Sinai-Ruelle-Bowen measures for piecewise hyperbolic maps with two directions of instability in three-dimensional spaces

A class of piecewise $C^2$ Lozi-like maps in three-dimensional Euclidean spaces is introduced, and the existence of Sinai-Ruelle-Bowen measures is studied, where the dimension of the instability is equal to two. Further, an example with computer simulations is provided to illustrate the theoretical results.


Introduction
The study of the existence of the invariant measure of a map is an interesting problem in dynamical systems. Sinai investigated the C 2 Anosov diffeomorphism on a compact connected Riemannian manifold, and showed that the measure has absolutely continuous conditional measure on unstable manifolds [18]. Bowen and Ruelle obtained similar results for Axiom A attractors [5]. Based on Sinai, Ruelle, and Bowen's work, the invariant Borel measure, which has absolutely continuous conditional measure on unstable manifolds with respect to the Lebesgue measure, is called the Sinai-Ruelle-Bowen measure (SRB measure).
For more information on SRB measure, please refer to Young's work [23].
Later, Pesin developed the non-uniformly hyperbolic theory [1,15]. For singular systems, Katok and Strelcyn investigated the existence of invariant manifolds and obtained some similar results [11]. And, there are lots of work on the billiard systems and so on [7,12]. In [10], Jakobson and Newhouse obtained some sufficient conditions for the existence of SRB measure for piecewise C 2 diffeomorphisms with unbounded derivatives. In [16], Sánchez-Salas provided some sufficient conditions for the existence of SRB measure for transformations with infinitely many hyperbolic branches.
In the research of two-dimensional maps, there are two important types of maps, one is the Hénon map, the other is the Lozi map. A series of work on the Hénon map obtained by Benedicks, Carlson, Young, Viana, and Wang, described the relationship between the 1 Email address: xuzhang08@gmail.com (X. Zhang).
The study of the SRB measure also inspires the study of the statistical properties of dynamical systems. The well-known Lasota-Yorke inequality and some generalization contributed to the development of the chaos theory greatly [6,13]. A powerful tool in the research of the statistical properties is the transfer operator approach [17,21,22].
In our present work, we apply the bounded variation function method [14] and Young's idea in two-dimensional maps [20] to study the existence of SRB measure for a class of three-dimensional maps, which can be thought of as the generalization of the Lozi map in three-dimensional spaces, where the dimension of the instability is equal to two. Our results and discussions can be easily generalized to study the maps with several directions of instability in high-dimensional spaces.
The rest is organized as follows. In Section 2, the main result is introduced. In Section 3, the proof of the main results is provided. In Section 4, an example with computer simulations is given to illustrate the theoretical results.

Main Results
In this section, the main results are introduced and some basic concepts and lemmas are given.
So, for any 1 ≤ k ≤ (p + 1)(q + 1), one has Consider a map f on R such that f (R) ⊂ R and it satisfies the following assumptions: have bounded first and second derivatives, respectively.
The assumption (A1) means that the action of Df when projected onto the x-axis and y-axis is uniformly expanding, respectively.
Remark 2.1. An example satisfying all the above assumptions is given in the last section.
(ii) each element of P n |V n is an open subset of some unstable manifold; (iii) if {µ c : c ∈ P n |V n } denotes the system of some unstable manifold and P n |V n and m c denotes Riemannian measure on c, then for almost every c ∈ P n |V n , one has µ c ≪ m c . Now, we introduce the bounded variation functions [9,14]. For any Ω ⊂ R 2 , the support of any function h = (h 1 , h 2 ) ∈ L 1 (Ω, m) is contained in Ω, where m is the Lebesgue measure.

The existence of SRB measure
In this section, it is to show Theorem 2.1.
It follows from the definition of the map f that the unstable manifold are piecewise smooth surface zigzag across R, which are turning around at unknown places. To avoid the singular set, the strategy is to construct an invariant measure µ with good dynamical behavior on a neighborhood of the singular set S. 1], suppose that (A1) and (A2) hold, it is to show that if the angle between the normal vector of the surface and the z-axis (including both the positive and negative axes) is less than 45 degrees, then the angle between the normal vector of f (graph(α)) and the z-axis is also less than 45 degrees, except points in the image of the singular set.

First, given any
The graph of α is (x, y, α(x, y)). The normal vector is the cross product of the vectors The cosine of the angle between the normal vector and the z-axis is The assumption that the angle between the normal vector and the z-axis is less than 45 degrees is equivalent to ), one has that the tangent vectors are So, the cross product is The absolute value of the cosine of the angle between the normal vector and the z-axis is |C| √ A 2 +B 2 +C 2 . By (A2), (A3), and (3.1), one has that |C| ≥ |A| + |B|, which implies that Hence, the angle between the normal vector and the z-axis is less than 45 degrees.
Let p x : R → [0, 1] and p y : R → [0, 1] be the projection onto the x-axis and y-axis, If N > 1, where N is specified in (A3), we could define the sets L i 1 i 2 ···i k and L i 1 i 2 ···i k j as above for the map f N . Without loss of generality, assume that N = 1 in the following discussions, that is, λ > 2.
In the following discussions, fix L i 1 i 2 ···i k and L i 1 By (A0) and (A1), one has that T is C 2 between B j+ and D j− , where T = (T 1 , T 2 ). Set T −1 := (T −1 1 , T −1 2 ). Next, it is to study the density of the invariant measure on the unstable manifolds. And, it is to show the following lemma. Suppose φ = (φ 1 , φ 2 ) with φ ∈ C 1 0 (R 2 , R 2 ) and φ ∞ ≤ 1. It follows from direct calculation that By direct computation, one has and It follows from (3.3), (A2), and (A3) that there exist δ > 0 and τ > 1 such that Further, set Hence, one has that Thus, by Lemma 2.1, one has that . One has that Hence, for any k, Thus, one has that 1 n n k=1ρ k BV ≤ M.
Hence, it follows from Lemma 2.1 that the sequence {n −1 n k=1ρ k } n∈N is precompact in L 1 ([0, 1] 2 , m). There exists a convergent subsequence, denoted by ρ, the corresponding measure is convergent in the weak star topology, which is a Borel probability measure µ.
This completes the proof.
where D(S, δ) is the δ-neighborhood of the singular set S, M is specified in the proof of Lemma 3.1. It follows from the Borel-Cantelli Lemma that w is in f k D(S, δλ −k ) for at most finitely many k, µ-a.e., that is, for µ almost everywhere w, there is δ(w) > 0 such that f −k w ∈ D(S, δ(w)λ −k ) for all k > 0, implying the existence of local unstable manifold W u δ(w) (w) by [11]. Now, it is to show Theorem 2.1.
Next, it is to define a sequence of measurable partitions P 1 ⊂ P 2 ⊂ P 3 ⊂ · · · . For any Fix a partition P n , it is to define a sequence of measures {μ k } k∈N as follows: since is defined on f k (graph(α)) and f k (graph(α)) is a finite union of smooth surfaces, letμ k be µ k annihilated on those parts of its support that only partially cross someŨ i,j , that is, the support ofμ k consists of all of the sets V 0 satisfying that there is a smooth component Next, it is to show that given any ǫ > 0, there is n = n(ǫ) > 0 such that for the fixed partition P n and sufficiently large k,μ k (R) > 1 − ǫ. For w ∈ (support(µ k ) − support(μ k )), w is either in a small piece, which only partially crosses someŨ i,j , or the distance between w and a cusp in f k (graph(α)) is less than 1/2 n(ǫ) . Hence, one has where M is specified in the proof of Lemma 3.1. The first term is very close to zero as n(ǫ) goes to positive infinite, the second term goes to zero as k goes to positive infinite. For the given ǫ, take a sufficiently large n(ǫ), the corresponding partition P n(ǫ) is denoted byŨ i,j , k=1μ k →μ in the weak topology. It follows from the definitions ofμ k and µ k that one has thatμ ≪ µ and 0 ≤ dμ/dµ ≤ 1. So, by taking n(ǫ) large enough, one has thatμ is equivalent to µ except on a set with the µ-measure less than ǫ.
It is to show that there is a transverse measureμ T for the measureμ such that forμ T -a.e. c ∈ P n(ǫ) , one has thatμ c ≪ m c . Denoteg i 1 ···i k as the density of (p x × p y ) * (μ k |L i 1 ···i k ). For 1 ≤ i, j ≤ 2 n(ǫ) ,Ũ i,j , any i 1 , ..., i k , it is to show that either where M 1 is a positive constant independent on the choice of i 1 , ..., i k . If L i 1 ,,,i k does not cross the fullŨ ij , theng Suppose w 1 = (x 1 , y 1 ) and w 2 = (x 2 , y 2 ). It follows from the assumption that f |(R − S) has bounded first and second derivatives that where C 1 is a positive constant. Further, since ∂T 1 ∂x − ∂T 1 ∂y ≥ λ, ∂T 2 ∂y − ∂T 2 ∂x ≥ λ, (A0), and (3.3), it follows from direct calculation that where C 2 is a positive constant. The last inequality implies that where M 1 is independent on i 1 , ..., i k .
This completes the whole proof.
Remark 3.1. It is easy to obtain similar results for maps with several directions of instability in high-dimensional spaces.
Remark 3.2. Some statistical properties of these maps could be obtained by using the transfer operator methods with some proper function spaces.
Fix k 1 = 2.4, k 2 = 0.08, and k 3 = 0.25. Figures 1 and 2 are the simulation diagrams with different initial values. In Figure 1, the initial value is taken as (0.2, 0.1, 0.5). In Figure   2, the initial value is taken as (0.5, 0.5, 0.5). In Figure 1, the chaotic dynamical behavior is observed. In Figure 2, a "regular" orbit is observed. From these simulations, we guess that there might exist two ergodic components. It is an interesting problem to prove or disprove the existence of at least two ergodic components.