1 Introduction

The symbolic dynamics plays an important role in the study of nonlinear dynamical systems. For the piecewise monotone interval maps,the kneading theory established by Milnor and Thurston^[1] is an effective tool to describe their dynamical behavior. The kneading sequences, which are topological invariant,determine all admissible itineraries of the systems. There are various applications of the kneading theory in mechanical,physical,and chemical systems^{[2, 3, 4]}.

As an analogue to the kneading sequences in the one-dimensional case,Cvitanović^[5] and Cvitanović et al.^[6] introduced the concept of pruning front for a class of horseshoe type maps on the plane such as the Hénon^[7] and Lozi maps^[8]. They regarded the strange attractor of a map of this type as an incomplete horseshoe and conjectured that all its allowed symbol sequences can be completely determined by its pruning front and primary pruned region,there are no sequences pruned out by other mechanisms,and all other forbidden regions of the symbol space are obtained by backward and forward iterations of the primary pruned region.

Unlike the one-dimensional case,there is no single “maximal point” for two-dimensional maps,and each sheet defines a locally largest allowed itinerary. The pruning front can be computed by determining the primary homoclinic tangencies. (Primary tangencies are those tangencies which lie closest to the x-axis.) The connecting line of the pre-images of all primary tangencies gives a suitable partition of the phase space. Thus,for each point of the attractor or some invariant set,there is a corresponding two-sided symbol sequence. A key step is to give an order on the symbol plane in terms of the dynamics nature of the local stable and unstable manifolds. Using these facts,one can obtain the relation between the phase space and symbol space. However,for the Hénon map,the pruning front conjecture is proved only in certain parameter regions^{[9, 10]},and most of the literatures focus mainly on the numerical investigation^{[11, 12, 13]}. One of the main difficulties is that it is unable to formulate the positions of all homoclinic tangencies. Fortunately,for the Lozi map and the map considered in this paper, primary tangencies lie on the x-axis. The study is thus comparatively easier. Ishii^{[14, 15]} proved the pruning front conjecture for the Lozi map and used that to study the dynamics of the map.

Non-smooth dynamical systems often come from the mechanical systems with the impact^{[16, 17, 18, 19, 20, 21]},the dry friction^{[22, 23, 24, 25, 26]},and so on. Non-smoothness is an important origin for the complexity of these systems. In the non-smooth case,even those piecewise linear systems can appear very complex dynamics behavior. The Belykh map (a two-dimensional discontinuous piecewise linear map) is a well-known example.

The Belykh map was introduced by Belykh^[27] as a simple model in the phase synchronization theory. In this paper,we rewrite the Belykh map in a convenient form as

where

Under certain conditions,the map f has a strange attractor as shown in Fig. 1. Tél^[28] calculated the Box dimension of the strange attractor of a similar map. The statistical properties of the attractors of a class of piecewise hyperbolic maps including the Belykh-type map,such as existence of Sinai-Ruelle-Bowen measures,exponential decay of correlations,and the central limit theorem were studied in Refs. [29]-[32].

The main aim of this paper is to prove the pruning front conjecture for the map f on an invariant set consisting of points with bounded orbits under a hyperbolicity condition. Furthermore,we construct a topological dynamics model and solve the boundary of horseshoes for f.

2 Hyperbolic cone field

In this section,we construct a hyperbolic cone field for the map f. We only construct the unstable cones. The construction of the stable ones is similar.

Suppose that a > b + 1. For all points that do not lie on the y-axis,let

where

We now determine the smallest constant α such that . Substituting and ,we obtain

The inequality (1) holds if Since |v| ≤ α|u|,the inequality (2) holds if

Thus,the smallest α is . Besides,we need to show that Df expands the vectors in the unstable cone. Since |v| ≤ α|u| and = u,we have |v| ≤ α||. By || ≤ α||,and = u,we obtain |u| ≤ α||.

Similar calculations show that there exists the largest such that if || ≥ β||,then |v| ≥ β|u|,|u| ≥ β||,and |v| ≥ β||. Note that α < 1 < β when a > b + 1.

For the map f,its Jacobian matrix is independent of points. Thus,when a > b + 1,if a point whose orbit does not lie on the y-axis,then its stable and unstable subspaces are the eigenspaces of stable and unstable eigenvalues,respectively. This is the reason why in Theorem 2,the slopes of the stable and unstable leaves are constants.

3 Map from symbol plane to K

In this paper,we focus on the symbolic dynamics of the map f on an invariant set K consisting of all points whose forward and backward orbits are bounded. Similar to the Lozi map,we regard f : K → K as an incomplete horseshoe and its symbol model as a subshift on two-sided symbol space endowed with the usual metric. The key step is to determine which sequence of the symbol space is disallowed. We use the ingredients developed by Ishii in Ref. [14] for the Lozi map to construct the pruning front and primary pruned region of f : K → K.

First,we show that the invariant K is contained in a square region.

Proposition 1 Suppose that a > b + 1,and let that R = 1/(a − b − 1). The invariant set K is contained in

Proof We define a partition of the plane by

First,it is easy to check that when a > b + 1,we have f(M₁) ⊂ M₃,f(M₃) ⊂ M₁,f ⁻¹(M₂) ⊂ M₄,and f ⁻¹(M₄) ⊂ M₂ (see Fig. 2).

We denote the image of the point (x₀,y₀) by (x₁,y₁) and use negative subscripts for preimages.

We will show any point outside the region M₀ whose orbit is unbounded. For any point (x₀,y₀) ∈ M₁,we have

For any point (x₀,y₀) ∈ M₂,we have

By the symmetry of the map f,we have |x₁| > |x₀| for any point (x₀,y₀) ∈ M₃ and |y₋₁| > |y₀| for any point (x₀,y₀) ∈ M₄. This completes the proof of the proposition.

For a point X = (x,y) ∈ K,we put its itinerary as

where

Here,Y_x denotes the x-component of point _Y .

We construct a map from the symbol space to the phase space which is the inverse of the projection map π on π(K). Note that the map f is equivalent to

Let us consider the Banach space f_∞ of all bounded sequences endowed with the supremum norm. For each sequence of two-sided symbol space ,we define the operator

where x _n denotes the nth-coordinate of and

Theorem 1 Suppose that a > b + 1. Then,

(i) for any sequence ,there exists a unique fixed point of ,for all satisfies

(ii) satisfies (3),if and only if for all ,

Proof Take any two different sequences . We have

When a > b + 1,the operator is contractive on the space f_∞. Therefore,by the contraction map principle,for each sequence has a unique fixed point The statement (ii) of Theorem 1 is obtained by comparing (3) and (4).

Write

which is the set of admissible symbol sequences for the map f : K → K.

Lemma 1 Let . The map is onto and gives the inverse of the map π on π(K).

Proof For each X ∈ K,let (x_n,x_n−1) = fⁿ(X),then,the sequence {x_n} ∈ f_∞ satisfies (3). Take any symbol sequence ,by the definition of itinerary,we know ε_nx_n ≥ 0. Thus,{xn} satisfies (4) and is the fixed point of ,therefore,the map is onto. Besides,in the above proof,the choice of the itinerary of π(X),X ∈ K is arbitrary,and the fixed point of the operator is unique,therefore,for each X ∈ K,we have .

4 Stable and unstable leaves

For a symbol sequence · · · ε₋₂ε₋₁ · ε₀ε₁ · · · of ,we call = · · · ε₋₂ε₋₁ and =ε₀ε₁ · · · the tail and head of ,respectively. We will show in this section that the map maps all sequences with a fixed tail (resp. head) into a line which we will denote by (resp. ),and its intersection with the x-axis is the point . Moreover, (resp. ) is expressed by a power series in terms of a,b,and (resp. ). We will use the series p() and q() to construct the pruning front and primary pruned region for the map f on K in the following section.

Lemma 2 The map is continuous on the symbol space .

Proof Take two sequences such that ε_n = ε'_n for all |n| ≤ N +1. For any initial sequence ,we have . As N becomes bigger, and will be arbitrarily close. Therefore,the map is continuous.

Let C^u (resp. C^s) be the set of all tail (resp. head) of the elements in and define two breaches of f as

Theorem 2 Suppose that a > b + 1. maps symbol sequences with the same tail (resp. head ) into a line whose slope and intersection with the x-axis are expressed by power series expansions using a,b,and the tail (resp. head) only.

Proof We see that the following diagram commutes:

The diameter of the cylinder set · · · ε₋₃ε₋₂ε₋₁ · C^s converges to zero under σ⁻ⁿ (n > 0). By the uniform continuity of ,the diameter of set converges to zero by the following iterations:

Since there exists a hyperbolic cone field for the map f (see Section 2),we see that the tangent vectors of are contained in

where X is a point in ,and T_Y denotes the tangent space at the point Y . Consequently,the set is contained in a line.

Next,let us determine the slope of the line and its intersection with the x-axis. Take two sequences with the same tail Let and be the fixed points of and ,respectively. From (4),the slope of the line containing and is

We consider the following continued fraction:

We have

when a > b + 1,since the map

is contractive on the interval

if a² > 4b and the ratio (x_−n −x' _−n)/(x_−n+1 −x' _−n+1) is in this interval as we saw above. Thus, we have proved that maps every ε with a fixed tail · · · ε₋₂ε₋₁· on a line for some p.

Finally,we determine the value of p. As above,we have (x_n − x' _n)/(x_n+1 − x' _n+1) = s for all ,and the point (x_n,x_n−1) lies on a line of the form: y = s(x − p_n−1) for some p_n−1. Substituting x_n−1 = s(x_n − p_n−1) and x_n = s(x_n+1 − p_n) into (4),we obtain the recursion p_n = ε_n − bsp_n−1. Thus,

Similarly,we see that maps sequences with a fixed head C^u · ε₀ε₁ε₂ · · · into a line : ry = x − q,where

and

Remark 1 When a > b + 1,the series s,r,p,and q uniformly converge,and they are the continuous functions of ε,a,and b.

Remark 2 If an orbit of a point X ∈ K does not lie on the y-axis (in this case,the itinerary π(X) is just one sequence ),then the line contains the local stable manifold W_loc^s(x) and local unstable mainfold W_loc ^u (x) of the point X (see Fig. 3). We remark that these lines can also be defined for symbol sequences which are not itineraries of points in the invariant K.

Definition 1 For a symbol sequence ,we call set an unstable (resp. stable) leaf.

5 Proof of pruning front conjecture for f : K → K

In this section,we first give the definition of the pruning front and the primary pruned region of f on K and interpret their meanings. Then,we prove the pruning front conjecture.

Definition 2 We call

and

the pruning front and the primary pruned regions of f : K → K,respectively.

For a , and . It is clear that the intersection of and is . By Theorem 2,we know that the slope of is negative, and that of is positive. Thus,ε₋₁(p− q) ≥ 0 if and only if . This gives another characterization of the pruning front and the primary pruned region of f : K → K,i.e.,

Lemma 3 For each ,we have ,where σ is the shift operator.

Proof It is clear that

The claim comes from this and the fact that every initial sequence of f_∞ converges to a fixed point of under the iteration of it.

Now,we begin to prove the pruning front conjecture for f : K → K.

Theorem 3 Suppose that a > b + 1. Then,for a sequence ,there exists a point X ∈ K such that if and only if does not lie in D_f for all n ∈ .

Proof By Theorem 1,we know that a symbol sequence is admissible if and only if for all . Using the other characterization of the pruning front and the primary pruned region of f : K → K above and Lemma 3,we obtain

Thus,all allowed symbol sequences of f : K → K are completely determined by P_f and D_f.

6 Partial order in symbol plane

Now,we explain the relation between the above definition of the pruning front and the primary pruned regions of f : K → K and the original idea of Cvitanović et al.^[6]. Represent as the direct sum of C^u and C^s,i.e.,. Recall that C^u (resp. C^s) is the set of all tails (resp. heads). First,we define a partial order in the symbol plane (see Fig. 4).

Definition 3

(i) Take two distinct tails and in C^u. We can find the biggest integer i < 0 such that

(ii) Take two distinct heads and in C^s. We can find the smallest integer i ≥ 0 such that

The following proposition shows that the map π preserves the order defined above.

Proposition 2 Suppose that a > b+1. Then,the map π is order preserving in the following sense: take two points X and Y on an unstable (resp. stable) leaf,and suppose that X_x < Y_x (resp. X_y < Y_y). Thus,for any ,we have .

Proof It suffices to show that if we take two points X and Y in an unstable leaf (resp. stable leaf) and suppose X_x < Y_x (resp. X_y < Y_y),then we have f(X)_x > f(Y )_x (resp. f(X)_y < f(Y )_y). This is true that,by Remark 2,for each point in the set K,its stable leaf (resp. unstable leaf) expresses the contracting (resp. expanding) direction at it,and the eigenvalues of the Jacobian matrix of f are and .

Remark 3 Under the partial order of the symbol plane defined above,the series p and q are the monotonically increasing function of tails and heads,respectively.

For a sequence ,if ,then the stable and unstable leaves determined by are “tangent” on the x-axis (here,the “tangency” means that one endpoint of the unstable leaf contacts with the stable leaf,as shown in Fig. 5). There exist two case: ε₋₁ = +1 or ε ₋₁ = −1. We only discuss the first case. The discussion of the second case is similar. The partial order of the symbol plane tells us that p (resp. q) is a monotonically increasing function of (resp. ). By this and Remark 1,for any sequence which satisfies or ,and ,the operator cannot determine a point in K. Thus,any sequence satisfying is disallowed (see Fig. 6).

7 Symbolic dynamics model

In this section,we construct a symbolic dynamics model for f : K → K. First,we define an equivalence relation in the set A_f.

Definition 4 For two symbol sequences ,we say that ε is equivalent to δ if .

Let be the quotient space of A_f with respect to the above equivalence relation. For any ,we put and ,where [ε] denotes the equivalent class of ε.

The following theorem says that there is a symbol model for the dynamics of the map f on K described by (P_f ,D_f) only,and the model is a subshift on the quotient space of A_f.

Theorem 4 is a homeomorphism from to K such that the following commutative diagram holds:

Proof First,we note that the symbol space is compact and that is open by the definition of the primary region D_f. Therefore, is a compact metric space,this also holds for its quotient space by the standard theory of topology. Lemmas 1 and 2 show that is a continuous surjection,and this also holds for . Moreover, is one-to-one. Therefore, is a continuous bijection from the compact metric space to the Hausdorff space K,therefore,it is a homeomorphism. The commutative diagram follows from Lemma 3.

8 Boundary of horseshoes

In this section,we solve the boundary of the parameter region where the map f : K → K has a horseshoe. Suppose that a > b + 1. Let H := {(a,b) : f : K → K is equivalent to the shift map of the two-sided symbol space}. Since the system is symmetrical,we only need to consider the case ε₋₁ > 0. Recall that and are monotonically increasing functions of and ,respectively. The smallest value of p is which is associated to the tail · · · − − − +·. The biggest value of q is which is associated to the head · + − + − · · ·. Parameter values (a,b) lie on the boundary of the region H (see Fig. 3) if and only if

Substitute (6) and (9) into (10). By a simple calculation,we obtain