A shift map with a discontinuous entropy function

Let $f:X\to X$ be a continuous map on a compact metric space with finite topological entropy. Further, we assume that the entropy map $\mu\mapsto h_\mu(f)$ is upper semi-continuous. It is well-known that this implies the continuity of the localized entropy function of a given continuous potential $\phi:X\to R$. In this note we show that this result does not carry over to the case of higher-dimensional potentials $\Phi:X\to R^m$. Namely, we construct for a shift map $f$ a $2$-dimensional Lipschitz continuous potential $\Phi$ with a discontinuous localized entropy function.


Introduction
Let f : X → X be a continuous map on a compact metric space with finite topological entropy, and let M denote the set of all f -invariant Borel probability measures on X endowed with the weak * topology. This makes M a compact convex metrizable topological space. For a continuous mdimensional potential Φ = (φ 1 , · · · , φ m ) : X → R m we define where rv(µ) = ( φ 1 dµ, · · · , φ m dµ). It follows that R(Φ) is a compact and convex subset of R m . The set R(Φ) is frequently referred to as the rotation set of Φ (see e.g. [3,8,9,10,12,17]), while in the context of multifractal analysis it is often referred to as the spectrum of (Birkhoff) ergodic averages (see e.g. [1,2,4]). The localized entropy function of Φ on R(Φ) is defined by where h µ (f ) denotes the measure-theoretic entropy of µ. We note that for various systems and potentials the localized entropy function coincides with the entropy of certain multifractal level sets (e.g. [2,4]). Recall that the measure-theoretic entropy is an affine function on M. This shows that w → H(w) is concave which implies its continuity on the interior of R(Φ), see e.g. [15]. If R(Φ) has empty interior we still obtain the continuity of H on the relative interior of R(Φ), i.e., the interior of R(Φ) considered as a subset of the affine hull of R(Φ). Another frequently considered condition is the upper semi-continuity of the entropy map µ → h µ (f ), which holds for example when f is expansive [16], when f is a C ∞ -map on a compact smooth Riemannian manifold [13] or when f satisfies entropy-expansiveness (as for example certain partial hyperbolic systems [5]). The upper semicontinuity of the entropy map immediately implies that the supremum in (2) is actually a maximum and more importantly that w → H(w) is upper semi-continuous. One might suspect that the latter actually even guarantees the continuity of the localized entropy function for all dimensions m. Indeed, it was stated by Jenkinson [9, p. 3723] that the upper-semi continuity of the entropy map implies the continuity of the localized entropy. This claim was restated by Kucherenko and Wolf in [10,11,12]. 1 However, it turns out that the argument in [9] is incomplete. While the continuity of every upper semi-continuous concave function with domain in R is immediate, the situation in higher dimensions is more delicate. Indeed, a striking result by Dale, Klee and Rockafellar [7] shows that for a compact convex set D ⊂ R m the property that every concave upper semi-continuous function on D is continuous is equivalent to D being a polyhedron. 2 We point out that R(Φ) being a polyhedron actually occurs in relevant situations, e.g. for subshifts of finite type (SFT) and locally constant potentials in [9,17], and for certain non-locally constant potentials in [9,10]. On the other hand, the results in [7] do not imply that w → H(w) can be discontinuous. After all H is a rather special upper semi-continuous concave function. In this note we show that the continuity of the localized entropy function can even fail in the case of shift maps and Lipschitz continuous potentials. More precisely, we have the following result (see Example 1 and Theorem 1 in the text).
Theorem. Let f : X → X be a shift map on a one-sided full shift with 3 symbols. Then there exists a Lipschitz continuous potential Φ : X → R 2 with the following properties: (i) The set R(Φ) has non-empty interior and countably many extreme points of which all but one are isolated; (ii) The localized entropy function w → H(w) is discontinuous at the non-isolated extreme point.
Further, one can show that the localized entropy function in the theorem is analytic on the interior of R(Φ). This follows from a more general analyticity result for so-called STP-maps (including SFT's, uniformly hyperbolic systems and expansive homeomorphisms with specification) and for Hölder continuous potentials, see [2,6,10]. We note that the reason for formulating our theorem for one-sided shift maps on a shift space with 3 symbols is for 1 We note that the theorems in [9,10,11,12] do not rely on the continuity of the localized entropy function. The only exception is Theorem A in [12] whose proof uses the continuity of H restricted to a line segment, i.e., m = 1. As noted above, for m = 1 the localized entropy is always continuous. 2 We note that the results in [7] are formulated in terms of lower semi-continuous convex functions.
the ease of presentation. Our techniques can be applied to obtain similar discontinuity results for more general SFT's in the one-sided and two-sided case.
We end the introduction with the discussion of a simple example of a upper semi-continuous concave function that fails to be continuous. Let x 2 + 1 for x 2 > 0 and g(0, 0) = 1. It is straight-forward to verify that g is concave and g(R) = [0, 1]. Further, g is continuous everywhere except at (0, 0) where g is only upper semi-continuous. The limit of g(x 1 , x 2 ) is 0 as (x 1 , x 2 ) approaches (0, 0) along the parabola x 2 = x 2 1 . However, the limit is 1 when (x 1 , x 2 ) approaches (0, 0) along any line segment in R. Moreover, g attains in each neighborhood of (0, 0) all values in [0, 1]. Indeed, if S x denotes a line segment joining a point x on the parabola x 2 = x 2 1 and (0, 0) then g(S x ) = [0, 1]. We note that while the function g is not lower semicontinuous, it does attain its infimum. We point out that there do exist bounded functions that are concave and upper semi-continuous but do not attain their infima, see [7,Lemma 1].
This paper is organized as follows. In Section 2 we recall some basic notation from symbolic dynamics and then construct in Section 3 an example of a discontinuous localized entropy function. The main ingredients of the proof are presented in Proposition 3 and Theorem 1.

Shift maps
We collect some basic notation and facts for shift maps. Let d ∈ N, and let A = {0, . . . , d − 1} be a finite alphabet with d symbols. The (one-sided) shift space X = X d on the alphabet A is the set of all sequences ξ = (ξ k ) ∞ k=1 where ξ k ∈ A for all k ∈ N. We endow X with the Tychonov product topology which makes X a compact metrizable space. For example, given 0 < θ < 1, the metric given by induces the Tychonov product topology on X.
The shift map f : Let M be the set of all invariant Borel probability measures endowed with the weak * topology, and let M E ⊂ M denote the subset of ergodic measures. Recall that M is a compact convex metrizable topological space. Given µ ∈ M we denote by h µ (f ) the measure-theoretic entropy of µ, see [16] for the definition and details. Clearly f is expansive and consequently the entropy map µ → h µ (f ) is upper semi-continuous. We say t = t 1 t 2 · · · t k ∈ A k is a block of length k and write |t| = k. Further, ε denotes the empty block. Moreover, we say Given ξ ∈ X, we write π k (ξ) = ξ 1 · · · ξ k ∈ A k . For ξ i ∈ A and k ∈ N we write ξ k i = ξ i · · · ξ i ∈ A k and define the concatenation of blocks t and s by ts = t 1 · · · t k s 1 · · · s l . Moreover, we denote by t k the k-times concatenation of the bock t. We denote the cylinder of length k generated by t by C k (t) = {ξ ∈ X : ξ 1 = t 1 , . . . , ξ k = t k }. Given ξ ∈ X and k ∈ N, we call C k (ξ) = C(π k (ξ)) the cylinder of length k generated by ξ. Further, we call O(t) = t 1 · · · t k t 1 · · · t k t 1 · · · t k · · · ∈ X the periodic point with period k generated by t. We denote by Per n (f ) the set of periodic points of f with prime period n and by Per(f ) and the set of periodic points of f . Let x ∈ Per n (f ). We call τ x = x 1 · · · x n the generating segment of x, that is x = O(τ x ). For x ∈ Per n (f ), the unique invariant measure supported on the orbit of x is given by where δ y denotes the Dirac measure on y. We also call µ x the periodic point 3. Construction of the example.
In this section we give an example of a shift map and a 2-dimensional Lipschitz continuous potential that exhibits a discontinuous localized entropy function. For convenience we consider here a one-sided shift map on a shift space with 3 symbols. We note that our construction can be modified to obtain discontinuous localized entropy functions for other shift maps with positive entropy. We begin by constructing a certain compact convex subset of R 2 that will become R(Φ) in our example.
Fix a, b > 0 and fix λ ∈ N with λ ≥ 3. Fix θ ∈ (0, 1). We consider a continuous function h : [0, a] → R which is strictly increasing and strictly concave. Further assume h(0) = 0 and h(a) = b. Let (x k ) k∈N be a strictly decreasing sequence with for all k ∈ N and some C > 0. The existence of such a sequence (x k ) follows from the continuity of h at 0. We define u k = (x k , 0) for all k ∈ N.
Since ||u k || ≤ ||v k ||, equation (5) also holds for u k . Let w ∞ = (0, 0) and w 0 = (a, 0). Further, for k ≥ 1 we define For k ≥ 1 let m k denote the slope of the line segment joining w k and w k−1 . Since (x k ) k is strictly decreasing it follows that the x-coordinates of (w k ) k are strictly decreasing. Thus, m k ∈ R for all k ≥ 1. We refer to Figure 1 for an illustration.
Proposition 1. The set R has the following properties: Proof. (i) That lim k→∞ w j = w ∞ follows from (5) and (6). Hence, w ∞ is the only accumulation point of V. We conclude that V is compact which implies the compactness of its convex hull R.
(ii) By (6), It now follows from an elementary induction argument that the points w k lie strictly below the graph of h. Therefore, the statement that m k is strictly decreasing follows from h being strictly increasing.
(iii) First notice that w 0 and w ∞ are extreme points of R. This holds since R has empty intersection with {(x, y) : x < 0}, {(x, y) : x > a} and {(x, y) : y < 0}. Finally, for k ≥ 1 that w k is an extreme point of R follows from statement (ii).
Next we consider the case l = min{j : By combining (9) and (10) we conclude that Φ is Lipschitz continuous with Lipschitz constant max{C 1 θ −λ , 2Cθ −λ }. Next we prove R(Φ) = R. Recall that R(Φ) is convex. Therefore, in order to prove R ⊂ R(Φ) it suffices to show that each extreme point of R (i.e. each point in V) coincides with the rotation vector of some invariant measure. For k ∈ N let ξ k = O(1 k+λ−1 2), that is ξ k is the periodic point whose generating segment τ k def = τ ξ k is given by k + λ − 1 1's followed by a 2. Hence ξ k ∈ Per k+λ (f ). It follows from equations (4), (6) and the definition of Φ (see (8)) that rv(µ ξ k ) = w k . Further, we clearly have rv(µ O(02) ) = w 0 and rv(µ Finally, we prove R(Φ) ⊂ R. It is well-known that the periodic point measures M Per are weak * dense in M, see [14]. Thus, by compactness of R it suffices to show that {rv(µ x ) : x ∈ Per(f )} ⊂ R. Let x ∈ Per n (f ) for some n ∈ N. Recall that τ x = x 1 · · · x n denotes the generating segment of It follows that if τ x does not contain a subblock in {τ k : k ∈ N} then rv(µ x ) ∈ [0, a] × {0} ⊂ R. It remains to consider the case when τ x contains at least one block in {τ k : k ∈ N}. By replacing x with a point in the orbit of x if necessary, we can write τ x as a finite concatenation of blocks of the form where the η i 's are blocks that do not have a subblock contained in {τ k : k ∈ N} and whose last symbol is either 0 or 2. The latter ensures that the blocks τ k i are of maximal length. We note that some of the η i 's in (12) may be the empty block. Let n i denote the length of η i . For each i there exists a m i such that Notice that n = lλ + l i=1 n i + k i . Therefore, (13) shows that rv(µ x ) is a convex combination of points in R which implies that rv(µ x ) ∈ R.
To complete the proof of the proposition it remains to show that Claim 3 holds for all the points in the orbit of ξ k . Let ζ k = f l (ξ k ) for some l = 1, · · · , k + λ − 1. Since f k+λ−l (ζ k ) = ξ k we conclude from Lemma 1 that Theorem 1. Let Φ be the potential defined in (8). Then H(w k ) = 0 for all k ∈ N and H(w ∞ ) = log 2. In particular, w → H(w) is discontinuous at w ∞ .