FLEXIBILITY OF LYAPUNOV EXPONENTS FOR EXPANDING CIRCLE MAPS

. Let g be a smooth expanding map of degree D which maps a circle to itself, where D is a natural number greater than 1. It is known that the Lyapunov exponent of g with respect to the unique invariant measure that is absolutely continuous with respect to the Lebesgue measure is positive and less than or equal to log D which, in addition, is less than or equal to the Lyapunov exponent of g with respect to the measure of maximal entropy. Moreover, the equalities can only occur simultaneously. We show that these are the only restrictions on the Lyapunov exponents considered above for smooth expanding maps of degree D on a circle.


1.
Introduction. Let S 1 = R/Z. We say that a map is a circle map if it maps the circle S 1 to itself. Let g be a smooth expanding circle map of degree D, where D is a natural number greater than 1. Let M(g) denote the set of g-invariant Borel probability measures. For any µ ∈ M(g) that is ergodic, its Lyapunov exponent is defined by λ µ (g) = S1 log |g |dµ, and we denote by h(µ, g) the metric entropy of g with respect to µ. We concentrate our attention on the Lyapunov exponent λ abs (g) with respect to the unique measure µ abs ∈ M(g) that is absolutely continuous with respect to the Lebesgue measure (see Theorem 5.1.16, Proposition 5.1.24 and Corollary 5.1.25 in [5] or Theorem 18 in [9]) and the Lyapunov exponent λ max (g) with respect to the unique measure µ max ∈ M(g) of maximal entropy (see Theorem 19 in [9]).
Throughout the paper, whenever convenient, we will consider a continuous expanding circle map as a piecewise continuous expanding map of the interval [0, 1] such that the values at 0 and 1 coincide mod 1.
Recall that two continuous circle maps f and g are said to be topologically conjugate if there exists a homeomorphism h : A continuous expanding circle map has degree D if and only if it is topologically conjugate to the ×D-map (see [5,Theorem 2.4.6]). Also, topological entropy is invariant under topological conjugacy [5,Corollary 3.1.4]. Therefore, it follows from all of the above that for any smooth expanding circle map g of degree D we have λ abs (g) ≤ log D ≤ λ max (g).
If we also have that λ abs (g) = log D, then we obtain that h(µ abs , g) = h top (g). Therefore, µ abs is the measure of maximal entropy for g and λ max (g) = λ abs (g) = log D. On the other hand, assume g is a smooth expanding circle map of degree D and λ max (g) = log D, i.e., h top (g) = S1 log |g |dµ max . Then, by Theorem III.1.2-3 in [7] we have that µ max is the measure which is absolutely continuous with respect to the Lebesgue measure implying that λ abs (g) = λ max (g) = log D. Thus, we have seen that the equalities in λ abs (g) ≤ log D ≤ λ max (g) can only hold simultaneously. It is a natural question if these inequalities are the only restrictions on the pair of values of the considered Lyapunov exponents. In the following theorem, we answer this question affirmatively by constructing smooth expanding circle maps of degree D that take on all possible values of pairs of Lyapunov exponents (λ abs , λ max ) corresponding to measures µ abs and µ max , respectively.
Main Theorem. For any positive numbers a, b such that a < log D < b, there exists a smooth expanding circle map g of degree D such that λ abs (g) = a and λ max (g) = b.
Main Theorem confirms in the setting of expanding circle maps the flexibility philosophy proposed by A. Katok. The flexibility program states that classical smooth systems (diffeomorphisms and flows) are quite flexible in comparison to actions of higher rank abelian groups. The first example in this direction was obtained in work of the author and A. Katok [3] which shows the flexibility for the values of the pair of metric entropy with respect to the Liouville measure and topological entropy for geodesic flow on surfaces of negative curvature with fixed genus greater than or equal to 2 and fixed total area. Another result in the flexibility program is due to J. Bochi, F. Rodriguez Hertz and A. Katok [1] who show how to vary the Lyapunov exponents with respect to the Lebesgue measure for volumepreserving Anosov diffeomorphisms with dominated splittings into one-dimensional bundles.
There are still many open questions related to the flexibility program and many properties of smooth dynamical systems whose flexibility is unknown. In particular, a natural extension of Main Theorem would be to consider a similar problem on the torus T 2 = R 2 /Z 2 . Let L A be an Anosov linear area-preserving automorphism of T 2 . For L A , we have that the Lyapunov exponent λ Leb (L A ) with respect to the Lebesgue measure, topological entropy h top (L A ), and the Lyapunov exponent with respect to the measure of maximal entropy λ max (L A ) coincide. Let us consider a smooth Anosov area-preserving diffeomorphism g on T 2 homotopic to L A . Then, we have that The question is if these inequalities are the only restrictions on the considered Lyapunov exponents for smooth Anosov area-preserving diffeomorphisms on T 2 homotopic to L A . We hope to answer this question in future work.
The rest of the paper consists of the proof of Main Theorem. First, in Section 2 we construct continuous piecewise linear circle maps of degree D which realize all possible values for the pairs of the considered Lyapunov exponents. We then give a procedure for smoothing these maps in Section 3. Finally, in Section 4 we prove the continuity of Lyapunov exponents for the constructed family of circle maps which implies Main Theorem in combination with Theorem 2.5.
2. Flexibility of Lyapunov exponents for SUSD-circle maps. In this section, we show in Theorem 2.5 that SUSD-circle maps (see Definition 2.1) take on all possible values of pairs of Lyapunov exponents corresponding to the measures µ abs and µ max (see Lemmas 2.2 and 2.3).
Definition 2.1. For any natural numbers D > 1 and n, k > 2 and any positive numbers δ ∈ [D −n−1 , D −n ) and ε ∈ [D −k−1 , D −k ), we define the SUSD-circle map (Speed Up Slow Down -circle map) f (x; n, δ; k, ε) of degree D by the formula: The graph of a SUSD-circle map of degree D = 2 is displayed in Figure 1. Note that a SUSD-circle map is a continuous piecewise linear expanding circle map with f (0; n, δ; k, ε) = 0 and f (x; n, D −n−1 ; k, D −k−1 ) ≡ Dx (mod 1) for any values of the parameters n, k, δ and ε that we have allowed. The derivative with respect to x of the SUSD-circle map f (x; n, δ; k, ε) outside of points of possible non-differentiability (x = 0, δ, D −n , D −1 − D −k , D −1 − ε) is given by the formula: Lemma 2.2. A SUSD-circle map has a unique invariant probability measure µ abs that is absolutely continuous with respect to the Lebesgue measure on S 1 and is ergodic. In particular, the invariant density q(x; n, δ; k, ε) for the SUSD-circle map f (x; n, δ; k, ε) of degree D is given by the formula and χ B (x) is the characteristic function of B ⊂ S 1 .
Proof. We fix parameters n, k, δ, ε and omit them from the notations for the rest of the proof. An integrable function q on S 1 is an invariant density of the SUSD-circle map f (x) = f (x; n, δ; k, ε) if and only if it is a fixed point of the Frobenius-Perron operator (see [6,Section 2]). In our case, it means that q has to satisfy the following equality a.e. with respect to the Lebesgue measure: Recall thatf (0) = 0. Let E = max{n,k} m=1f −m (0) be the collection of endpoints of a partition P of S 1 (the interval [0, 1]). Notice thatf (E) ⊂ E by the definition off . Moreover, for every P ∈ P there exists a natural number l such that f l (P ) = S 1 becausef is an expanding circle map with expansion rate at least min{(D n δ) −1 , D k ε −1 }. As a result, it belongs to the class C (see [2,Definition 3]). Therefore, by Theorems 1 and 3 in [2], there exists a unique probability measure µ abs absolutely continuous with respect to the Lebesgue measure, and the invariant density q is piecewise constant. In particular, µ abs is ergodic. We define the following intervals: , Using (1) and (3), we find the invariant density q in the following form: where a 1 , a 1 , a 1 , a D , a D , a D and a j for j = 2, . . . D − 1 are positive constants such that µ abs is a probability measure and χ B (x) is the characteristic function of B. To guarantee (3), we need to satisfy the following equalities.
As a result, we obtain that Therefore, using (a), we Using (a) and (b), we have a 1 = a 1 = a 1 (c), we obtain a j = a 1 . By (c) and (e), we have . Using (c), we see that we obtain the same equality for a D as in (f).
The total measure of the circle S 1 for the measure µ abs is equal to 1 if i.e., .
The Lyapunov exponent λ abs (g) = S1 log g dµ abs with respect to µ abs is equal to Proof. The statement follows from the definition of the Lyapunov exponent λ abs and Lemma 2.2.
Corollary 2. Fix natural numbers D > 1 and n > 2 and a positive number Proof. Notice that, by construction, g(x; k, D −1 ) acts as the ×D-map everywhere except the interval (0, D −n ) and is independent of k on that interval. Therefore, g(x; k, D −1 ) is independent of k. By Corollary 1, we have The function E(y) is monotonically decreasing and its values vary from log D to 0 as y varies from

FLEXIBILITY OF EXPONENTS FOR CIRCLE MAPS 2331
Both bounds on λ abs (g(x; k, v)) are independent of v and tend to Therefore, we obtain the desired result.
where E(y) = y log 1 y + (1 − y) log D−1 1−y . In particular, λ abs (ĝ(x; u)) is monotonically decreasing and its values vary from log D to 0 as u varies from D −1 to 1.

Lemma 2.3.
A SUSD-circle map has a unique invariant probability measure µ max of maximal entropy that is ergodic. In particular, for the SUSD-circle map f (x; n, δ; k, ε) of degree D we have Proof. A SUSD-circle map of degree D is topologically conjugate to the ×D-map and the conjugacy h can be constructed via coding (see Theorem 2.4.6 and its proof in [5]). In particular, for the SUSD-circle map f (x; n, δ; k, ε) of degree D, we have The Lebesgue measure is the unique measure of maximal entropy for the ×D-map that is ergodic [5,Sections 4b,4c, 5a]. A measure of maximal entropy is mapped to a measure of maximal entropy under the topological conjugacy. From that, we get the statement of the lemma. The Lyapunov exponent λ max (g) = S1 log g dµ max with respect to µ max is equal to Proof. The statement follows from the definition of the Lyapunov exponent λ max and Lemma 2.3.  1). Then, λ max (g(x; v)) is monotonically increasing to ∞ as v varies from D −1 to 1.
In particular, Proof. By Corollary 4, we have Proof. Fix a positive number ε < α 1+log D−α . By (4), to guarantee item 1 in the lemma, it is enough to show that for any n > 2 there existsū =ū(n) First, E(u) is independent of n and monotonically decreasing, and its values vary from log D to 0 as u varies from D −1 to 1. Therefore, there existsǔ ∈ [D −1 , 1) such that for any n > 2 and u ∈ [ǔ, 1) we have E(u) ≤ ε. By Corollary 4, we obtain λ max (g(x; n, u)) = M (u) Moreover, for a fixed n, we have λ max (g(x; n, u)) is monotonically increasing because the function M (u) is monotonically increasing and its values vary from D log D to ∞ as u varies from D −1 to 1. To guarantee items 1 and 2, it is enough to show that there existsN > 2 such thatū ( β . Therefore, we need to show that there existsN > 2 such that Finally, sufficiently largeN andû(N ) =ū(N ) = 1−ε D−1 D(D n −1) provide Lemma 2.4 because λ abs (g(x;N , u)) is a continuous function of u, λ abs (g(x;N , D −1 )) = log D, and λ abs (g(x;N ,û)) ≤ α.
All the inequalities for Lyapunov exponents with respect to µ abs and µ max discussed in the introduction will still hold for a SUSD-circle map. In Theorem 2.5, we show that SUSD-circle maps take on all possible values of pairs of Lyapunov exponents (λ abs , λ max ) corresponding to measures µ abs and µ max , respectively. Theorem 2.5. Let D > 1 be a natural number and a, b, δ be any numbers such that It is easy to see from Corollaries 1 and 4 that the map . Therefore, using the topological fact that there is no retraction of the square onto its boundary, we obtain that for any

Smoothing a SUSD-circle map.
In this section, we show how to smoothen a SUSD-circle map to obtain a smooth expanding circle map. Proof. Observe that the collection of points of non-differentiability forf (x) is a countable set because by Definition 2.1 we have that f (x) has at most 6 points of non-differentiability andf (x) is the lift of f (x) to R. Let {p i } ∞ i=−∞ be an indexing of the points of non-differentiability forf (x) such thatf (x) is linear on (p i , p i+1 ). Choose α sufficiently small such that ( By the properties of convolution, we have thatf α is a smooth map on R and for any x ∈ [p i − α, p i + α]. Therefore, using the fact that α −α θ α (y) = 1, we obtain is a smooth expanding map on R because L i > 1 for any i. Moreover,f (x + N ) =f (x) + N D for any integer number N . Therefore, for any integer number N . As a result, f α (x) =f α (x) (mod 1) is a smooth expanding circle map.
By a property of convolution, f α can be made arbitrarily close to f with respect to the sup norm for sufficiently small α. As a result, for sufficiently small α the degree of f α is D (see [5,Lemma 2.4.5]). We By Definition 2.1, we obtain thatf (x) = L i x + C i for some constants L i > 1 and C i if x ∈ (p i , p i+1 ). Therefore, for any x ∈ (p i + α, p i+1 − α) and y ∈ (−α, α) we have f (x − y) = L i (x − y) + C i . Consequently, using the properties of θ α , for any Finally, using the fact that f α (x) =f α (x) (mod 1), we have f α (x) = f (x) outside of α-neighborhoods of the points of non-differentiability of f .

4.
Proof of main theorem. Throughout this section, we fix the following notations that we will use in the statements of the results.
Let D > 1 be a natural number and a, b be positive numbers from the statement of Main Theorem. Choose any positive number δ such that 0 < a − δ < a < a + δ < log D < b − δ < b < b + δ.
By Theorem 2.5, there exist natural numbersK,N > 2 and constantsû,v ∈ [D −1 , 1) such that we have a continuous family of SUSD-circle maps Choose a smooth positive even function θ α (x) on R such that R θ α (y)dy = 1 and By the topological fact that there is no retraction of the square onto its boundary, to prove Main Theorem it is enough to show that there exists α > 0 such that: are smooth expanding circle maps of degree D. Therefore, they are topologically conjugate. Furthermore, if (u, v) is close enough to (u 0 , v 0 ), then f u,v and f u0,v0 are close in the sup norm. This implies that f α u,v and f α u0,v0 are close in the sup norm and the conjugating homeomorphism h u0,v0 u,v ) can be chosen close to the identity map (see [5,Theorem 2.4.6] and its proof). Let µ u,v max and µ u0,v0 max be the measures of maximal entropy for f α u,v and f α u0,v0 , respectively. Under the topological conjugacy, the measure of maximal entropy is mapped to the measure of maximal entropy, i.e., µ u,v max is the pushforward of µ u0,v0 max by h u0,v0 u,v . Furthermore, we have The second term in the inequality above can be made arbitrarily small because d dx f α u0,v0 is a smooth function on S 1 and if (u, v) is close enough to (u 0 , v 0 ), then h u0,v0 u,v y is close to y for any y ∈ S 1 . Now we show that the first term in the inequality above can be made small if (u, v) is close enough to (u 0 , v 0 ).
Recall that for any (u, v) In particular, we have d dxf α u,v x=z is equal to: 1.
As a result, we obtain the lemma.
To prove Lemma 4.2 and Lemma 4.5, we will need the following fact.
The proof follows the ideas in [4,Lemma 4].
Proof. It is enough to prove the convergence to 0 for any continuous function. Therefore, let F be a continuous function.
Recall that for any t ∈ T we have that g t is topologically conjugate to the ×D-map on S 1 . In particular, each g t has a unique fixed point p t . Let P t be a partition of S 1 into D left-closed semi-intervals t and the right endpoint of I D t , and the right endpoint of I i t is the left endpoint of I i+1 t . Note that g t I i t : I i t → S 1 is one-to-one. Furthermore, the endpoints of I i t tend to the corresponding endpoints of I i t0 as t → t 0 . Denote . As a result, we have where (φ i t0 ) (x) does not exist but left and right derivatives are finite, and w F is the modulus of continuity of F .
To show convergence to 0, we used convergence of the endpoints of corresponding elements of the partitions P t and P t0 for all terms. Moreover, for the first term in the sum we used the L 1 convergence of the derivative of g t to the derivative of g t0 , and for the second term in the sum we used the uniform convergence of g t to g t0 . Assume that q u,v does not converge to q u0,v0 in the L 1 norm. Then, there exists ε > 0 and a sequence (u n , v n ) → (u 0 , v 0 ) as n → ∞ such that S 1 |q un,vn − q u0,v0 |dx ≥ ε for any n.
Using the precompactness of the set {q u,v } in L 1 , we have a convergent subsequence of {q un,vn } n≥1 in L 1 . Without loss of generality, assume that q un,vn → q in the L 1 norm as n → ∞.
Let P u,v be the Frobenius-Perron operator of f α u,v and denote by · 1 the L 1 norm on S 1 . Then, P u0,v0 q − q 1 ≤ P u0,v0 q − P un,vn q 1 + P un,vn q − P un,vn q un,vn 1 + P un,vn q un,vn − q un,vn 1 + q un,vn − q 1 .
The first summand tends to 0 by Lemma 4.3 as n → ∞. The second and the fourth summands converge to 0 by the assumption that q un,vn → q in L 1 as n → ∞ and P un,vn 1 = 1. The third is equal to 0 because the invariant density is the fixed point point of the corresponding Frobenius-Perron operator. Therefore, P u0,v0 q = q and q is the density for f α u0,v0 of an invariant probability measure that is absolutely continuous with respect to Lebesgue. But f α u0,v0 has a unique invariant probability measure that is absolutely continuous with respect to Lebesgue. Therefore, q u0,v0 = q and that contradicts (5). As a result, we have that q u,v converges to q u0,v0 in the L 1 completing the proof of Lemma 4.2.
Now we formulate and prove lemmas that will allow us to finish the proof of Main Theorem.