ON PIECEWISE AFFINE INTERVAL MAPS WITH COUNTABLY MANY LAPS

We study a special conjugacy class F of continuous piecewise monotone interval maps: with countably many laps, which are locally eventually onto and have common topological entropy log 9. We show that F contains a piecewise affine map fλ with a constant slope λ if and only if λ ≥ 9. Our result specifies the known fact that for piecewise affine interval leo maps with countably many pieces of monotonicity and a constant slope ±λ, the topological (measure-theoretical) entropy is not determined by λ. We also consider maps from the class F preserving the Lebesgue measure. We show that some of them have a knot point (a point x where Df(x) = D−f(x) = ∞ and D+f(x) = D−f(x) = −∞) in its fixed point 1/2.


Introduction, main results
In their interesting article [5] the authors showed, among other results, that laws ruling piecewise monotone interval maps do not work when we admit countably many pieces of monotonicity. They showed Theorem 1.1. [5] For λ > 2 and every α > log 2 there exists a continuous map T λ : [0, 1] → [0, 1] with the following properties: (i) f has countably many turning points.
(v) For every ergodic T λ -invariant Borel probability measure the partition into the laps of T λ has finite entropy.
That is, for piecewise affine interval leo maps with countably many pieces of monotonicity and a constant slope ±λ, the topological (measuretheoretical) entropy is not given by λ.
In Section 5 of the same work they also gave an example of a continuous (not piecewise monotone) locally eventually onto map satisfying sup x∈ [0,1] |T (x)| ≤ r, sup x∈I |T (x)| < r for some interval I ⊂ [0, 1], but h top (T ) = log r.
In this paper we provide one specific completion of the results cited above (we do not discuss the item (v) of Theorem 1.1). We show, roughly speaking, that all maps under consideration can be taken from one conjugacy class (with common entropy value log 9) containing an "optimal representative", i.e., the map f such that |f (x)| = 9 for all x ∈ (0, 1), except at the turning points of f .
As a by-product of our construction we show that there is an element of our conjugacy class preserving the Lebesgue measure and having a knot point (a point

Definition of class F
In what follows we introduce a conjugacy class F. Later on, this class will be used to demonstrate several interesting features of piecewise monotone maps with countably many laps.
We will call a pair of real increasing sequences Using admissible sequences V, X we define a continuous map by (see Figure 1) The property (c) can be satisfied since for admissible V, X by (a),(b), We denote by F(V, X) the set of all continuous interval maps fulfilling (a)-(d) for a fixed admissible pair V, X. Finally, we put Proof.
Let f ∈ F . For i ≥ 1 denote u 2i−1 , resp. u 2i the unique solution (because of (c)) of the equation An open interval (a, b) with a, b ∈ D and (a, b) ∩ D = ∅ will be called D-basic (for f ).
For two maps f, f ∈ F and sets D(U, V ), X and D( U , V ), X there exists the unique increasing bijection Two basic intervals (a, b) and (π(a), π(b)), resp. two points t and π(t) will be then called corresponding.

F is a conjugacy class
As we have already announced in Introduction, in Theorem 3.3 of this section we prove that any two elements of the class F are topologically conjugated.
One can see that by our construction In what follows, for f ∈ F we will need the complete backward orbit where By J(t) ∈ J we denote the D-basic interval that contains a point t.
The following is true.
(i) If D-basic intervals J, J, resp. K ∈ P(J), K ∈ P( J) are corresponding then K is increasing, resp. decreasing for P(J) if and only if K is increasing, resp. decreasing for P( J). (ii) For every J ∈ J and corresponding J ∈ J , the preimages P(J), P( J) contain corresponding intervals. (iii) For every J ∈ J , the preimage P(J) contains either 0 or 2 nonmonotone D-basic intervals. (iv) For any two corresponding intervals J, J and points t ∈ J, t ∈ J and Proof. The properties (i), (ii) directly follow from the definition of functions f, f and the corresponding intervals.
(iv) Number from the left to the right all intervals Using (iii) assume that ∈ {0, 2} of them are nonmonotone. Then for every t ∈ J we have where each monotone K, resp. nonmonotone K corresponds to one t i ∈ K, resp. to two consequent t i , t i+1 ∈ K with (t i , t i+1 ) ∩ s(X) = {x}. Thus, the coordinate i 1 ∈ {1, 2, . . . , k + } of t(i 0 , i 1 ) uniquely determines an interval K from P(J) with t(i 0 , i 1 ) ∈ K and, if K is nonmonotone, also the position of t(i 0 , i 1 ) with respect to {x} = s(X)∩ K. Using this fact repeatedly for f, t, resp. f , t, we get the conclusion.
Let us show (v). We will proceed by induction. Arguing as in (iv) we can see that the conclusion is correct when m = 0 (or by the symmetry, n = 0), since then u 0 ∈ s(X) and u 0 ∈ s( X) is corresponding.
Let m, n ≥ 1 and m ≥ n, consider points Assume that for some k ∈ {1, 2, . . . , n} the equality If K is monotone for P(J(u m−k )) and also for P(J(v n−k )) then by (i)-(iii) the corresponding K has the same type of monotony for P( J( u m−k )) and P( J( v n−k )), hence we get sgn( If K is nonmonotone for P(J(u m−k )), resp. P(J(v n−k )) (K can be nonmonotone for one of them only), the last coordinates i m−k+1 , j n−k+1 determine the effective connected components of f −1 (J(u m−k )) ∩ K (containing u m−k+1 ), resp. f −1 (J(v n−k )) ∩ K (containing v n−k+1 ) and for f the order of effective component is the same. Thus, Since we have shown at the beginning of this part that a k ∈ {1, 2, . . . , n} satisfying (7) has to exist, our proof is finished.

Maps with constant slope in F
In this section we will check if the class F contains piecewise affine maps with constant slopes, i.e., to a given λ > 1 a map f = f λ such that |f λ (x)| = λ for all x ∈ (0, 1), except at the turning points of f λ (clearly, if it exists, then it is unique). The reader can easily verify by standard computation that such a map f λ would be uniquely determined by a Proof. By our construction we are interested in increasing solutions {w n = (v 2n−1 , v 2n )} n≥1 of (8) (w n+1 > w n for each n ∈ N) and such that lim n→∞ w n = (1/2, 1/2). Denote by A(λ) the matrix from the equation (8). Direct computations show that: • An increasing solution of (8) that converges to the (1/2, 1/2) exists if and only if λ ≥ 9 and it is if and only if all eigenvalues of A(λ) are real positive. • For λ = 9 the matrix A(9) has the unique eigenvalue 1/4 of multiplicity two; the solution of (8) is then given by the explicit as one can easily check substituting (9) into (8).
Since the topological entropy is a conjugacy invariant (see [8]) and Theorem 3.3 holds true, we can speak about entropy value h top (F) of the class F. We know that f 9 ∈ F and f 9 is 9-Lipschitz hence by [3, Theorem 3.2.9], h top (F) ≤ log 9. In order to show that it equals to log 9 we will use standard tools developed for interval maps.
Let f : [0, 1] → [0, 1] be a continuous interval map, and Q = {q 1 < q 2 < · · · < q n } be a finite subset of [0, 1] (Q need not be f -invariant). The matrix of Q (with respect to f ) is the (n − 1) × (n − 1) matrix A Q , indexed by Q-basic intervals and defined by A JK is the largest non-negative integer l such that there are l subintervals J 1 , . . . , J l of J with pairwise disjoint interiors such that f (J i ) = K, i = 1, 2, . . . , l.
An interval map f : The following lemma is needed in the proof of Theorem 4.4. One of well-known results from one-dimensional dynamics is the following.  Let us formally denote A n = {a 0 , a 1 , . . . , a n }. Using two admissible sequences V, X (introduced in Section 2) corresponding to the map f 9 , put for k ∈ N where as before, s(Y ) denotes the symmetric extension Y ∪ (1 − Y ) of Y . Let G k be "connect-the-dots" map given by the pair (Q k , g k ). One can use a similar way as in Proposition 2.1 to show that G k is locally eventually onto hence also transitive. Since the set Q k is G k -invariant and G k is affine on each Q k -basic interval, Lemma 4.2 applies. From that lemma we get the topological entropy is lower semicontinuous on the space of all continuous interval maps equipped with the supremum norm (see [4]) and f 9 = lim k G k . This fact together with (10) imply Let α k be a map guaranteed by Theorem 4.3, i.e., a piecewise affine map from [0, 1] into itself which has slope ±e htop(G k ) on each affine piece. From above we know that lim k e h top (G k ) = e h top (f 9 ) = β ≤ 9; by our definitions of α k s and the class F, where f β is piecewise affine which has slope ±β. Using Proposition 4.1 we get β = 9, i.e., h top (f 9 ) = log 9 = h top (F).  There is a map f ∈ F and a union of two intervals I ⊂ (0, 1) such that if f (x) exists then either |f λ (x)| = 9 when x / ∈ I or |f λ (x)| = 23/7 for x ∈ I.

Proof.
To the map f 9 correspond the sequences V and U introduced in Section 2. In particular, from (8),(9) we get u 3 = 3/10 and v 3 where r is affine, preserves orientation and maps the unit interval onto [1/4, 3/4] -see Figure 3. The reader can verify by direct computations that f and I = r −1 (s([u 3 , v 3 ])) satisfy the conclusion.

Maps preserving the Lebesgue measure in F
The class F contains also maps preserving the Lebesgue measure. One possible way how to see it follows from Figure 4. It shows a piecewise affine map with countably many laps which is uniquely determined by a sequence {k i } i≥1 of reals from the interval (2, ∞). One can easily verify that such a map preserves the Lebesgue measure if and only if We recall that by a knot point of a function f we mean a point x where D + f (x) = D − f (x) = ∞ and D + f (x) = D − f (x) = −∞. It was discussed elsewhere [1] that for the problem of understanding of relationship of two characteristics of an interval (or tree) map -its topological entropy and cardinalities of level sets -it could be useful to understand the role of knot points of Lebesgue measure preserving maps, for example, to evaluate the topological entropy of such a map having a knot point at its fixed point. The best estimate is not clear at all, but using elements of the class F we obtain the following.