following statement where the term negative slope region of a piecewise uniformly expanding map refers to a maximal interval in the complement of the critical points on which the map reverses orientation. The straightforward proof is left to the reader.   (a, b) ∈ B f is a critical periodic corner point such that f is positive at a. Then there is a neighborhood U ⊆ ∆ of (a, b) and a sequence of maps (f n ) n∈N converging uniformly to f so that B fn ∩ U = ∅ for every n ∈ N. The same holds true if f is negative at b.
To see this, note that for x ∈ I with y = (f p + t)(x) ∈ J we have y ∈ I + , due to (1). Since f p + t is uniformly expanding on I + (by (2)), the existence of the above m x follows. Now, for big enough n ∈ N, there is an orientation preserving homeomorphism g n with g n = Id on I \ [a − ε 1 − δ, a + ε 2 + δ], g n (x) = x + 1/n on (a − ε 1 , a + δ) and d ∞ (g n , Id) = 1/n. Set f n = g n • f . On the interval I, we have (g n • f ) p = (f p + 1/n), due to (1) and (3). If 1/n < ε 2 , the above observation implies that for every x ∈ I we have m x ∈ N such that Consider now a neighborhood U of (a, b) ∈ ∆ such that for (a , b ) ∈ U we have a ∈ I, (a , b ) ⊃ J, and f n b,a (b ) ∈ I. Then, given (a , b ) ∈ U we have for large enough n that (g n • f ) p·m a (a ) ∈ (a , b ) and (g n • f ) p·mz (z) ∈ (a , b ) where z = f n b,a (b ). Hence, U ⊆ ∆ \ B gn•f for big enough n which proves the statement.
For the next statement, we consider the space of continuous self-maps on I equipped with the supremum metric d ∞ and the space of non-empty closed subsets of ∆ endowed with the Hausdorff metric.  (1) If Per(f ) ∩ Cri(f ) = ∅, then every step is isolated from below. Further, in case Cri(f ) is empty or only consists of transitive points, we get that f is a continuity point of the map g → B g . (2) If Per(f ) ∩ Cri(f ) = ∅, then there is at least one step accumulated from below or f is a discontinuity point of g → B g .

Proof of Theorem C
In order to emphasize the applicability of our results, we now make use of the statements and techniques of the previous sections to describe the dependence of the bifurcation set on the parameter of a particular family of interval maps. The specific family we are interested in is given by the collection of restricted tent maps (T s ) s∈ (1,2] which are defined via    Proof. Observe that for all s ∈ (1, 2] the point x s = 1 − 1/s − 1/s 2 + 1/s 3 verifies x s ∈ (0, c s ), T s (x s ) ∈ (c s , 1) and T 2 s (x s ) = c s . By choosing y s sufficiently close and to the right of x s , we can guarantee that 0 < y s < T 2 s (y s ) < c s < T s (y s ) < T 3 s (y s ) < 1. In particular, T s is order preserving in y s as well as in T 2 s (y s ) and order reversing in T s (y s ) as well as in T 3 s (y s ). Given s ∈ [ √ 2, 2], by Theorem 5.1 there is an arbitrarily close s such that c s is transitive. Observe that there is n ∈ N with 0 < T n s (c s ) < T n+2 s (c s ) < c s < T n+1 s (c s ) < T n+3 s (c s ) < 1 (pick n such that T n s (c s ) is sufficiently close to y s ). Now, Theorem 5.1 allows to pick s such that c s is periodic and such that s is sufficiently close to s to guarantee 0 < T n s (c s ) < T n+2 (c s ). Note that either T s is negative at a , positive at b , negative at c and positive at d or the other way around, that is, T s is positive at a , negative at b , positive at c and negative at d .
In the first case, choose a ∈ O(c s ) ∩ [a , b ] to be such that T s is negative at a and T s is positive at each element of (a, b ] ∩ O(c s ). Choose b to be the smallest element of (a, b ] ∩ O(c s ). Then (a, b) is a periodic corner point with T s negative at a and positive at b. In the second case (when T s is positive at a etc.), we obtain a similar statement by dealing with b instead of a and c instead of b .
As s ∈ [ √ 2, 2] is arbitrary and s can be chosen arbitrarily close to s, the statement follows.