Geometric Barycentres Of Invariant Measures For Circle Maps

Oliver Jenkinson

UPR 9016 CNRS, Institut de Mathématiques de Luminy, 163 avenue de Luminy, case 907, 13288 Marseille, cedex 9, France. email : omj@iml.univ-mrs.fr

Abstract. For a continuous circle map T, define the barycentre of any T -invariant probability measure μ to be b(μ) = ∫_{S^{1}} z dμ(z). The set Ω of all such barycentres is a compact convex subset of . If T is conjugate to a rational rotation via a Möbius map, we prove Ω is a disc. For every piecewise-onto expanding map we prove that the barycentre set has non-empty interior. In this case, each interior point is the barycentre of many invariant measures, but we prove that amongst these there is a unique one which maximises entropy, and that this measure belongs to a distinguished two-parameter family of equilibrium states. This family induces a real-analytic radial foliation of int(Ω), centred around the barycentre of the global measure of maximal entropy, where each ray is the barycentre locus of some one-parameter section of the family. We explicitly compute these rays for two examples. While developing this framework we also answer a conjecture of Z. Coelho

[6] regarding limits of sequences of equilibrium states.

1991 Mathematics Subject Classification: 58F11, 58F15, 58F03, 52A10

Section 1. Introduction.

Let T : S^{1} → S^{1} be a continuous map of the circle. The set M of T -invariant Borel probability measures is non-empty, by the Krylov-Bogolioubov Theorem. Geometrically M is a simplex, and its extremal points are precisely the ergodic measures. For many circle maps M has a rich structure, and is hard to describe properly. In particular it may well be infinite-dimensional, allowing rather counter-intuitive properties such as density of extremal points (in which case M is the so-called Poulsen simplex, see Lindenstrauss et al. [12], Glasner & Weiss [8]). We hope to gain insight into M by considering finite dimensional projections. In particular, to each measure μ ∈ M we define its barycentre (or first Fourier coefficient) to be b(μ) = ∫_{S^{1}} z dμ(z), the point in the plane describing the average position of μ-almost every orbit of T. This is geometrically the most natural two-dimensional projection. We define Ω = b(M) to be the set of all such barycentres. If w ∈ Ω then we let M(w) denote the convex set b^{−1}(w) ⊂ M.

We will study the geometry of Ω, and its relation to M. For certain well-known maps the boundary ∂Ω has rather pathological differentiability properties (see Bousch [2], Hunt & Ott [10], Jenkinson [11], and also §7, §8). In this article we mainly focus on the interior of Ω. We highlight a two-parameter family of measures whose barycentres cover int(Ω), and study the behaviour of these measures near the boundary ∂Ω.

We begin by proving some basic properties common to all continuous circle maps.

Proposition 1. Let T : S^{1} → S^{1} be a continuous circle map.

(a) The barycentre set Ω is a compact convex subset of the unit disc in .

(b) If M(w) is a singleton set containing an ergodic measure then w is an extremal point of Ω.

(c) If w is an extremal point of Ω then any extremal point of M(w) is ergodic.

Proof. (a) Ω is clearly a subset of the unit disc. We recall (see Walters [27]) that M is a convex set, compact in the weak^{∗} topology. The corresponding properties of Ω follow since the barycentre map b is continuous and affine.

(b) Recall that the extremal points of M are precisely the ergodic measures. If w is not extremal in Ω then there exist μ_{1}, μ_{2} ∈ M, μ_{1} ≠ μ_{2}, and α ∈ (0, 1), such that w = αb(μ_{1}) + (1 − α)b(μ_{2}) . Thus the non-ergodic measure αμ_{1} + (1 − α)μ_{2} belongs to M(w).

(c) Let μ be an extremal point of M(w). We will show that in fact μ is extremal in M, and therefore ergodic.

Suppose, for a contradiction, that we can find distinct measures μ_{1}, μ_{2} ∈ M and α ∈ (0, 1) such that

μ = αμ_{1} + (1 − α)μ_{2} .  (1)

Applying b to (1) gives w = αb(μ_{1}) + (1 − α)b(μ_{2}). But w is extremal in Ω, so b(μ_{1}) = b(μ_{2}) = w. That is, μ_{1}, μ_{2} ∈ M(w). So (1) implies μ is not extremal in M(w), a contradiction.

The convexity of Ω means it is either a point, a line segment, or has interior. If T is uniquely ergodic then clearly Ω is just a point. This occurs, for example, if T is an irrational rotation - the unique invariant measure is normalised Lebesgue measure, which gives barycentre at 0. In fact even if the rotation is rational then the barycentre set is still the origin - in this case the ergodic measures are each concentrated on single periodic orbits, and each have zero barycentre. The fact that each measure in this one-parameter family is evenly distributed is a rather special property of a pure rotation. More generally one might consider circle diffeomorphisms with rational rotation number. We prove:

Theorem A. If T : S^{1} → S^{1} is a diffeomorphism, conjugate to a rational rotation via a Möbius map, then its barycentre set is a disc.

An interesting open question is to determine exactly which compact convex subsets of the disc can arise as the barycentre set of a circle map. The only way Ω can be the whole disc is if T is the identity, since b(μ) ∈ S^{1} if and only if μ is the Dirac measure concentrated on b(μ). In fact any segment (the convex hull of an interval I = {e^{2πit} : a ≤ t ≤ b}) can arise as a barycentre set. We just take T to be the identity on I, then join up its graph in a monotone way (to ensure S^{1} \ I contains no T -invariant subsets). Note that such maps are not topologically transitive. If T : z ↦ e^{2πiα}\overline{z} is reflection through the line segment joining e^{πiα} and e^{πi(α+1)}, then Ω is precisely this line segment.

For most of this paper we consider the more complicated dynamics of piecewise-onto expanding maps (defined in §3). We prove the following result.

Theorem B. Let T : S^{1} → S^{1} be a piecewise-onto expanding map. The barycentre set Ω has non-empty interior.

For a given w ∈ Ω it is natural to consider the entropy of those measures with barycentre w, and to study the concave function w ↦ sup_{μ∈M(w)} h(μ). More precisely, for w ∈ int(Ω) we solve the problem of identifying those m ∈ M(w) such that h(m) = sup_{μ∈M(w)} h(μ).

Theorem C. Let T : S^{1} → S^{1} be a piecewise-onto expanding map, with barycentre set Ω. Let K = [0, 1). If w ∈ int(Ω) then

(a) M(w) contains a unique equilibrium state m_{tf_{θ}} of a function of the form tf_{θ}, where t ≥ 0, θ ∈ K, and f_{θ}(x) = cos 2π(x − θ),

(b) this equilibrium state m_{tf_{θ}} is the unique measure of maximal entropy amongst all measures in M(w),

(c) this maximal entropy h(m_{tf_{θ}}) is positive and equal to the pressure of the function t (f_{θ} − ⟨w, e^{2πiθ}⟩),

(d) for any θ ∈ K this maximal entropy h(m_{tf_{θ}}) is a strictly decreasing function of t ≥ 0.

The existence and uniqueness part of Theorem C means the barycentres b(m_{tf_{θ}}) give a natural and geometrically interesting radial foliation of int(Ω), which moreover is realanalytic. The rays are the curves t ↦ b(m_{tf_{θ}}), which we call equilibrium rays, and they are centred around their common intersection point b(m_{0}), the barycentre of the measure of maximal entropy for T. In §6 we address the problem of accurately computing this radial foliation. We express each equilibrium ray in terms of an adapted coordinate system, the components of which are readily computable in terms of poles of zeta functions. We work with the reciprocals of these zeta functions, and consider their power series expansions. Rapid decay of the coefficients in these expansions means we can work with low degree polynomial approximations and still get accurate results. In §7 and §8 we apply our techniques to two specific examples, the doubling map and the tent map. For the doubling map we also prove (Proposition 13) that the radial foliation is conformal at its centre.

In developing this framework we also consider more general questions regarding the limiting behaviour of equilibrium states of the form m_{tf}. For a general Hölder function f, Coelho [6] conjectured that as t → ∞ the equilibrium states m_{tf} always converge in the weak^{∗} topology. He conjectured further that this limiting measure always takes a particular form, namely that it is the evenly-weighted centroid of so-called extremal measures. In §9 we show this second conjecture is false.

Acknowledgments. I would like to thank Eli Glasner, Mark Pollicott, and Hans Henrik Rugh for interesting conversations, and Anthony Manning for conjecturing Proposition 13. I thank Thierry Bousch for a careful reading of a preliminary version of this paper.

Section 2. Diffeomorphisms with rational rotation number.

In order to re-state Theorem A more precisely, first recall ([17], page 178) that the Möbius homeomorphisms of S^{1} are of the form

M_{a}^{φ}(z) = e^{iφ} \frac{z − a}{āz − 1} and N_{a}^{φ}(z) = e^{iφ} \frac{az − 1}{z − ā} = e^{iφ}/M φ ; ā(z),

where a ∈ with |a| < 1, and φ ∈ [0, 2π).

Theorem A. Suppose T = MRM^{−1} : S^{1} → S^{1}, where R is the rational rotation R(z) = e^{2πip/q}z, and M = M_{a}^{φ} or N_{a}^{φ} is a Möbius homeomorphism.

Then the barycentre set for T is a disc D. Its centre is

c = e^{i(φ+Arg(a))} (\frac{|a| − |a|^{2q−1}}{1 − |a|^{2q}} )

and its radius is

r = \frac{|a|^{q−1} − |a|^{q+1}}{1 − |a|^{2q}} .

Proof. We assume M = M_{a}^{φ}, the case M = N_{a}^{φ} being analogous. The only T -ergodic measures are the images under M of the R-ergodic measures, namely the atomic measures supported on single periodic orbits. Now any extremal point of the barycentre set must be the barycentre of at least one ergodic measure. We will show the barycentre locus of the 1-parameter family of atomic measures is precisely the boundary of D.

Now the 1-parameter family of atomic measures m_{t} for R is given by

m_{t} = \frac{1}{q} ∑_{j=0}^{q−1} δ(e^{2πit}e^{2πij/q}) , 0 ≤ t < 1/q,

where δ(·) denotes Dirac mass. Writing z = e^{2πit} and μ = M_{∗}(m_{t}), the corresponding barycentre is

b(μ) = \frac{e^{iφ}}{q} (∑_{j=0}^{q−1} \frac{ze^{2πij/q} − a}{āze^{2πij/q} − 1}) .

A calculation involving the relation between the roots and coefficients of a polynomial leads us to the formula

b(μ) = e^{iφ} \frac{ā^{q−1}z^{q} − a}{ā^{q}z^{q} − 1}.

Note in particular that b(μ) = B(z^{q}), where B is the Möbius map

B(z) = e^{iφ} \frac{ā^{q−1}z − a}{ā^{q}z − 1}.

The barycentre locus L is the path b(M_{∗}(m_{t})), 0 ≤ t < 1/q. But z ↦ z^{q} maps the segment e^{2πit}, 0 ≤ t < 1/q, to the whole circle S^{1}. So the barycentre locus is the image of S^{1} under the Möbius map B, and hence is itself a circle.

Now we determine the centre and radius of the circle L. Let us write B = E ∘ C, where E(z) = e^{iφ}z and

C(z) = \frac{ā^{q−1}z − a}{ā^{q}z − 1}.

Then

C^{−1}(z) = \frac{z − a}{ā^{q}z − ā^{q−1}} .

Suppose w lies on the barycentre locus L. Writing z = e^{−iφ}w we have that C^{−1}(z) ∈ S^{1}, so that

|z − a| = |ā^{q}z − ā^{q−1}| = |a|^{q}|z − 1/ā|.

Now 1/ā lies outside the unit circle, and has the same argument as a. In particular this implies that the centre of the circle E^{−1}(L) lies on the line through the origin of angle Arg(a). Moreover, E^{−1}(L) cuts this line at the two points

(\frac{|a| + |a|^{q−1}}{1 + |a|^{q}} ) e^{iArg(a)} and (\frac{|a| − |a|^{q−1}}{1 − |a|^{q}} ) e^{iArg(a)}.

A computation gives that E^{−1}(L) has centre at

e^{iArg(a)} (\frac{|a| − |a|^{2q−1}}{1 − |a|^{2q}} )

and radius

\frac{|a|^{q−1} − |a|^{q+1}}{1 − |a|^{2q}} .

Applying E we arrive at the desired expressions for the centre and radius of the barycentre locus L.

Remark 1. The barycentre set is independent of the numerator p of the rotation number p/q, since as a set, a periodic orbit of R_{p/q} is independent of p.

Remark 2. Every interior point of the unit disc can arise as the centre of a barycentre set, by the above construction (though note 0 can only be the centre in the degenerate case where T is the identity map). However, for any given centre, the set of possible radii is discretely parametrised, so that the above construction does not guarantee that all subdiscs of the unit disc can arise as barycentre sets.

Section 3. Expanding Maps.

It will be convenient to use the additive circle K = [0, 1), with addition defined modulo one, and usual distance function ρ. From now on we assume that T : K → K is a piecewise-onto expanding map.

Definition 1. We say a continuous map T : K → K is a piecewise-onto expanding map if there exists a finite number of disjoint open intervals P_{1}, …, P_{r} ⊂ K with K = ∪_{i=1}^{r}\overline{P_{i}}, and

(a) there exist ϵ > 0 and γ > 1 such that for all i, if x, y ∈ P_{i} satisfy ρ(x, y) < ϵ, then ρ(T x, T y) > γρ(x, y),

(b) \overline{T P_{i}} = K for each i.

If x ∈ K then we let O(x) denote its forward orbit under T. T has a countable infinity of periodic points, and these are dense in the circle. Let Fix(n) = {x ∈ K : T^{n}x = x}.

Let H = H(K, ) denote the space of all real-valued α-Hölder functions, equipped with the usual Banach norm ∥ · ∥ = ∥ · ∥_{α} + ∥ · ∥_{∞} . Here

∥f∥_{α} = sup_{x≠y} \frac{|f(x) − f(y)|}{ρ(x, y)^{α}} ,

while ∥f∥_{∞} = sup_{x∈K} |f(x)| is the uniform norm.

Notation. Given a function f, we write f^{n}(x) = ∑_{i=0}^{n−1} f(T^{i}x), and Ω(f) = {∫ fdμ : μ ∈ M} .

Since μ ↦ ∫ f dμ is continuous and affine, and M is compact and convex, then Ω(f) is also compact and convex. That is, Ω(f) is a closed interval contained in the closed interval [inf f, sup f].

If u : K → is Borel measurable and bounded, and a ∈ , then we say the function u · T − u + a is an essential coboundary for T. We call u the transfer function. We say two functions f, g ∈ H are essentially cohomologous if they differ by an essential coboundary.

Given a continuous function g : K → we define its pressure P (g) by P (g) = sup_{μ∈M} (h(μ) + ∫ g dμ) , where h(μ) is the entropy of T with respect to μ. An equilibrium state for g is a measure m_{g} ∈ M satisfying h(m_{g}) + ∫ g dm_{g} = P (g).

If g is Hölder then m_{g} is unique. Moreover, it is ergodic, fully supported, and has positive entropy (see Walters [27]). In fact the resulting measure-preserving map is an exact endomorphism with Bernoulli natural extension (see Walters [26]). Two Hölder functions have the same equilibrium state if and only if they are essentially cohomologous.

For a fixed f ∈ H, the map t ↦ P (tf) is an analytic function of the real variable t. If f is not an essential coboundary then t ↦ P (tf) is strictly convex. Its derivative is given (see Ruelle [24], Parry & Pollicott [18]) by the formula

\frac{d}{dt}P (tf) |_{t=t_{0}}= ∫ f dm_{t_{0}f} .  (2)

Lemma 2. Let T : K → K be a piecewise-onto expanding map. If f ∈ H(K, ) is not an essential coboundary, then

int(Ω(f)) = \{\frac{d}{dt}P (tf) |_{t=t_{0}}: t_{0} ∈ \} .

Proof. Let D = \{\frac{d}{dt}P (tf)|_{t=t_{0}} : t_{0} ∈ \}. From (2) we see that D ⊂ Ω(f). Note that D is connected, being the continuous image of , and in fact the strict convexity of t ↦ P (tf) means that D is an open interval. Thus D ⊂ int(Ω(f)).

We now show that Ω(f) ⊂ \overline{D}. The definition of pressure means that for any μ ∈ M we have

P (tf) ≥ h(μ) + ∫ tf dμ.  (3)

The strict convexity of t ↦ P (tf) guarantees the existence of the asymptotic derivatives lim_{t_{0}→∞} \frac{d}{dt}P (tf)|_{t=t_{0}} and lim_{t_{0}→−∞} \frac{d}{dt}P (tf)|_{t=t_{0}}. Moreover, from (3) we know the graph of t ↦ P (tf) lies above the line t ↦ h(μ) + t ∫ fdμ. Comparing asymptotic derivatives gives

lim_{t_{0}→∞} \frac{d}{dt}P (tf) |_{t=t_{0}}≥ ∫ fdμ ≥ lim_{t_{0}→−∞} \frac{d}{dt}P (tf) |_{t=t_{0}} .

Since μ ∈ M was arbitrary, we have Ω(f) ⊂ \overline{D}.

Notation. If a ∈ Ω(f), we denote M(f, a) = {μ ∈ M : ∫ f dμ = a} .

Lemma 3. Let T : K → K be a piecewise-onto expanding map. Suppose f ∈ H is not an essential coboundary for T, and a ∈ int(Ω(f)). Then

(a) the equilibrium state m_{t_{0}f} belongs to M(f, a), where t_{0} ∈ is the unique value satisfying a = \frac{d}{dt}P (tf)|_{t=t_{0}} = ∫ f dm_{t_{0}f},

(b) m = m_{t_{0}f} is the unique measure in M(f, a) satisfying h(m) = sup_{μ∈M(f,a)} h(μ),

(c) h(m_{t_{0}f}) = P (t_{0}(f − a)),

(d) the entropy map t ↦ h(m_{tf}) is analytic, strictly increasing on (−∞, 0] and strictly decreasing on [0, ∞).

Proof. (a) is clear by (2) and Lemma 2.

(b) Suppose μ ∈ M(f, a) with μ ≠ m_{t_{0}f}. The equilibrium state m_{t_{0}f} satisfies

∫ t_{0}f dm_{t_{0}f} + h(m_{t_{0}f}) > ∫ t_{0}f dμ + h(μ) .  (4)

Now μ, m_{t_{0}f} ∈ M(f, a), so ∫ t_{0}fdμ = t_{0}a = ∫ t_{0}fdm_{t_{0}f}. Thus (4) reduces to h(m_{t_{0}f}) > h(μ).

(c) Since m_{t_{0}f} = m_{t_{0}(f−a)} we have

P (t_{0}(f − a)) = ∫ t_{0}(f − a)dm_{t_{0}(f−a)} + h(m_{t_{0}(f−a)})
= ∫ t_{0}(f − a)dm_{t_{0}f} + h(m_{t_{0}f})
= h(m_{t_{0}f}).

(d) Analyticity follows from the analyticity of pressure and the relation

\begin{aligned}h(m_{t_{0}f}) & = P (t_{0}f) − t_{0} ∫ f dm_{t_{0}f} \\ & = P (t_{0}f) − t_{0} \frac{d}{dt}P (tf) |_{t=t_{0}} . \end{aligned}

A calculation gives

\frac{d}{dt}h(m_{tf}) |_{t=t_{0}}= −t_{0} \frac{d^{2}}{dt^{2}} P (tf) |_{t=t_{0}},

and the result follows from strict convexity of pressure.

Section 4. The barycentre set has interior.

In this section we prove (Theorem B) that if T is a piecewise-onto expanding map then its barycentre set has non-empty interior, or equivalently that not every barycentre lies on the boundary ∂Ω. We parametrise ∂Ω by θ ∈ K, which we think of as indexing a normal to ∂Ω in the 2πθ direction.

We let q(θ) = sup_{w∈Ω}⟨w, e^{2πiθ}⟩ denote the maximal component in the 2πθ direction, and w(θ) = {w ∈ Ω : ⟨w, e^{2πiθ}⟩ = q(θ)} the set of barycentres achieving this maximum. Here we are identifying with ^{2}, and ⟨·, ·⟩ denotes the usual inner product in ^{2}.

Clearly any w(θ) must lie in ∂Ω, and is either a point or a line segment, and conversely any point of ∂Ω must belong to some w(θ).

We remark that q(θ) = sup_{μ∈M} ∫ f_{θ}dμ, where f_{θ} : K → is defined by f_{θ}(x) = cos 2π(x − θ) = ⟨e^{2πix}, e^{2πiθ}⟩. This motivates the following definition.

Definition 2. Given f ∈ H(K, ) we define its optimal ergodic average Q(f) by

Q(f) = sup_{μ∈M} ∫ f dμ = sup Ω(f).

A measure m ∈ M is called f-optimal if ∫ fdm = Q(f).

The weak^{∗} compactness of M guarantees that an optimal measure always exists, though it need not be unique. By the same proof as used in Lemma 2 we know that

Q(f) = lim_{t_{0}→∞} ∫ f dm_{t_{0}f} = lim_{t_{0}→∞} \frac{d}{dt}P (tf)|_{t=t_{0}} .  (5)

Thus any weak^{∗} limit point of m_{tf}, as t → ∞, is an optimal measure. Bousch [3] has a proof that a generic f ∈ H(K, ) has a unique optimal measure (in fact his argument holds for a wider class of f and T than considered here). In this case the optimal measure is precisely lim_{t→∞} m_{tf}.

The following type of lemma was first proved by Mañé [15] in the context of Lagrangian systems. Contreras, Lopes and Thieullen [7] proved the same result for orientationpreserving C^{1+ϵ} expanding circle maps. The result for piecewise-onto expanding maps is essentially contained in Bousch [2]. Corollary 5 was already known (see Coelho [6]).

Lemma 4. Let T : K → K be a piecewise-onto expanding map, and suppose f ∈ H(K, ). A measure μ ∈ M is f-optimal if and only if f is cohomologous to the optimal ergodic average Q(f) on the support of μ.

Corollary 5. Let T : K → K be a piecewise-onto expanding map. Suppose f ∈ H(K, ) is not an essential coboundary for T. Any optimal measure for f cannot be fully supported.

Proof of Theorem B. Now Q(f_{θ}) = sup_{μ∈M} ∫ f_{θ} dμ = q(θ), the maximal component in the 2πθ direction. Thus b(μ) ∈ ∂Ω if and only if μ is f_{θ}-optimal for some θ ∈ K.

Now if Ω had empty interior then every barycentre would lie on ∂Ω, including the barycentre of the measure of maximal entropy m_{0}. So m_{0} would be f_{θ}-optimal for some θ ∈ K. But m_{0} is fully supported, so by Corollary 5, f_{θ} must be an essential coboundary. That is,

f_{θ} − a = u · T − u  (6)

for some bounded measurable function u : K → and a ∈ . Then a must be the value of f_{θ} at the fixed point(s) of T, so the left hand side of (6) has exactly 2 zeros. We will argue that the right hand side has at least 3 zeros.

This is immediate in the case when T has at least 3 fixed points. If T has either one or two fixed points, we first use the Livsic regularity theorem [13], [14] (see also [19], [25]) to show that the transfer function u is (Hölder) continuous. We then use the following intermediate value argument to guarantee the remaining zeros.

First suppose T has just one fixed point. Then T must be of topological degree two, and in particular is orientation-preserving. The fixed point is a zero of u · T − u, so it remains to find two more zeros.

Whenever we say a point is a maximum of u we will mean that u achieves its global maximum at that point. Suppose x_{0} is a maximum of u and is not the fixed point (if the maximum is unique and is the fixed point then we just apply an analogous argument to the global minimum). Then T x_{0} ≠ x_{0}, and T^{−1}x_{0} = {x_{1}, x_{2}}, where x_{1} < x_{0} < x_{2}. We will treat eight separate cases.

If x_{1} and x_{2} are maxima of u then they are both zeros of u · T − u, and we are done. If neither of x_{1}, x_{2} is a maximum, and T x_{0} is not a maximum, we obtain the inequal-

ities

(v · T − v)(x_{1}) > 0 > (v · T − v)(x_{0}) and (v · T − v)(x_{2}) > 0 > (v · T − v)(x_{0}).

The intermediate value theorem then gives zeros in both the intervals (x_{1}, x_{0}) and (x_{0}, x_{2}), and we are done.

If neither of x_{1}, x_{2} is a maximum, but both T x_{0} and T^{2}x_{0} are maxima, then both x_{0} and T x_{0} are zeros for u · T − u, and we are done.

If neither of x_{1}, x_{2} is a maximum, but T x_{0} is a maximum, but T^{2}x_{0} is not a maximum, then x_{0} is a zero of u · T − u, so it remains to find one more zero. Let x′_{0} ∈ T^{−1}T x_{0} satisfy x′_{0} ≠ x_{0}. Then either x_{0} < T x_{0} < x′_{0}, or x′_{0} < T x_{0} < x_{0}. If v(x′_{0}) = v(T x_{0}) then x′_{0} is a zero of u · T − u, and we are done. If not, then we have

(v · T − v)(x′_{0}) > 0 > (v · T − v)(T x_{0}),

so the intermediate value theorem guarantees a zero between x′_{0} and T x_{0}, and the previous inequalities imply this zero is not the point x_{0}. So we are done.

If x_{1} is a maximum, but x_{2} and T x_{0} are not maxima, then x_{1} is a zero of u · T − u, and similarly to previous cases we find another zero between x_{0} and x_{2}, so we are done.

An analogous argument applies if x_{2} is a maximum, but x_{1} and T x_{0} are not maxima. If x_{1} and T x_{0} are maxima, but x_{2} is not a maximum, then x_{1} and x_{0} are both zeros

of u · T − u, and we are done.

An analogous argument applies if x_{2} and T x_{0} are maxima, but x_{1} is not a maximum. This completes the case where T has a single fixed point.

Now suppose T has exactly two fixed points. These fixed points are both zeros of u · T − u, so it remains to find a third zero. The above arguments are easily modified to suit this case unless u has a unique global minimum, a unique global maximum, and these are attained precisely at the fixed points. In such a case we first note that one of the fixed points must lie on the boundary of the partition. So without loss of generality, let us suppose 0 is the fixed point, and the partition is given by 0 = a_{0} < a_{1} < … < a_{r} = 1. In particular note that T a_{i} = 0 for i = 0, …, r. Let x_{0} ≠ 0 be the other fixed point. Without loss of generality suppose x_{0} is the maximum of u and 0 is the minimum. We can choose some pre-image a_{j} ≠ 0 of 0, and some pre-image x_{1} ≠ x_{0} of x_{0} such that neither 0 nor x_{0} lies in between a_{j} and x_{1}. Then (v · T − v)(x_{1}) > 0 > (v · T − v)(a_{j}), so that v · T − v has a zero between a_{j} and x_{1}, and this zero is not one of the fixed points.

Section 5. Equilibrium rays.

Definition 3. For a fixed θ ∈ K we call the curve r_{θ} : t ↦ b(m_{tf_{θ}}), t ≥ 0, an equilibrium ray. We call

\begin{matrix}B : _{≥0} & × K → \\ (t, & θ) ↦ b(m_{tf_{θ}}) \end{matrix}

the radial map.

Note by Lemma 3 (d) that the entropy of the corresponding equilibrium states m_{tf_{θ}} is strictly decreasing along each ray.

Proposition 6. Let T : S^{1} → S^{1} be a piecewise-onto expanding map. The radial map B : (t, θ) ↦ b(m_{tf_{θ}}) is real-analytic.

Proof. We just note that m_{tf_{θ}} is a fixed point for the dual of the normalised transfer operator corresponding to the function tf_{θ} (see Parry & Pollicott [18]). Since the eigenvalue 1 is simple and isolated, it follows by standard perturbation theory (see Appendix V of [18]), that both the eigenvalue and the eigenvector are analytic functions of the parameters. So fixing θ, for example, we have that m_{tf_{θ}} is a weak^{∗} analytic function of t. That is, for any continuous function g the map t ↦ ∫ g dm_{tf_{θ}} is analytic. Similarly if we fix t then θ ↦ ∫ g dm_{tf_{θ}} is analytic.

Since b(m_{tf_{θ}}) = ∫ z dm_{tf_{θ}}(z), then choosing g(z) = z gives the result.

Corollary 7. Let T : S^{1} → S^{1} be a piecewise-onto expanding map. Each of the equilibrium rays r_{θ} is real-analytic.

Clearly all equilibrium rays r_{θ} share the same basepoint r_{θ}(0) = b(m_{0}), the barycentre of the unique measure of maximal entropy for T. The substance of the following Proposition 8 is that this is the only point where distinct rays intersect. However (see §7, §8) there are examples where rays intersect at infinity. That is, distinct rays can land at the same point on ∂Ω.

Proposition 8. Let T : S^{1} → S^{1} be a piecewise-onto expanding map. Then the restriction of the radial map B to ^{+} × K is injective. That is, no equilibrium ray selfintersects, and two distinct rays can only intersect at their common basepoint.

Proof. Suppose (t, θ) ≠ (t′, θ′). We claim the functions tf_{θ}, t′f_{θ′} are not essentially cohomologous with respect to T. To see this, suppose for a contradiction that

tf_{θ} = t′f_{θ′} + u · T − u + a

for some a ∈ and bounded measurable function u. Then

t ∫ f_{θ}dμ − t′ ∫ f_{θ′}dμ = a

for all μ ∈ M. That is,

⟨b(μ), te^{2πiθ} − t′e^{2πiθ′}⟩ = t⟨b(μ), e^{2πiθ}⟩ − t′⟨b(μ), e^{2πiθ′}⟩ = a

for all μ ∈ M. So all barycentres project to the same value a in the te^{2πiθ} − t′e^{2πiθ′} direction. This is a contradiction, since by Theorem B we know that Ω has an interior. Since tf_{θ}, tf_{θ′} are not essentially cohomologous, the corresponding equilibrium states m_{tf_{θ}}, m_{t′f_{θ′}} are not equal (see Parry & Pollicott [18], Prop. 3.6).

Now suppose, for a contradiction, that

b(m_{tf_{θ}}) = w = b(m_{t′f_{θ′}}).  (7)

Projecting w onto the 2πθ direction gives

∫ f_{θ} dm_{tf_{θ}} = ∫ f_{θ} dm_{t′f_{θ′}} = ⟨w, e^{2πiθ}⟩ =: c ,

so that m_{tf_{θ}} and m_{t′f_{θ′}} both belong to M(f_{θ}, c). By Lemma 3 (b) we know that m = m_{tf_{θ}} is the unique measure in M(f_{θ}, c) satisfying h(m) = sup_{μ∈M(f_{θ},c)} h(μ). Thus h(m_{tf_{θ}}) > h(m_{t′f_{θ′}}).

If instead we project (7) onto the 2πθ′ direction, a similar argument gives h(m_{t′f_{θ′}}) > h(m_{tf_{θ}}), giving the required contradiction.

Proposition 9. Let T : S^{1} → S^{1} be a piecewise-onto expanding map. The radial map

\begin{matrix}B : _{≥0} & × K → \\ (t, & θ) ↦ b(m_{tf_{θ}}) \end{matrix}

is a surjection onto the interior of the barycentre set Ω.

Proof. By Proposition 6 we know each equilibrium ray r_{θ} is a real-analytic, and hence continuous, curve. Note that r_{θ}(0) = b(m_{0}) is the barycentre of the measure of maximal entropy of T, which lies in the interior of Ω.