ENTROPY OF DIFFEOMORPHISMS OF LINE

. For diﬀeomorphisms of line, we set up the identities between their length growth rate and their entropy. Then, we prove that there is C 0 -open and C r -dense subset U of Diﬀ r ( R ) with bounded ﬁrst derivative, r = 1 , 2 , ··· , + ∞ , such that the entropy map with respect to strong C 0 -topology is continuous on U ; moreover, for any f ∈ U , if it is uniformly expanding or h ( f ) = 0, then the entropy map is locally constant at f . Also, we construct two examples: 1. there exists open subset U of Diﬀ ∞ ( R ) such that for any f ∈ U , the entropy map with respect to strong C ∞ -topology, is not locally constant at f . 2. there exists f ∈ Diﬀ ∞ ( R ) such that the entropy map with respect to strong C ∞ -topology, is neither lower semi-continuous nor upper semi- continuous at f .


Mathematics and Science College, Shanghai Normal University
Shanghai 200433, China

(Communicated by Shaobo Gan)
Abstract. For diffeomorphisms of line, we set up the identities between their length growth rate and their entropy. Then, we prove that there is C 0 -open and C r -dense subset U of Diff r (R) with bounded first derivative, r = 1, 2, · · · , +∞, such that the entropy map with respect to strong C 0 -topology is continuous on U ; moreover, for any f ∈ U , if it is uniformly expanding or h(f ) = 0, then the entropy map is locally constant at f . Also, we construct two examples: 1. there exists open subset U of Diff ∞ (R) such that for any f ∈ U , the entropy map with respect to strong C ∞ -topology, is not locally constant at f . 2. there exists f ∈ Diff ∞ (R) such that the entropy map with respect to strong C ∞ -topology, is neither lower semi-continuous nor upper semicontinuous at f .

1.
Introduction. Topological entropy and metric entropy are very important invariants. In this paper, we only consider topological entropy. The entropy of diffeomorphisms induces the natural map from diffeomorphisms space to R, which is said to be entropy map. By structural stability of hyperbolic diffeomorphisms of compact manifolds, the entropy map is locally constant. For partially hyperbolic diffeomorphisms of compact manifolds with one dimensional center bundles, the entropy map is continuous at the known examples and is conjectured that [7]: The topological entropy is continuous on the space of C 1 partially hyperbolic diffeomorphisms of compact manifolds with the dimension of the center equal to one.
During the visiting in University of Paris-Sud, J.Buzzi told us another related question that Question 1. Is the topological entropy locally constant on a dense (necessarily open) subset of the space of C 1 partially hyperbolic diffeomorphisms with one-dimensional center (on compact manifolds)?
Our paper on the entropy of diffeomorphisms on line is motivated by these problems.
For C 1 endomorphisms without degenerated critical points on compact interval, the entropy is continuous with C 1 -topology [4,5], and locally constant on its' C 1dense and open subset satisfying the structural stability [1]. As we know, the dynamics on non-compact spaces are quite different with dynamics on compact spaces. For example, topological stability preserves entropy of diffeomorphisms of compact manifolds, but it is not valid for the case of non-compact manifolds. Though topological expanding diffeomorphisms on line are conjugated each other, they can have zero entropy, positive entropy, and infinity entropy.
Let Diff r b (R) be the set of C r -diffeomorphisms of line with bounded first derivative, r = 1, 2, · · · , ∞. Our results in this section are worked under the strong topology of Diff r b (R).
such that for any f ∈ U, the entropy map with respect to strong C ∞ -topology, is not locally constant at f .
such that the entropy map with respect to strong C ∞ -topology, is neither lower semi-continuous nor upper semi-continuous at f .
The first example shows that Buzzi's corresponding problem is not valid for diffeomorphisms on line. For generic diffeomorphisms on line, we still have the continuity of entropy map.
, +∞, such that the entropy map with respect to strong C 0 -topology is continuous on U; moreover, for any f ∈ U satisfying that it is uniformly expanding or h(f ) = 0, the entropy map is locally constant at f . By the above theorem and the corollary 2, we give the characteristic of the diffeomorphisms having robustly zero entropy. And by the above theorem, for any uniformly expanding diffeomorphism on line, the entropy map is locally constant at it. By the proof of theorem 4, we can construct non uniformly expanding diffeomorphism on line with positive entropy, where the entropy map is locally constant. So, it is naturally ask the problem: Question 2. How is the union of diffeomorphisms on line where the entropy map is locally constant?
There is an interesting phenomea that "there is C 1 -dense subset U of diffeomorphisms space on compact manifold such that for any f ∈ U, the entropy map at f is either robustly positive or robustly zero ", hidden in the Palis weak C 1 -density conjecture [2]. In our case, we show that such phenomea is C r -dense, r = 1, 2, · · · , +∞. It is a directed consequence of the above theorem on the continuity of entropy map. Theorem 1.4. Let U * be the union of diffeomorphisms where the entropy map is robustly positive or robustly zero. Then U * is C r -dense in Diff r b (R), r = 1, 2, · · · , +∞. This paper is organized as follows. In section 2, we introduce length-growth rate of diffeomorphisms and set up the identity between length-growth and entropy.
In section 3, we show that the entropy map with respect to weak topology, is not continuous.
In section 4, we mainly introduce uniformly topologically expanding diffeomorphisms on positive or negative diffeomorphisms, and set up a simpler identity of entropy.
In section 5, we prove the continuity of entropy map and length-growth rate of generic diffeomorphisms on line.
In last two sections, we construction two examples of theorem 1.2 and theorem 1.1.
2. Entropy of diffeomorphisms on line. In this section, we review the definition of entropy of metric space defined in Bowen's way [8], and introduce the definition of the length growth rate of diffeomorphisms on line. Then we show that lengthgrowth rate equals entropy for diffeomrphisms with positive entropy.
Let f be continuous endomorphism on metric space X, take compact subset K ⊂ X. For any ε > 0 and integer n > 0, a set A is said to be (n, ε)-separated if for any and The topological entropy of f is defined as Through the paper, I denotes some interval of R, and (I) denotes the length of I. For general interval J, take over bounded interval I ⊂ J, the length growth rate of f on J is defined as In particular l(f, R) is simplified as l(f ).
Remark 1. For diffeomorphisms on compact manifolds, the volume growth rate is study by Yomdin and Newhouse to pursue the continuity of entropy [6,9]. The above identity has the same flavor with Misiurewicz-Szlenk's identity on the entropy of continuous piecewise monotone maps of the interval [5].
Proof. Firstly, we show that for any bounded interval I, any integer n > 0 and any ε > 0, we have the inequality that Obviously, there exists [ (f n−1 I) ε ] + 1 points such that the distance between f (n−1)iteration of these points each other is not smaller than ε. Then by the definition of (n, ε)-separated set, we have the left part of the inequality. Let x 1 < x 2 < · · · < x k be the (n, ε)-separated set, and . By the definition of (n, ε)-separated set, we have that for any i ∈ [1, k −1], Then, So, we have the right part of the inequality.
By the definition of l(f, J) and h(f, J) , and the left part of the above inequality, we have that l(f, J) ≤ h(f, J).
Fix any bounded interval I ⊂ J. By the definitions of l(f, J), we have that for any δ > 0, there exists N δ such that for any n ≥ N δ , the following inequality satisfies ln( (f n I)) n < l(f, J) + δ.
By the right part of the inequality about #s(n, ε, I), we have that #s(n, ε, I) < 1 + B δ (I) + e n(l(f,J)+δ) −1 Remark 2. In the above proof, we have the more accurate equality that for any ε > 0 and any bounded interval I, Then, f can be said to be entropy-expansive.
3. Entropy map on strong topology and weak topology. As we know, Diff r (R) has weak topology and strong topology. The strong C r -topology, r = 0, 1, 2, · · · , is induced by And the strong topology on Diff ∞ (R) is the union of the topologies induced by Diff ∞ (R) → Diff r (R) for r finite.
For any compact subset K ⊂ R, define .

ENTROPY OF DIFFEOMORPHISMS OF LINE 4757
Note that weak topology does not control the infinity of R. While the entropy concerns the infinity of R. So, we have that entropy map on weak topology, is not continuous.
Proposition 1. For any f ∈ Diff r (R) with positive entropy,r ≥ 0, the entropy map is neither upper semi-continuous nor lower semi-continuous at f on the weak C r -topology of Diff r (R).
It can easily deduced by the identity between entropy and length growth rate, and the following basic lemma. Also the proof of the lemma is quite basic. So, we omit these proof.
In the rest part of the paper, we only consider entropy map on the strong topology of Diff r (R). 4. Topologically expanding and topologically contracting maps. In this section, we introduce the definitions of the uniformly topologically expanding or contracting diffeomorphisms in positive or negative orientation. For these diffeomorphisms, we give the simpler identity of entropy which will be used in the following sections.  On the other side, by the definition of length growth rate, we have that for any ε > 0, there exists N such that for any n ≥ N , ln(f n+1 (a) − f n (a)) n < l(f, [p, +∞)) + ε.
Remark 3. If we replace K by 0 in the above two definitions, these maps are said to be topologically expanding in positive or negative orientation. The above two propositions are also valid for these more general maps.  In the end, we give some examples on these definitions.
5. The continuity of length growth rate and entropy map. In this section, we prove the continuity of the length growth rate on E + ∪ E − . Then by the identity of entropy in above section, we prove the continuity of the entropy map on (E + ∪ C + )∩(E − C − ). Moreover, we show that continuity of the entropy map is a generic property on Diff r b (R). Proof. For any f ∈ E + , there exists p, λ > 1, K > 0 such that for any ε > 0 and any g ∈ E + with d 0 (f, g) = ε, we have that • for any interval I with (I) > ε, min{ (f (I)), (g(I))} < λ (I).

2
. For any small perturbation g of f , we have that f n (p) < g n (p ) < f n+1 (p) for any n > 0. By the identity in Proposition 2, we have that l + (.) is locally constant at f . Let ϕ(x) = −x. For any f ∈ E − , ϕ −1 f ϕ ∈ E + , and l − (f ) = l + (ϕ −1 f ϕ). Therefore, we can deduce the same result of E − and l − (.), by the result of E + and l + (.). Now we give the proof of Theorem 1.3.
For any C r -diffeomorphism f , suppose f 2 ∈ E + . Then, For any ε > 0, let g = f + εα be the perturbation of f . Then, It is not difficult to deduce that εf (ξ)α(x) + εα(f (x) + εα(x)) < −ε for big enough x. Then, g 2 ∈ C + . So, {f 2 : f ∈ Diff r b (R)} ⊂ C + ∪ E + , and then U + is C r -dense subset of Diff r b (R). Similarly, we have that . For any f ∈ (E + ∪ C + ) ∩ (E − C − ), it has the following four cases: Then by proposition 4 and h(f 2 ) = 2h(f ), we get the same properties of entropy map on U with the properties of l + (.) and l − (.) on ( Remark 4. For diffeomorphisms whose first derivatives are away from zero (can have unbounded first derivatives), the corresponding proposition 4 and theorem 1.3 are still valid. But for general diffeomorphisms on line, the proposition is not valid. To show the continuity of their entropy map and length-growth rate, we need additional condition [3].
In the above proof, we show that Corollary 1. C + ∪ E + = C − ∪ E − is the set of orientation-preserving diffeomorphisms on line with bounded first derivatives.
is the directed property of the above theorem. Fix r. By the corollary 1, we have that Then for any diffeomorphism f ∈ Diff r b (R), f 2 is contained in C + ∩ C − or C + ∩ E − or E + ∩ E − or E + ∩ C − . Note that for any g ∈ C + ∪ C − , the corresponding length growth rate is zero. Then by proposition 4, we have that if h(f ) = 0, there exists g arbitrarily C r -closed to f such that g has robustly zero entropy and g 2 ∈ (E + ∪ C + ) ∩ (E − C − ).
6. Proof of Theorem 1.2. In this section, we construct diffeomorphism where the entropy map is neither lower semi-continuous nor upper semi-continuous. At first, we give a sensitive diffeomorphism on interval, which has a non-hyperbolic fixed point. And for any 0.1 > ε > 0, α w * +ε (0) > 1.9. Proof. Note that the curves α w are convex. Then, w * is the value such that the curve α w * is tangent with the line y = x. The other properties are quite trivial.
Before giving the example of Theorem 1.2, we talk about the main idea of the construction. It is constructed by induction. In its n-th "periodic", it is uniformly expanding, then uniformly contracting, and behaves the same dynamics with some α wn (w n > w * ) in the end, besides very few connecting iterations. By the above lemma, the iteration number of the dynamics α wn can become suddenly smaller by small perturbation. This makes the entropy suddenly become bigger.
Fix g and g ε . For any n > 1, there exists unique m n such that g n (0) ≤ g mn ε (0) < g n+1 (0). Note that for any n > 100 and any g ∈ U, 2 n+1 < t n (g) < 2 n+2 . Then for big enough n, by induction, it is not difficult to deduce that m n n < 1 − 10 −2 ε.