Markov partition in the attractor of Lozi maps

In this paper, we study iterations of two-dimensional maps, in particular iterations of Lozi maps in the region of the parameter space where it has a strange attractor. Using symbolic dynamics techniques for two-dimensional maps, based on the kneading theory of Milnor and Thurston and also in the symbolic dynamic formalism developed by Sousa Ramos, through the kneading sequence for the Lozi maps, we characterize the region in the parameter space that contains the kneading curves and present a method to define a Markov partition for the Lozi attractors. Consequently, the topological entropy for the Lozi map is computed.


Introduction
Consider the following parametrized family of Lozi maps: ( 1 ) These piecewise linear plane invertible maps were introduced in 1978 by René Lozi [5] as a simplified version of the parametrized family of Hénon maps, without loosing the capacity to show chaotic behaviour.Since then, Lozi maps has been an important tool to understand complex behaviour of iterated maps on the plane.Since it is an invertible map, it is possible to define its inverse, L −1 ab , given by In 1992 [6] Liu et al. showed that for parameters (a, b) satisfying every map L ab has indeed a strange attractor.Let L denote the corresponding region of the parameter space; see Figure 1.From this point the Lozi maps considered will have parameter values a and b belonging to L .For simplification purposes, L ab will be denoted by L.
Earlier, Michał Misiurewicz [7] introduced a region ∇ L in R 2 satisfying allowing to present the Lozi map attractor, A L , from its successive forward iteration, Considering the fixed point that lies in the first quadrant and computing its local stability by evaluating the eigenvalues of the jacobian matrix of the map at the fixed point, in the domain of the parameters for a > |b| + 1, the fixed point is a saddle point.Let L s (A) and L u (A) be the stable and unstable manifolds of A, respectively.

Definition 1.1:
The Lozi map fundamental domain ∇ L is the triangle with vertices I, L(I), and L 2 (I), where is the intersection point of the unstable manifold of the fixed point A, L u A , with the horizontal axis.
Next, some definitions of Lozi maps symbolic dynamics are presented, following [4].
where the symbols δ n , for n ∈ Z, are defined as (7) where X x corresponds to the x component of X.
By the previous definition, it(X) is a bi-infinite symbolic sequence of {−1, * , +1} Z .It is important to distinguish the itinerary for the past and the itinerary for the future of a point

Critical set
Let V be the vertical axis, such that where V +1 is the nonnegative part of the vertical axis and V −1 is the negative part of the vertical axis.
According to the construction process suggested by Ishii [4] and Baptista et al. [3], some points belonging to A L ∩ Vplay a very important role in the study of Lozi's maps dynamics, especially the point L −1 (I).The orbit associated with the forward itinerary of the point I is the orbit whose itinerary, for the future, corresponds to the maximum symbolic sequence within the set of all admissible symbolic sequences for the future.Definition 2.1: Given a Lozi map L, its kneading sequence K(L) is the itinerary for the future of I, i.e.
Given a Lozi map L, its pruned kneading sequence, k n (L), is the initial subsequence of symbols of K(L) such that the first symbol appears at position n, that is, L n−1 (I) x = 0.
From the inverse map L −1 , two maps can be introduced, L −1 +1 and L −1 −1 , as follows: and Using these two maps and a pruned kneading sequence, k n (L), we will define a certain type of line segments, in ∇ L , as follows: After the definition of the line segments and the points L(I) and I.
The following proposition presents the slope of each line segment be the pruned kneading sequence of L with dimension n.Consider the jacobian matrix of L −1 at the point X ∈ R, defined as follows: where X y corresponds to the y component of X.
Let v = (0, 1) be the vector of the line segment V δ n−2 .The vector of a line segment Depending on the value of δ n−2 , the point L n−1 (I) will belong to that passes through the point (x i , y i ) = L i (I) has the vector given by and slope given by The intersection of the line segment with the x-axis is the point (ξ i , 0) where

Proof:
The line segment S * (when i = n − 1) is the vertical line, and its intersection with the x-axis occurs at the point Therefore, the line segment that passes through the points (0, −1) and L n−2 (I) has slope m n−2 and can be given by y = m n−2 x − 1.Since this line segment contains the segment S δ n−2 * , the intersection of the line segment S δ n−2 * with the x-axis occurs at the point (ξ n−2 , 0) = ( 1 m n−2 , 0).Continuing this process, the pre-image of the point (ξ n−2 , 0) is the point (0, ξ n−2 − 1).Therefore, the line segment S δ n−3 δ n−2 * is contained in a line segment that passes through the point (0, ξ n−2 − 1) with slope m n−3 and intersects the x-axis at the point (ξ n−3 , 0) = ( 1 m n−3 (1 − ξ n−2 ), 0) .In general, the pre-image of a point (ξ , 0) is the point (0, ξ − 1) and consequently the line segment S δ i •••δ n−2 * with 2 ≤ i ≤ n − 2, for n > 3, intersects the x-axis at the point (ξ i , 0), where Considering Propositions 2.5 and 2.6, the next result follows.
Proposition 2.7: for n > 3, and the vertical line segment S * do not intersect.

Order relation in the set {−1, * , +1} N
From [4], the attractor points with the same fixed itinerary for the past ε u lie in a segment L u , called the unstable leaf, completely characterized by the parameter values (a, b) and ε u .Moreover, attractor points with the same fixed itinerary for the future ε s lie in a segment L s , called the stable leaf, also completely characterized by (a, b) and ε s .If a point X ∈ A L has a fixed itinerary ε, X is the intersection of the line segments previously defined, denoted by L u (X) and L s (X), respectively.In order to rewrite the order relation established by Ishii [4] for the set {−1, * , +1} N , consider the following definition.Definition 2.8: Let ε s and δ s be two symbolic subsequences for the future.ε s < s δ s if one of the following conditions is satisfied: (1) ε 0 < δ 0 ; (2) where the order on the symbols is −1 < * < +1.
From the previous definition it can be establish that, for any point X ∈ ∇ L , Considering the set {σ i (δ s )} n−1 i=0 , where σ is the shift-operator, the set {ξ i } n−2 i=2 ∪ {I x , L(I) x } and denoting by p as a permutation in the set {2, . . ., n − 1} such that it follows that and where A B means that A is on the left side of B, and consequently Figure 2 illustrates the critical set, the attractor and the invariant triangle ∇ L , considering the pruned kneading sequence δ s = +1 − 1 + 1 + 1 * .

Markov partition on A L
Using the elements of the critical set C δ s (L), where δ s = k n (L), it is possible to construct a partition, with n − 1 regions, for the set ∇ L and therefore to A L .Considering the Definition 2.8 and the order relations given in (9), ( 10) and (11) we define the n − 1 regions on the triangle ∇ L as follows: let p be a permutation in the set {2, . . ., n − 1} such that The n − 1 regions on the triangle ∇ L are defined as follows: . . .
where A (x, y) means that (x, y) is on the right side of A or (x, y) belongs to A and x, y A means that (x, y) is on the left side of A.
Considering the previous construction the next result follows: Theorem 2.9: Let X belong to the interior of a set R i .Suppose that where R j is single region or a union of two or more regions of the topological partition P. Assume, without loss of generality, that it is a single region.Thus, Moreover, by Ishii [4], the unstable leaf and the stable leaf of X lie in a segment denoted by L u (X) and L s (X), respectively, and X is the intersection point of the segments previously defined.Therefore, and due to the Lozi map linearity, we can state that Thus, by Bowen [2], it follows: Theorem 2.10: The topological partition {R 1 , . . ., R n−1 } provides a Markov partition of ∇ L associated to δ s = k n (L).The transition matrix over ∇ L is a square matrix A δ s , of order n − 1, such that

Transition matrix and topological entropy
Following [8] and [1], the topological entropy of L can be calculated using the corresponding transition matrix.This result can be stated as follows: The itinerary given by the pruned kneading sequence δ for (a, b) ≈ (1.518, 0.185728) is given in Figure 3. Applying the shift map to the pruned kneading sequence δ s , it follows: By Definition 2.8, the following order relation is valid and consequently The partition of ∇ L is given through the regions (Figure 4) which can be associated to the following transition matrix A δ s given by Applying the shift map to δ s that verifies the following order relationship: The partition of ∇ L is given through the regions (Figure 5)  δ) x, y , which can be associated to the following transition matrix A δ s : 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 1 1 1 1 1 1 0 0 0 0 The transition matrix A δ s has a maximum eigenvalue λ max ≈ 1.9363 and the topological entropy is approximately h top = log (1.9363) ≈ 0.66078.

Isentropics curves for Lozi maps
From [3], a pruned kneading sequence corresponds to an algebraic condition on the parameters and, therefore, to a curve on the parameter plane.For example, a simple computation shows that the kneading curve for the pruned kneading sequence δ s = +1 − 1 Following the work [3], and considering Proposition 3.2, the topological entropy of L along the curve (δ s ) is equal to log(a δ s ), where (a δ s , 0) is the point of the horizontal axis given by the limit, as b → 0, of (δ s ) and therefore 0 ≤ h top (L) ≤ log (2) .Figure 6 illustrates the kneading curves (isentropic curves) for pruned kneading sequence up to dimension 7.

Figure 1 .
Figure 1.The parameter space region for which L ab has a strange attractor.
and (a, b) ∈ L .Let A δ s be the transition matrix associated to δ s .The topological entropy of L is given byh top (L) = log λ max A δ s ,where λ max (A δ s ) is the spectral radius of A δ s .