Directional complexity and entropy for lift mappings

We introduce and study the notion of a directional complexity and entropy for maps of degree 1 on the circle. For piecewise affine Markov maps we use symbolic dynamics to relate this complexity to the symbolic complexity. We apply a combinatorial machinery to obtain exact formulas for the directional entropy, to find the maximal directional entropy, and to show that it equals the topological entropy of the map. Keywords: Rotation interval, Space-time window, Directional complexity, Directional entropy;


Introduction
There are problems where the behavior of orbits in a compact phase space is determined by the dynamics of a system generated by the lift map on the universal cover. The classical one is a classification of rotation of the circle by using the Poincaré rotation number. Such problems possess space-temporal features, where the universal cover serves as "the space". One can study average speed of diffusion, transport properties, etc, and relate them to the dynamics on the original compact phase space.
This approach was profitable enough in the past and, as seems to us, is far from being completely exhausted. Our article follows this way. We shall exploit notion of the ǫ-separability introduced by Kolmogorov and Tikhomirov [KT] in the context of [AZ]. A notion of space-time window introduced in [M86, M88] for cellular automata and used in [ACFM, AMU, CK] for lattice dynamical systems we apply here for maps on R 1 that are lifts for maps of the circle of degree 1. If such a map generates the dynamical system with non-zero topological entropy then, very often, it has a rotation interval different from a single point. It implies the existence of trajectories with different rotation numbers, i.e. with different spatio-temporal features. We suggest here to measure the amount of trajectories with a given rotation number using the notion of a directional entropy. Roughly speaking if X is a subset of a the circle such that the trajectories going through X have the rotation number, say, cot θ, then the (ǫ, n)-complexity of X behaves asymptotically (n >> 1) as exp(nH θ ). We call the number H θ the directional entropy in the direction θ. The greater H θ the greater the rate of instability manifested by trajectories with the rotation number cot θ. But one has to be careful. It can happen (and occurs for mixing systems) that for any fixed rotation number cot θ inside the rotation interval the set of initial points, say X θ , corresponding to this rotation number is dense in the circle. So, the topological entropy on X θ coincide with the topological entropy of the whole system. To avoid it we approximate X θ by sets of initial points which trajectories stay in a space-time window, calculate the entropy on this window, and obtain H θ as the limit of these entropies.
In this article we study mainly piecewise affine Markov maps of the circle. For such maps it is possible to replace the calculation of the (ǫ, n)-complexity by that of the symbolic complexity of some subsets of a corresponding topological Markov chain (TMC). The TMC is determined by the Markov partition of the circle and the subsets -by the admissibility condition formulated according to the value of the rotation number. After that the problem becomes purely combinatorial. We use the approach of [PW1,PW2] adjusted for our situation to obtain the explicit formulas for H θ . The formulas depend only on the entries of the transition matrix of the TMC and on the weights of the edges of the corresponding oriented graph, where the weights are determined by the Markov partition and the lift map.
The article is organized as follows. In Section 1 we give the definitions of the directional complexity and the directional entropy H θ . In Section 2 we show that H θ = 0 only if cot θ belongs to the rotation interval. In Section 3 we define piecewise affine Markov maps and show how to calculate the (ǫ, n)-complexity in terms of symbolic dynamics. Section 4 is devoted to the description of the combinatorial machinery. In Section 5 we describe a specific example where all can be explicitly seen. In Section 6 we construct some invariant probabilistic measures for which measure theoretical entropies coincide with the directional entropies. By using this we show that the topological entropy coincides with a directional entropy for some specific direction. We present a formula for this direction. Section 7 contains some concluding remarks.

Definitions
Let f : S 1 → S 1 , S 1 = {x mod 1} be a continuous mapping of degree one, i.e. there is a lift mapping F : R 1 → R 1 of the form where h is 1-periodic function such that be the "window" in R × R + . Denote by e the vector (cos θ, sin θ).
Definition 1. [AZ] 1) Two points x, y ∈ R are (ǫ, W, T )-separated if (F n x, n) ∪ (F n y, n) ⊂ W for each n ≤ T , and there exists 0 ≤ n ≤ T such that |F n x − F n y| ≥ ǫ.
3) The number is called the directional ǫ-complexity in the direction e with respect to the window W .
Here, card X is the cardinality (the number of points) of X.
4) The number is called the directional entropy in the direction e with respect to the interval [l 1 , l 2 ]. The limit is called the directional entropy in the direction e.
Roughly speaking, C ǫ and H θ are quantities reflecting the number of orbits "moving" with the velocity cot θ along the circle. Indeed, to be in the window W , the point (F n x, n) must satisfy the inequality l 1 + n cot θ ≤ F n x ≤ l 2 + n cot θ, thus the "velocity" F n x n is approximately cot θ if n >> 1. 5) Given a window W , an (ǫ, W, T )-separated set X is optimal if card X = C ǫ (W, T ).

Rotation intervals and directional entropy
The ratio F n x n is not only the velocity but also is related to the rotation number of the orbit going through the point x.
Definition 2. [NPT], [I]. The set i. e., the set of all points of accumulation for all initial points x ∈ [0, 1] (the upper topological limit), is called the rotation interval of f.
It is known ( [I], [NPT], [BMPT]) that the rotation interval is a closed interval and for every Proof. Denote by a (b) the left (right) endpoiont of the segment I. It is known (see [ALM]) that there are functions F 1 , F 2 : R → R such that: i) F 1,2 are weakly monotone, i.e. the inequality x < y implies F 1,2 (x) ≤ F 1,2 (y); ii) there exist limits for any x ∈ R; iii) for any The properties i) and ii) imply that for every x ∈ R and n ∈ N. Assume now that H θ > 0 and cot θ > b. It means that there exists ǫ > 0 and l 1 < l 2 such that H θ (l 1 , l 2 ) > b + ǫ. Therefore there exists x ∈ R such that the inequalities (2) hold for each n ∈ N. The inequalities (2) and (3) imply that Taking the limit as n → ∞ we obtain a contradiction. In the same way we prove that H θ cannot be positive if cot θ < a.

Piecewise affine Markov maps
In this section we consider arbitrary piecewise affine Markov maps on the circle. For that, we represent S 1 as {x, mod 1, x ∈ R} or as the interval [0, 1] with the identified endpoints. Let D = {d 0 = 0, d 1 , . . . , d p = 1}, d i < d i+1 , i = 0, . . . , p − 1, be an ordered collection of points on S 1 . We introduce the following class of maps f : S 1 → S 1 : (i) f is a continuous map of degree 1, Remark that the condition (ii) says that the points D determine a Markov partition for f on S 1 , and the condition (iv) claims that f is expanding on each element of this partition. Let us emphasize that this class of maps is interesting and large enough: first of all, Markov maps are dense in the space of expanding maps endowed with the topology of uniform convergence, and second, any Markov expanding map is semi-conjugated to a piecewise affine Markov map (is conjugated in the transitive case), see, for instance, [ALM].
Given f of this class, let us choose the lifting map F : R → R such that F (0) ∈ [0, 1], F (1) ∈ [1, 2]. Since f is of degree 1, such a lift always exists.
Because of the condition (iv), the diameter of an element ξ (n) goes to 0 as n → ∞, so one may find out n 0 , such that diam F (ξ n 0 j ) < 1 for every element ξ (n 0 ) j ∈ ξ n 0 and treat ξ (n) as the original partition ξ. Because of that, one may see that, As usual, we identify the elements ξ i with the symbols i, consider the p × p-matrix A = (a ij ), so, the shift map σ : Ω A → Ω A , (σω) k = ω k+1 , k ∈ Z + , will be continuous. The coding map consists of the only one point.

Estimates from above
We introduce an oriented graph Γ A having p vertices such that there exists an edge starting at the vertex i and ending at j iff a ij = 1. By L * Γ we denote all Γ-admissible finite words (paths: (ω 0 , ω 1 , . . . , ω n−1 ) ∈ L * Γ iff (ω j−1 , ω j ) is a Γ-edge for all j = 1, . . . , n − 1). As the graph Γ is normally fixed we sometimes omit the subscript Γ. We relate a weight k ij ∈ Z to every edge (i j) of the graph Γ A as follows: . . , s 0 + ρ}, s 0 ≤ 0. Now we want to estimate C ǫ (W, T ) through the cardinality of different sets of words generated by Γ A . Let us start with some notation and definitions. For a finite word w = w 0 . . . w n−1 ∈ L * Γ we denote: • |w| = n, the length of the sequence.
, the weight of w.
• L n = {w ∈ L * Γ | |w| = n}, the collection of all admissible words of length n.
• L n m = {w ∈ L n | v(w) = m}, the collections of admissible n-words of the weight m.
Proof. In fact, the statement directly follows from the definition of k ij . Indeed, if 0 ≤ x ≤ 1 then F x ∈ [k w 0 w 1 , k w 0 w 1 + 1] and so on.
The following proposition is an easy implication of the definition of B n,α,r .
Let P be a an (ǫ, W, n)-separated optimal set.
Theorem 2. The following estimate holds ..w n−1 is an interval of the length less than 1, the number of points of P inside

An estimate from below
Let m ∈ N. The set {∆ w | w ∈ L m } is a partition of [0, 1] by intervals. Let ǫ m be the minimal length of the intervals ∆ w , w ∈ L m .
Proof. Let S ⊂ L m satisfy the following property: Fix a maximal S satisfying this property. One can check that x ∈ ∆ w and y ∈ ∆ v are (ǫ, W, km)- Proof. Let lim n→∞ ln C ǫ (W (θ, [−r, r]), n) n = H θ (ǫ, r).
Theorem 2 and Theorem 3 together say that for ǫ ≤ ǫ m . Taking ln( km √ ·) from all parts of the above inequality and directing k → ∞ one gets The smaller ǫ is the larger m can be taken (ǫ ≤ ǫ m → 0 when m → ∞). So, Finally we obtain the formula (6) Remark 1. We believe that formula (6) can be obtained by using the technique developed by M.
Misiurewicz (see, for instance [ALM]). But, since we deal generally with non-invariant sets, this technique should be adjusted to the "non-invariant situation". So, we decided to make a direct proof here.

Combinatorial part.
Let e α = log lim n→∞ n |L n αn |. The aim of this subsection is to show that e α = lim r→∞ e α,r = H θ (α = cot θ) and to explain how to calculate e α .
Let D ⊂ L * be finite subset. Let the matrix M(D) ∈ Mat p×p (N) be such that M(D) ij is the number of words in D starting from i and ending by j. Given X, Y ⊂ L * and B The following proposition is a direct corollary of the above definitions.
Recall that L n is the set of admissible words related to matrix A. It is known that M(L n ) = A n−1 , see, for instance, [AH]. Let us represent the matrix A in the form  So, Proposition 3 implies the statement.
The following proposition is a consequence of definition of B n,α,r and L n m . Proposition 5.
Proof. The words of the set B ct,α,r are the words such that the weights of their initial subwords are in the [-r,r]-strip with slope α. In the words from l.h.s. of the equation (8)  where m j = ⌊jtα⌋ − ⌊(j − 1)tα⌋. Moreover, m j = ⌊tα⌋ or m j = ⌊tα⌋ + 1.
For a positive sequence a n we call lim n→∞ n √ a n the exponent of a n (if exists). The relation between exponents of D n and M(D n ) is clear: lim n |D n | = max ij {lim n m ij (n)}, where m ij are matrix entries of M(D n ). Using this fact and estimates of Proposition 7 one gets e α,r ≤ lim ǫ→0 sup{e β | β ∈ [α − ǫ, α + ǫ]}. So, the following lemma holds.
The estimates from below may be more tricky to obtain. We overcome this difficulty by imposing a rather general sufficient condition. In the next subsection we explain how to calculate M(L n ⌊αn⌋ ).

Generating function.
Let S = {s 0 , s 0 + 1, ..., s 0 + ρ}. We define the matrix generating function for M(L n m ) as We chose this type of generating function to avoid negative powers and to keep track of the number of total transitions. Proof. Taking into account the formula (E − X) −1 = E + X + X 2 . . . it suffices to show that We prove it by induction on n. For n = 0 the equality holds by the statement i) of Proposition 4. Supposing the equality for n − 1 we obtain In the last equality we use the simple fact that L n m = ∅ for m < (n − 1)s 0 and m > (n − 1)(s 0 + ρ).

The induction step follows because of Proposition 4.
Let H(x, y) = det(E − x ρ j=0 y j A s 0 +j ). It follows from the formula of an inverse matrix that HG is a polynomial matrix. In order to calculate the asymptotics we need to study the zeros of H, particularly, we need the so called minimal solutions, see [PW1,PW2,PW3].
The following proposition describes the minimal solutions for H(x, y) = 0 (11) Proposition 8. Let A be a primitive matrix. Let (x 0 , y 0 ) ∈ C 2 , y 0 = 0 be a minimal solution of the equation (11). Then the maximal (by the absolute value) eigenvalue of the matrix A(x 0 , y 0 ) = x 0 j y j 0 A s 0 +j is 1. Moreover, if rank of (A(1, e iφ )) > 1 for all φ ∈ R then (x 0 , y 0 ) ∈ R 2 + and (x 0 , y 0 ) is strictly minimal.
Proposition 9. Let b(A) be the greatest real eigenvalue of a matrix A. Let A be primitive and Av > v for some v > 0. Then b(A) > 1.
Proof. There exists n such that all entries of A n are positive. Observe that if u > v then

Asymptotics for 2-variable generating functions.
In this section we suppose that A is primitive and the rank condition of Proposition 8 is satisfied. All entries of G(x, y) have the form f (x,y) H(x,y) , where f is a polynomial. We are interesting in asymptotics of a n,⌊αn⌋ where a n,m are the coefficients of the expansion f (x, y) H(x, y) = a n,m x n y m We estimate a n,m using the Wilson-Pemantle technique [PW1,PW2]. The asymptotics depend on minimal points. Under the conditions of Proposition 8 all minimal points are strictly minimal and we may adapt Theorem 3.1 of [PW1] (see also [PW3,PW2]) as follows Theorem 4. Let (x 0 , y 0 ) ∈ R 2 + be the unique (in R 2 + ) solution of such that 1 is a maximal eigenvalue of A(x 0 , y 0 ). Then (x 0 , y 0 ) is a strictly minimal solution of the equation (11) and the following asymptotics takes place: Particularly, it implies that lim n→∞ ln(a n,⌊αn⌋ ) n = −ln(x 0 ) − α ln(y 0 ), if f (x 0 , y 0 ) = 0 and Q(x 0 , y 0 ) = 0.

Important example
In this section we consider an example that, in fact, contains all main features of systems on the circle possessing a Markov partition.
Consider the map f for which The map f has the Markov partition ξ of 3 intervals: ξ 1 = 0, 1 3 , ξ 2 = 1 3 , 2 3 , ξ 3 = 2 3 , 1 (see Fig. 5), and the corresponding topological Markov chain is determined by the transition matrix  One can see that the transition (3, 1) corresponds to the change of the integer part of F . So, we represent the transition matrix A = A 0 + A 1 where A 0 corresponds to all transitions without (31) and A 1 corresponds to (31): We calculate the generating function: Now we can find the asymptotics using Theorem 4. Let H = −x 3 y − x 2 y − x + 1 We have to find positive solutions of the system H = 0 αxH x = yH y Using SAGE (see [SA]) we have found: x = α ± √ 5 α 2 − 4 α + 1 (2 α − 1) .
In this example α is a fraction of (31)-transition (A 1 -transition). If α > 1/2 then 2 consecutive A 1 transitions should appear. But there is no word with consecutive (31)-transition. So, we have to consider the interval 0 < α ≤ 1/2 only. The positive branch for 0 < α < 1/2 is The dependence of the entropy on α is given by the formula h = − ln(x) − α ln(y) shown on the figure 3. One can see that our case satisfies Theorem 5, so, H(θ) = h(θ).

Measures and entropy
In this section we construct a measure on the subshift generated by the matrix A such that it's measure theoretical entropy coincides with H θ . This measure turns out to be a Markov measure constructed using the matrix A(x 0 , y 0 ) = x 0 j y j 0 A s 0 +j with (x 0 , y 0 ) ∈ R 2 being the solution of system (12) satisfying the condition of Theorem 4. First of all, we construct a stochastic matrix Π and measure µ Π , as it is described, for example, in [KH]. Then we show by direct computation that µ Π -entropy (h(µ Π )) of the subshift coincides with the directional entropy H θ , where α + s 0 = cot θ and α is the parameter of the system (12).

Construction of the measure.
Recall, that under the conditions of Theorem 4 the matrix A(x 0 , y 0 ) has 1 as the greatest simple eigenvalue. Let l be a row-vector (r be a column-vector) such that lA(x 0 , y 0 ) = l (A(x 0 , y 0 )r = r). By the Perron-Frobenius theorem l and r are positive. Normalize l and r such that lr = 1. Let A(x 0 , y 0 ) = {a jk }. Define (see [KH]) the matrix Π = Π(x 0 , y 0 ) as Π jk = a jk r k r j . Let q j = l j r j and q = q 1 , q 2 , . . . , q p . Observe that Π is a stochastic matrix and q is its left 1-eigenvector. The measure µ Π of the cylinder [w 1 , w 2 , w 3 , ..., w n ] is defined as µ Π ([w 1 , w 2 , w 3 , ..., w n ]) = q w 1 Π w 1 w 2 Π w 2 w 3 . . . Π w n−1 wn .
The entropy of the subshift with respect to µ Π can be calculated by the formula see [KH].
We are going to show that h(µ Π ) = ln(x 0 ) + α ln(y 0 ). In our situation the equation (13) can be rewritten as −h(µ Π ) = ik l i a ik r k ln( a ik r k r i ) = ik l i a ik r k ln(a ik )+ ik l i a ik r k ln(r k ) − ik l i a ik r k ln(r i ).
Observe that the last line of the equation is 0. (Indeed, evaluating the first sum over i and the second one over k and taking into account that l (r) is a left (right) 1-eigenvector of A we obtain that k l k r k ln(r k ) − i l i r i ln(r i ) = 0.) Let A j = {(i, k) | (A s 0 +j ) ik = 1}. Now we can write: whereÃ(x 0 , y 0 ) = y 0 A y (x 0 , y 0 ) = j jx 0 y j 0 A s 0 +j . So, in order to prove the equality H θ = h(µ Π ) we should show that (lÃ(x 0 , y 0 )r) = α, of course, under the condition that lA(x 0 , y 0 ) = l, A(x 0 , y 0 )r = r, lr = 1, (x 0 , y 0 ) is the solution of the system (12) satisfying the condition of Theorem 4. To this end we need the following result (recall that H(x, y) = det(E − A(x, y))). Observe that γ = trace(B). Let D = diag(−1, 1, −1, 1 . . . , (−1) p ). The matrix D −1B D is the matrix of the minors of B. By a theorem due to Kronecker (see [Gan]) the eigenvalues of D −1B D (as well as ofB) are products of p − 1 eigenvalues of B. So, trace(B) = β, the unique non-zero eigenvalue ofB.
Remark 2. The direct computation shows that Ω A v(w[: 1])dµ Π (w) = α + s 0 (the function v(·) is defined in Section3.1). With shift invariance of µ Π it probably implies that the support of µ Π consist of initial words with rotation number cot θ.

When H θ = h top .
Theorem 6 implies that H θ = h top if µ Π is the measure of the maximal entropy. Observe that A(x, 1) = xA. So, our construction of µ Π in the case of y 0 = 1, in fact, coincides with the construction of the measure of maximal entropy in [KH]. Substituting y 0 = 1 to the system (12), we can find α = cot θ − s 0 and x 0 . It is clear that, in fact, x 0 = e −htop , the inverse value of the greatest eigenvalue of A since A(x 0 , 1) = x 0 A. We can formulate the procedure of finding the angle, corresponding the topological entropy in the form of the following Theorem 7. Let λ be the greatest eigenvalue of A; l (r) be its left (right) λ-eigenvector. Let cot θ = lA(1, 1)r lÃ(1, 1)r + s 0 , π 2 < θ < π 2 .
Then H θ = h top .

Concluding remarks
Following ideas of Milnor [M86, M88] and also [AZ, ACFM, AMU, CK] we have introduced and studied the directional complexity and entropy for dynamical systems generated by degree one maps of the circle. In particular, we have considered the maps that admit a Markov partition and have positive topological entropy. For them we have reduced the calculation of the (ǫ, n)complexity on a set of initial points having a prescribed rotation number to that of symbolic complexity of admissible cylinders of a topological Markov chain (TMC). The admissibility of the cylinders is constructively determined by the rotation number. To calculate the symbolic complexity we have used a combinatorial machinery developed in [PW1,PW2] adjusted to our situation. As a result we have obtained exact formulas for the directional entropy corresponding to every rotation number. Using these formulas we have shown that the directional entropy coincides with the measure-theoretic entropy related to a Markov measure (different for different direction). In particular, we have proved that the measure of maximal entropy determines the direction in which the directional entropy equals the topological entropy of the original dynamical system and, also, we have found an exact formula for this direction. Acknowledgement. V.A. and L.G. were partially supported by PROMEP, UASLP CA-21.