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Critical points in higher dimensions, I: Reverse order of periodic orbit creations in the Lozi family

We introduce a renormalization model which explains how the behavior of a discrete-time continuous dynamical system changes as the dimension of the system varies. The model applies to some two-dimensional systems, including H\'enon and Lozi maps. Here, we focus on the orientation preserving Lozi family, a two-parameter family of continuous piecewise affine maps, and treat the family as a perturbation of the tent family from one to two dimensions. First, we give a new prove that all periodic orbits can be classified by using symbolic dynamics. For each coding, the associated periodic orbit depends on the parameters analytically on the domain of existence. The creation or annihilation of periodic orbits happens when there is a border collision bifurcation. Next, we prove that the bifurcation parameters of some types of periodic orbits form analytic curves in the parameter space. This improves a theorem of Ishii (1997). Finally, we use the model and the analytic curves to prove that, when the Lozi family is arbitrary close to the tent family, the order of periodic orbit creation reverses. This shows that a forcing relation (Guckenheimer 1979 and Collett and Eckmann 1980) on orbit creations breaks down in two dimensions. In fact, the forcing relation does not have a continuation to two dimensions even when the family is arbitrary close to one dimension.


Introduction
Studies show that the dimension of a dynamical system may affect the behavior of the trajectories. One example from continuous-time dynamical systems is the Poincaré-Bendixson theorem [Poi81,Poi82,Ben01]. It implies that there is no chaos in two dimensions, whereas the Lorenz attractor [Lor63] gives an example of a chaotic system in three dimensions. This means that being two dimension forces the system to not have chaotic trajectories.
In discrete-time dynamical systems, there are also examples indicating such constraints relief as the dimension increases from one to two. First, the dimension may affect the number of attracting cycles. Singer [Sin78] showed that a sufficiently smooth map on a compact interval has at most a finite number of periodic attractors. On the contrary, Newhouse and Robinson [New74,New79,Rob83] showed that a smooth map on a topological disc can have infinitely many periodic attractors. Second, for continuous maps on a compact interval, the possible trajectories are restricted by some forcing relations. A forcing relation means that if a map has an orbit of type , then it forces the map to have an orbit of type whenever satisfies a prescribed forcing condition ( ). Sharkovsky [Sha64] introduced a forcing relation on the periods of periodic orbits given by the Sharkovsky ordering. A boundary case of such ordering shows that the existence of a period 3 cycle forces the map to have cycles of any period. In fact, Li and Yorke [LY75] showed that the boundary case imposes the existence of uncountably many points which are not asymptotically periodic. For unimodal maps, the result was sharpened by using symbolic dynamics, which is called the kneading theory [MT88,CE80]. The phase space is partitioned by the critical point, and orbits are encoded by the partition. The encoding of an orbit is called an itinerary. A forcing relation on unimodal maps is described by the itinerary of the critical orbit [Guc79,CE80]. In contrast to one dimension, for smooth maps on a topological disc, it is not hard to show that there exists a Hénon map [Hén76] having only period one, two, and three cycles, and all bounded orbits tend to one of the cycles [Ou21]. Third, the dimension may affect how the maps are classified. In one dimension, it is possible to classify maps by a finite-parameter family [Guc79,MT88]. In two dimensions, a finite-parameter family is not enough to solve the classification problem of a class of maps [HMT17,CP18,BŠ21]. What make one-dimensional systems different from higher dimensional systems are the number of critical (or turning) points. In one dimension, the critical orbits govern the dynamical behavior of an interval map. The number of periodic attractors is finite because the basin of each attractor contains a critical point [Sin78]. Interval maps can be classified by the itineraries of critical orbits [Guc79,CE80,MT88]. However, they are no longer true in two dimensions. This suggests that a map in one dimension has only finitely many critical values, while the number of them grows to infinitely many as the dimension increases from one to two.
In this paper, we introduce a renormalization model to visualize the critical values in two dimensions (Section 2). The idea was first announced by the author in a conference talk [Ou21]. The model applies to systems that mimic an unfolding of a homoclinic tangency. This includes the Lozi [Loz78] and Hénon [Hén76] families. The model gives explanations of aspects involving the change of dimension, e.g., the number of sinks, the classification problem, etc [Ou21]. Here, we apply the model to orientation preserving Lozi maps to explain that the forcing relation for unimodal maps no longer holds in two dimensions.
The Lozi family is a two-parameter family of maps where , ∈ R are the parameters. A Lozi map is orientation preserving (resp. reversing) if > 0 (resp. < 0). It is a generalization of the tent family ( ) = − | | + ( − 1) from one to two dimensions. A Lozi map is degenerate if = 0. When = 0, a degenerate Lozi map is identified with the tent map having the same parameter . The parameter serves as the amount of perturbation that is applied to the tent family. We study how the dynamical behavior changes as we perturb the parameter near = 0. When > 0, a Lozi map has infinitely many critical values { } ∞ =2 , which depend continuously on the parameters. When = 0, all the critical values degenerate into one: 2 = 3 = · · · . Since the Lozi family is a two-parameter family, the critical values { ( , )} ∞ =2 form a system of rank two. In other words, we can fully control two critical values by perturbing the two parameters and . To illustrate the ideas, we use 2 and 3 to study when periodic orbits appear, and prove the following theorem (a reformulation of Theorem 7.1).
1. On , there exists two analytic maps −, , +, : → R 2 such that −, ( , ) and +, ( , ) are periodic points of , with the same period for all ( , ) ∈ . In fact, on the boundary = ( ), the border collision bifurcation occurs and creates the two periodic points.
2. On \ , the periodic points do not have a continuation.
Moreover, the curves 2 and 3 have a unique intersection, and the intersection is transversal. Itinerary ι±,m,2 ι±,m,3 Figure 1: Intersections of ±, , -bifurcation parameters. The positive-slope and negative-slope curves are the ±, ,2 -and ±, ,3 -bifurcation parameters respectively. For each , the curves from left to right are the ones with = 4, 5, · · · , 14 in ascending order. For each , the two curves have a unique intersection.
In summary, the theorem says that the order of bifurcations reverses for any small > 0. For unimodal maps, the forcing relation [Guc79,CE80] implies that the creation of periodic orbits obeys a particular ordering. See Section 3.4 for an explanation. However, in the Lozi family, the ordering is reversed when = . Therefore, the forcing relation does not have a continuation in two dimensions, even when the maps are arbitrary close to one dimension.
To prove the theorem, first we use the fact that all periodic orbits of a Lozi map have an analytic continuation on the parameter space whenever they exist. Milnor [Ish97a,Proposition 3.1] showed that all bounded orbits can be identified with itineraries by using symbolic dynamics. A point is labeled by "−" and "+" according to the sign of the -coordinate. A periodic point with period is encoded by an itinerary with length according to the labeling of successive iterates. For each itinerary , , has at most one -periodic orbit . In addition, Ishii [Ish97a,Section 4] showed that is saddle and depends analytically on the parameters ( , ). Here, we give a new proof (Theorem 3.2) by using the universal stable and unstable cones [Mis80,Kuc21].
Third, we introduce a geometrical criterion to search for the bifurcation parameters (Proposition 4.3). This is a method different from the pruning conditions introduced by Ishii [Ish97a]. The pruning pair in Ishii's paper is defined by the candidates of the stable and unstable manifolds, whereas here the geometrical criterion is prescribed by the forward and backward iterates of the critical locus {( , ); = 0}.
Fourth, when is large enough, we show that the ±, , -bifurcation parameters form an analytic curve = , ( ) near = 0, and there are only creations but no annihilation as the parameter increases (Theorem 6.6). This gives an improvement of a theorem of Ishii [Ish97b, Theorem 1.2(i)] for some types of periodic orbits by using a different approach. Ishii used the pruning conditions to prove that, for all types of periodic orbits and near = 0, there are only creations but no annihilation of periodic orbits as the parameter increases. He did not show that the bifurcation parameters define a continuous curve. Here, we used the geometrical criterion to prove that the ±, , -bifurcation parameters are actually the graph of an analytic curve.
Finally, by using the renormalization model and the parameter curves, we show that, when > 0 is arbitrary small, there exists > 0 such that the two curves ,2 and ,3 have a unique intersection on (0, ). Figure 1 is an illustration of such curves. The proof demonstrates how to control the two critical values 2 and 3 by perturbing the two parameters. An outline is given in Section 2.
Nevertheless, we still can find some patterns of the bifurcation order from the renormalization model. (a) The ±, ,2 -bifurcation parameters. The curves are the bifurcation parameters with = 3, 5, · · · , 14 from left to right. (b) The ±, ,3 -bifurcation parameters. The curves are the bifurcation parameters with = 4, 5, · · · , 14 from left to right. If we fix the value , we showed in the main theorem and Figure 1 that the bifurcation curves ,2 and ,3 intersect. Instead, if we fix the value , we see that the bifurcation curves of { ±, , } ∞ = +1 do not intersect. See Figure 2. This suggests that there might be a generalization of the forcing relation that holds for some types of orbits. For example, Misiurewicz and Štimac [MŠ16] used countable many kneading sequences to describe all possible itineraries of an orientation reversing Lozi map. ( ) sets of a periodic point are unions of connected line segments. We still call them stable and unstable manifolds, even though they are not differentiable manifolds. For ∈ { , }, let 0 ( ) be the line segment of ( ) containing .
Here, we introduce vertical segments and vertical strips to study the geometry of a Lozi map. A line segment is called a vertical segment on an interval if there exists an affine map ℎ : → R such that ∩ (R × ) = {(ℎ ( ), ); ∈ }. The vertical slope of is the slope of ℎ . Suppose that and are disjoint vertical segments on the interval . The vertical strip ( , ) ⊂ R × is the region bounded between and , including the boundaries and .
Proof. The corollary follows immediately from Lemma 2.3.
Proof. The vertical segments ∞ and ∞ exist by Lemma 2.3. We claim that exists and ≤ by induction on ≥ 1, where ( , 0) is the intersection point of and the -axis. The base case follows from Lemma 2.6. Suppose that is a vertical segment and ≤ for some ≥ 1. By Proposition 2.5, +1 = Π − ( ) exists and +1 ≤ 0 ≤ . Consequently, this proves the claim by induction.

1.
< < ∞ for all 1 ≤ < .  Remark 2.10. If the map is orientation reversing and the renormalization model exists, then the order of the turning points will be flipping sides alternatively: Moreover, we define a subpartition { , } 2≤ <∞ on . Let , = − ( ∩ ) for , ≥ 2. See Figure 6 for an illustration.
Proposition 2.11. Suppose that ( , ) ∈ mod and , ≥ 2. If ≤ , then , is the union of two disjoint vertical strips. The vertical strips are bounded between vertical segments which are subsets of ( + ). Let , and , be the left and right components respectively. Proof. The proof is similar to Proposition 2.5. The details are left to the reader.
Thus, the ( + )-fold iterate + : , → , forms a "Lozi-like map" in a microscopic scale. If , ∩ , ≠ ∅, then the orbit of a point in , may have a recurrence in the set. This is called the renormalization defined by two returns to .
An outline of the proof of the main theorem (Theorem 7.1) We consider the two pairs of periodic points −, ,2 , +, ,2 ∈ ,2 and −, ,3 , +, ,3 ∈ ,3 created by the renormalization defined by two returns to . The points depend analytically on the parameters ( , ) whenever they exist (Theorem 3.2). For each pair ( −, , , +, , ), the two periodic points are created when there is a border collision bifurcation [Leo59,NY92]. The bifurcation parameters ( , ) form an analytic curve = , ( ) on the parameter space (Theorem 6.6). The existence of the curve is proved by using a geometrical characterization of the bifurcation given in Proposition 4.3. Our goal is to show that the two curves ,2 and ,3 have a unique intersection and the intersection is transverse.
Moreover, the intersection is unique and transverse because when is small (Corollary 6.7).

Symbolic dynamics and formal periodic orbits
We consider periodic orbits obtained from the two affine branches − and + . They are candidates of the periodic orbits of the Lozi map. An itinerary = ( 1 , · · · , ( ) ) is a sequence of alphabets in with a length ( ) ∈ {1, · · · , ∞}. An itinerary is finite if ( ) < ∞. When and are itineraries and is finite, we write for the concatenation of and , * for itineraries that start with and end with any tail, and ∞ = · · · . When is a finite itinerary, we write Its formal -orbit is called a formal -periodic orbit.
Next, we introduce the notion of admissibility. Let be a point and be an itinerary. Let 1 ( , ) = and When is finite, the -admissibility function is defined as Thus, admissible formal periodic points are periodic points of the Lozi map. We will show that for any given itinerary , there exists a unique formal -periodic point.

Existence and uniqueness of formal periodic orbits
To prove the existence of a formal periodic point, we write ( ) = − , where is a finite itinerary, ≡ D is the linear term of the affine map , and is a vectored-polynomial in and . Then (3.1) can be rewritten as a linear equation If the eigenvalues of are not 1, then (3.3) has a unique solution, which yields the desired periodic point.
To ensure that the spectrum of − is away from 1, we restrict our parameters to full in this section. Finally, by using the fact that the spectrum is away from 1, this gives a new proof of the existence and uniqueness of a formal periodic point. Proof. Let 1 and 2 be the eigenvalues of . Without loss of generality, we assume that | 1 | = ( ). By Lemma 2.1, Corollary 3.1, and det(D ) = for ∈ , we get | 1 | ≥ > 1 and | 2 | ≤ < 1 where = ( ). Thus, (3.3) has a unique solution , and is a saddle fixed point of . The solution is a rational function because the entries of and are polynomials in and .
It follows immediately by the theorem and Theorem B.1 that formal periodic points are the only candidates having periodic itineraries.
Corollary 3.4. Suppose that ( , ) ∈ full and is a finite itinerary. If has a ∞ -admissible point ∈ R 2 , then exactly one of the following holds.
2. If has bounded orbit, then is the formal -periodic point.

Hyperbolicity and the border collision bifurcation
A formal -periodic point is called hyperbolic if it is -admissible and 1 • ( ) ≠ 0 for all 0 ≤ < ( ). It is hyperbolic at a parameter ( , ) if and only if ℎ ( , , ( , )) > 0. Thus, a hyperbolic formal -periodic point persists under a small perturbation of a parameter.

Symbolic dynamics on the renormalization model
In this section, we apply symbolic dynamics to periodic orbits created by the renormalization defined by two returns to . We find the coding of periodic orbits in , . Recall that , , = (+(−) −2 + +(−) −2 ) and , . is the formal , , -periodic point for ∈ and , ≥ 2.

The forcing relation from the kneading theory
For completeness, we give a brief review of the forcing relation on itineraries for unimodal maps. The materials are based on [CE80]. We explain why Theorem 7.1 gives a counterexample to the one-dimensional forcing relation. The remaining part of this paper is independent of this section. Lemma 3.11. Suppose that and are finite itineraries such that = for some ≥ 1. If a unimodal map has an ∞ -admissible point, then it has a -periodic point.
Proof. The set containing all points with the same itinerary is a closed interval. In fact, is monotone on the interval for all ≥ 1. The interval is called a homterval. Let be the homterval consisting points that are ∞ -admissible. The interval is nonempty by the assumption. We have ( ) ⊂ where = ( ). Thus, has a fixed point in , which yields the desired -periodic point.
To apply the kneading theory and take care of the critical point, we extend the symbolic space by letting = {−, 0, +}. A U-itinerary is an infinite sequence of alphabets in . The modified coding : We define a total order on the space of U-itineraries. Let and be distinct U-itineraries. Then the U-itineraries can be expressed as the form = * and = * , where is a finite itinerary and , ∈ such that ≠ or = = 0. For each finite itinerary , let ( ) = (−1) , where is the number of + in . We say that ≺ if ( ) < ( ) . Let ∼ be an equivalence relation such that ∼ if = or = = 0. For each finite itinerary , ( ) is strictly monotone on the cylindrical set { * } of U-itineraries with the orientation ( ). Moreover, the coding map • is orientation preserving, i.e. • ( ) We are ready to state the forcing relation given by the itinerary of the critical orbit.  for all ≥ 0. If a unimodal map has an -periodic point, then it has a -periodic point.
Suppose that there exists ≥ 0 such that ( ∞ ) ≺ (+ ) ∞ , but The case when ( ) = +1 is similar. Thus, Finally, we apply the forcing relation to periodic orbits created by renormalization defined by two returns to . By Lemma 3.5, the itineraries of such orbits are given by , , for ∈ and , ≥ 2. Therefore, Theorem 7.1 shows that the one-dimensional forcing relation cannot be extended to two dimensions.
Remark 3.17. In terms of the renormalization model, the vertical strips ,2 , ,3 , · · · are aligned from left to right whenever they exist. When a Lozi map is degenerate, the vertical strips share the same critical value . Corollary 3.16 is true because the critical value moves from left to right as the parameter increases.

Criteria of admissibility
In this section, we find conditions such that has a periodic orbit with periodic + in , for > ≥ 2. We study the case when intersects , i.e. −1 ≤ . Since < and the map is orientation preserving, we have ≥ ≥ −1 ≥ . Thus, , exists. We fix the value of ≥ 0 and vary . The signs of the quantities − and − −1 divide the parameter space of into three regions. , . Proof. By [KY01,Gal02], the map has two periodic points in , with disjoint orbits. By Proposition 3.5, each satisfies one of the itineraries −, , and +, , . This yields that −, , and +, , are admissible by the uniqueness of formal periodic points (Theorem 3.2).     Small values of . Finally, we consider the case when is small such that < −1 . See Figure 8a for an illustration. Then ∩ = ∅ and hence , , is not admissible. Therefore, the border collision bifurcation happens in the intermediate region.
Proof. The corollary follows from Proposition 3.9.
We give a geometrical criterion that determines when the bifurcation happens in + −1 ( , ) ⊂ .  Figure 9 for an illustration. The points and are similar to the pruning conditions defined by Ishii [Ish97a, Definition 1.1] but not the same. The pruning conditions in [Ish97a] are defined by the candidates of the stable and unstable manifolds using the formal iterates. In Theorem 6.6, we will use the values and to find bifurcation parameters, and prove that the bifurcation parameters form an analytic curve in the parameter space. Suppose that there exist a periodic point ∈ , with period + such that 1 • + −1 ( ) = 0. Then 2 ( ) = 1 • + −1 ( ) = 0 and ∈ , . This implies that = ( , 0). By (4.2), we get = .

The geometry of the Lozi family
In this section, we consider the forward and backward iterates of lines by the branches − and + . We show that the quantities and used in Proposition 4.3 can be estimated by using ( − ) and ( − ).

The forward iterates of a line
We consider lines parameterized by its slope and the intersection point of and ( , ), where ∈ . We iterate by the branches − and + . We derive the corresponding transformations for the slope and the intersection point. By using the transformations, we prove a version of the inclination lemma (Propositions 5.5 and 5.7) for the Lozi maps. The inclination lemma shows that we can use ( − ) to approximate the value of .

The transformation for the slope
where ∈ . We note that − 1 is the stable fixed point of .
Lemma 5.1. Let be a line and be the slope of . If ≠ − , then ( ) is a line with the slope ( ).
The right hand side has an upper bound 1 8 on [0, 1].

The transformation for the intersection point
We now consider the orbit of a point on ( , ) for ∈ .
Proof. Let ( 0 , 0 ) ∈ and be a complex convex neighborhood of ( 0 , 0 ) such that 0 has an analytic continuation on . We may also assume that is small enough such that 0 is bounded and | | < on for some ∈ (0, 1). Then also has an analytic continuation and lim →∞ = 0 uniformly on by Lemma 5.6. Therefore, the conclusion follows from the Weierstrass' convergence theorem [SS10, P.73].

Estimation of , .
We show that the turning point of , can be estimated by a turning point of the unstable manifold ( , − ). Proof. The proposition follows from the chain rule, Lemmas 5.1, 5.2, 5.6, 5.8, and 5.9, and Propositions 5.5 and 5.7.

The backward iterates of a line
We consider lines parameterized by its vertical slope and the intersection point of and ( , ), where ∈ . We take the preimages of by the branches − and + . We derive the corresponding transformations for the slope and the intersection point. By using the transformations, we prove a version of the inclination lemma (Propositions 5.15 and 5.17) for the Lozi maps. We first show that −( + −2) ( ∩ , ) → ( − , − ) uniformly as → ∞. Then we can use ( − ) to estimate the value of .

The transformation for the slope
We note that − is the stable fixed point of .
Lemma 5.13. Let be a line and be the vertical slope of . If ≠ − , then the vertical slope of −1 ( ) is ( ).

The transformation for the intersection point
We now consider the backward orbit of a point on ( , ) for ∈ .

Estimation of , .
Recall that ( , 0) is the intersection point of and the -axis for 1 ≤ ≤ ∞. . We note that , is also the intersection point ofˆ, and the -axis because , exists andˆ, = , ∩ . By Lemmas 5.13, 5.14, 5.16, and 5.18, and are analytic functions on mod , and | | ≤ . Therefore, the corollary follows from Proposition 5.20.

The exponential convergence of the stable manifolds
We proved in Corollary 5.21 that lim →∞ = ∞ locally uniformly. In fact, here we show that the convergence is exponential.
Proof. Let be the slope of and ( + − , −1 + − ) be the intersection point of and ( − , − ). Then To estimate the second term, by the mean value theorem and Lemmas 5.14 and 5.26, we have We note that ∞ . (5.5) To find the lower bound of (5.5), we compute The last inequality follows from Lemma 2.2. The term Proof. The value 1 has a closed form The bounds follow immediately from 1 − 1 = −1 +1 and Lemma 2.1. Proof. Compute ℎ ( , , The last part of the inequality is obtained by Lemma 5.3.
Lemma 5.27. We have Proof. The lemma is clear when = 0. Suppose that > 0. By Lemma 2.2, we have The right hand side has an upper bound 1 3 on ∈ [0, 1].

The parameters exhibiting a full horseshoe
We find parameters when a Lozi map forms a full horseshoe. By Proposition 4.1, these are admissible parameters. Thus, the bifurcation parameters are contained in a compact subset of parameters ( , ) ∈ [1, 4] × [0, 1].
Proof. We have

The parameters of period-doubling renormalizable maps
In this section, we find parameters such that a Lozi map is period-doubling renormalizable. For these parameters, only the phase space 2 has interesting dynamical aspects. In fact, these are non-admissible parameters by Corollary 4.2.
6 The geometry near the tent family In this section, we study the geometry of degenerate maps ( = 0). By continuity, we can then extend some properties to a neighborhood of = 0 in the parameter space.

Intersections of the critical value and the stable manifolds
First, we study how the critical value intersect the stable manifolds in terms of the parameter . When a Lozi map is degenerate, all of the variables ∞ , , 2 , · · · , , 2 , · · · have the same value = − 1, and the variables have a closed form ∞ = 1 and Proposition 6.1. Let = 0. There exists an increasing sequence { } 1≤ ≤∞ with 1 = 1, 2 = √ 2, and ∞ = 2, such that the following holds. 1. = when = for 2 ≤ ≤ ∞.
We note that the map → −1 has a maximal value 1 ln . We get

The parameters of renormalization defined by two returns to
For ∈ (0, 1), let nbd ( ) = mod ∩ {0 ≤ ≤ }. The set nbd ( ) is called a neighborhood of the tent family.

The parameter curves of the boarder collision bifurcation
Ishii [Ish97b, Theorem 1.2(i)] proved that there are only creations of periodic orbits but no annihilation as increases. Here, we improve his theorem for some types of itineraries by showing that the bifurcation parameters are analytic curves.

The reverse order of bifurcation
In this section, we prove our main theorem.
Finally, the intersection of ,2 and ,3 is unique and transverse because (6.5) holds.

B The global dynamics
In this section, we study the global dynamics of the phase space. We prove a theorem similar to [BSV09]. They studied the global dynamics of orientation reversing Lozi maps when one fixed point does not have a homoclinic intersection, and gave an explicit description of the basin of the Misiurewicz's strange attractor [Mis80]. Here we generalize their theorem to the orientation preserving maps. We show that an orbit is either unbounded or eventually trapped inside a trapping region. Our theorem is not restricted to the condition of no homoclinic intersection. On the other hand, we do not study the geometrical structure of the basins to shorten the proof. Denote by ( ) the omega limit set of .