A Context free language associated with interval maps

For every interval map with ﬁnitely many periodic points of periods 1 and 2, we associate a word by taking the periods of these points from left to right. It is natural to ask which words arise in this manner. In this paper we give two different characterizations of the language obtained in this way.


Introduction
A topological dynamical system is a pair (X, f ), where f is a continuous self map on the topological space X.An element x ∈ X is called periodic if f n (x)= x for some positive integer n and the least such n is called the f −period of x.Here we consider dynamical systems on the interval [0, 1], which are generally called interval maps.For an interval map f with finitely many periodic points, say x 1 < x 2 < .... < x n we define w(f ) = p 1 p 2 ...p n , where p i is the f -period of x i .A known theorem of Šarkovskii [6] implies that each p i has to be a power of 2. A known fixed point theorem guarantees that one of the p i 's has to be 1 and infact, by a corollary of intermediate value theorem, it has to be in between the first 2 and the last 2 in this word (if at all there are 2's).If f and g are order conjugate then w(f ) = w(g).Now, it is natural to ask which words arise as w(f ).The main objective of this paper is to answer the above question.Our main theorem is one about the location of fixed points vis a vis points of period 2 for interval maps.The results of this paper form the starting point of answering a grand question: Which dynamical systems arise as the restriction of interval maps f to the set of all f -periodic points?

Main Results
Let G be the collection of continuous self maps on I such that the periodic points of period 1 and 2 are finitely many.Let F be the sub collection of G consisting of all those maps for which all periodic points are of periods 1 or 2. We denote by L p (the language of periods) the set of all words w(f ) where f ∈ F. This L p is a language over {1, 2} and we seek a description of L p .Let L be the set of all words w(f ) where f ∈ G.In this paper we prove that it coincides with L p .Some easy observations: • The number of 2's has to be even in every word.
• If there is only one 1 in a word, then it has to be exactly in the middle.
• It is less obvious but true that the words like 222121 and 212212 are not in L p .

Main theorem:
The following are equivalent for a word over {1, 2}: 2. w belongs to the smallest language, say L g satisfying:

Theorem 1
The language over {1, 2} is context free, but the similarly defined language over {1, 2, 4} is not context free.

A Context free language
To prove the main theorem, we introduce another language L c , the set of words over {1, 2}, satisfying the condition below: For w ∈ L c , there exists a permutation σ from {1, 2, 3, ..., |w|} to itself (where |w| is the length of w) (C 3 ) If σ(i) < i and j < σ(j) then there exists a l between i and j such that σ(l) = l.The map σ in the above theorem is called a companion map; σ(i) is called the companion of i. Condition C 3 can be restated as follows: there should be a '1' between any two 2's whose companions are in opposite directions.
Proof: Let w ∈ L c .If C 4 fails, then there exist i, k such that i < k < σ(i) < σ(k); we say that C 4 fails at each of the four points.We also say that i and k form a twining pair.Let i be the smallest point where C 4 fails and j be the largest point where C 4 fails.Then we call j − i as twining length.For the given word w, there may be many permutations satisfying C 1 , C 2 , C 3 .Choose σ among them such that its twining length is minimal.Let i 1 > i be the smallest point forming a twining pair with i. Define σ 1 as and C 3 .Thus, we have removed the twining in the first pair of twining cycles.Repeat the process to define σ 2 , σ 3 , ..., σ k , each time removing one twining at i, until the condition σ k (i) > σ(l) for every i < l < σ k (i) is satisfied for some k ∈ N. Since at every stage, the image of i is increasing and |w| is finite, this shall take only a finite number of steps.In this way we get σ such that its twining length is less than j − i, which contradicts the minimality of twining length.Therefore for every w ∈ L c there exists a σ satisfying C 4 . 2 Lemma 2 u ∈ L p if and only if u1, 1u ∈ L p .
Proof: Let u ∈ L p .Then there exists a continuous function g In other words f is the transfer of g on [0, 1  2 ], f = 1, and f is linearly extended elsewhere.
In fact we prove that if either 1u ∈ L p or u1 ∈ L p then u ∈ L p .Let u1 ∈ L p .Then there exists a function f = 0, and f is linearly extended elsewhere.) ] is invariant, and so x is not periodic.If ] and so x is not periodic.Therefore w(f Remark 1 The converse of the above lemma is not true.221221212 It is easy to observe that C 1 and C 2 are satisfied.Now we prove that C 3 is satisfied. If σ(i) < i < j < σ(j), then f (x i ) < x i < x j < f (x j ).By Intermediate value theorem there is a fixed point between x i and x j .So there is a 1 between w i and w j .Similarly, if i > σ(i) > σ(j) > j or i > σ(j) > σ(i) > j or i > σ(j) > σ(i) > j or i > σ(j) > j > σ(i), there is a fixed point between i, j.Therefore C 3 is satisfied and so w ∈ L c .Therefore L p ⊂ L c .Now we prove that L c ⊂ L g .Suppose it is not true.Let w ∈ L c be the word of smallest length, which is not in L g .Then there exists a σ : {1, 2, ..., |w|} → {1, 2, ..., |w|} such that the conditions C 1 , C 2 , C 3 and C 4 are satisfied (by Lemma 1).It is clear from the definition of That means there exists u ∈ L c such that w = 1u or u1 or 2u2 or u1v (where v ∈ L c ).By the minimality of length of w, we have that u, v ∈ L g .Therefore w ∈ L g , a contradiction.Hence L c ⊂ L g .This implies that L p ⊂ L g .
Step 2: L g ⊂ L p .This inclusion follows from Lemma 2, 3 and 4. Therefore L g ⊂ L p . 2 Remark 3 Theorem 1 completely describes the location of fixed points between the points of period 2.
In fact, the following theorem says that L = L p .
Theorem 3 L = L p .
Proof: It is obvious that L p ⊂ L and now we prove that L ⊂ L p .To prove this we recall L c and prove that L ⊂ L c , since L p = L c .Let w ∈ L and let f ∈ G be such that w = w {1,2} (f ).Define σ : {1, 2, .., |w|} → {1, 2, .., |w|} by σ(i) = j if f (x i ) = x j .It is easy to observe that C 1 and C 2 are satisfied.Now we will prove that C 3 is satisfied.If σ(i) < i < j < σ(j), then f (x i ) < x i < x j < f (x j ).By the Intermediate value theorem, there is a fixed point between x i and x j .So there is a 1 between w i and w j .Therefore C 3 is satisfied and so w ∈ L c .Therefore L ⊂ L c . 2 Remark 4 From the above theorem we can observe: • If w ∈ L, the new word obtained by inserting any number of 1's, anywhere in w, will also be in L.
Here is a description of all words with six 2's: the three words 2221222, 212122122, 221221212 and all words by insertion of additional 1's any where.There are no other words in L with exactly six 2's.
• Here we give the list of all words in L whose length is less than 7. Remark 5 For w = w 1 w 2 ...w n ∈ L, there exists an interval map f such that w(f ) = w.Let x 1 < x 2 < ... < x n be the periodic points of f and w i be the period of x i .Now we define that w i is equal to * when f (x i ) = x i , to [ when x i < f (x i ) and to ] when f (x i ) < x i .Note that w(f ) is a well-formed parentheses word as every ] to the right is in correspondence with a [ to the left and it is well known that well-parentheses words are exactly those with as many [ as ] and such that every prefix has at least as much [ as ].Consider any i < j such that w i and w j are different kinds of parentheses, either x i < f (x i ) and f (x j ) < x j or f (x i ) < x i and x j < f (x j ) thus by the intermediate value theorem f admits a fixed point in the interval [x i , x j ], i.e., there is a * somewhere in between every pair of different parentheses in w(f ), or equivalently the word w(f ) cannot contain the factor [] nor the factor ][. To summarize, we have proven that w(f ) is included in the language defined by the following grammar: S → * |[S]|S * S| * S|S * .
To prove that every word in the language can be produced, one can easily proceed by induction using piecewise linear functions (f (x) = x/2 realizes the * and lemmas 2, 3 and 4 in the paper give short explicit constructions of the inductive steps).The language L is the projection of this language, that is the language with the following grammar: S → 1|2S2|S1S|1S|S1. (i)   L is a context free language with the grammar defined above.
Remark 6 Using the pumping lemma for regular languages [5], we can prove that this language is not regular.
Remark 7 It can be proved that, for every word w ∈ L p , there is a piecewise linear map f such that every critical point of f is a periodic point (the proofs of our lemmas can be modified to yield this).This f is post critically finite [2].From Thurston's theorem( [2]) for real polynomials there exists only one polynomial p up to affine conjugation with the same kneading data as f .Here we have the open question: Is w(p) the same as w(f )?
Remark 8 The main result of this paper leads us to the question: What are all the finite subsystems f | P (f ) , if f ∈ F ?This question can be restated as follows: Given a word from the above language L, what are all the "pairings of 2's" that are allowed for interval maps?
4 A Language that is not context free Let C be the class of interval maps f such that P (f ) is finite.For f ∈ C, let x 1 < x 2 < ... < x n be the periodic points of f , whose period is in A. We denote w A (f ) by w 1 w 2 ..w n , where w i ∈ A is the f -period of x i for 1 ≤ i ≤ n and L A denote the collection of all words w A (f ) over A, where f ∈ C. For w j ∈ N, w k j means w j w j ...w j (k times).For f ∈ C 0 (I), P is a periodic orbit of period 2 n .We say P is simple if for any subset {x 1 , x 2 , ..., x k } of P where k ≥ 2 divides 2 n , for any positive integer r which divides 2 n , such that {x 1 , x 2 , ..., x k } is periodic orbit of f r with x 1 < x 2 < ... < x k , we have f r ({x 1 , x 2 , ..., x k 2 }) = {x k 2 +1 , ..., x k }.Theorem 4 [4] Let f ∈ C(I) and suppose that f has a simple periodic orbit of period 2 n .Then there exists n + 1 simple periodic orbits of periods 1, 2, 4, ..., 2 n such that the 2 k periodic points of period 2 k are separated by the 1 + 2 + ... + 2 k−1 = 2 k − 1 fixed points of f 2 k −1 for any k = 1, 2, ..., n.
(i) We are thankful to the referee for suggesting this remark and for avoiding our inclusion of a proof of another characterization of L.