Systèmes dynamiques/Dynamical systems

Directional Entropy of Rotation Sets

Oliver JENKINSON

Abstract.- For (X, T) a mixing subshift of finite type, and f : X → ℝ^{d} of summable variation, we give a characterisation of directional entropy (see [4]) at interior and exposed points of the corresponding generalised rotation set.

Entropie Directionnelle des Ensembles de Rotation

Résumé.- Pour (X, T) un sous-shift de type fini mélangeant, et f : X → ℝ^{d} une fonction de variation sommable, on caractérise l’entropie directionnelle (voir [4]) aux points intérieur à l’ensemble de rotation, et aux points exposés de l’ensemble de rotation.

Version Française Abrégée - Soit (X, T) un système dynamique donné par une surjection continue T : X → X où X est compact et métrisable. Soit f_{1}, …, f_{d} les fonctions coordonnées de la fonction continue f : X → ℝ^{d}. Si M est l’ensemble des mesures de probabilité boréliennes qui sont invariantes par T, alors f permet de définir l’application f_{∗} : M → ℝ^{d} :

f_{∗}(μ) = (∫ f_{1} dμ, …, ∫ f_{d} dμ) .

On appelle f_{∗}(μ) le vecteur de rotation de la mesure μ ∈ M. L’image f_{∗}(M), appelée ensemble de rotation, est compacte et convexe. La terminologie est motivée par le cas classique des vecteurs de rotation d’une application du tore (voir [9], [13]). Ici l’on définie f = F π^{−1} − π^{−1}, où F est un relèvement de l’application au revêtement universel ℝ^{d}, et π^{−1} est une branche inverse du revêtement.

D’autres exemples intéressants d’ensembles de rotation sont les suivants : la boule unité pour la norme stable de l’espace d’homologie réelle de certains surfaces [11], [12]; l’ensemble des barycentres ∫_{S^{1}} z dμ(z) des mesures de probabilité invariantes sous une application du cercle [1], [5], [6], [7]; le spectre des exposants de Lyapunov pour une application unidimensionnelle [2].

Geller & Misiurewicz [4] ont récemment introduit les notions de mesures ergodiques directionnelles et perdues. Une mesure ergodique est appelée directionnelle si toutes les mesures ayant leur support contenu dans le support de μ ont même vecteur de rotation que μ. Sinon, une mesure ergodique μ est appellée perdue. L’entropie directionnelle H(ϱ) d’un vecteur de rotation ϱ est ainsi définie comme supremum des entropies des mesures directionnelles dans f_{∗}^{−1}(ϱ), (où on définit le supremum de l’ensemble vide comme nul). On définit aussi H(ϱ) comme le supremum des entropies sur toutes les mesures dans f_{∗}^{−1}(ϱ).

Geller & Misiurewicz [4] considèrent en détail le cas d’un sous-shift de type fini mélangeant, où la fonction f est constante sur les cylindres de longueur un. Fan & Feng [3] ont traité le cas plus général où f est n’importe quelle fonction localement constante.

Pour énoncer nos résultats, il faut d’abord se rappeler de quelques notions du formalisme thermodynamique. La pression d’une fonction continue g : X → ℝ est définie par P (g) = sup_{μ∈M} (h(μ) + ∫ g dμ), et toute mesure m ∈ M qui atteint ce suprémum est appelée un état d’équilibre pour g. Pour v ∈ ℝ^{d}, soit v.f la fonction v_{1}f_{1} + … + v_{d}f_{d}. On définit p : ℝ^{d} → ℝ par p(v) = P (v.f). Soit M_{f}(v) l’ensemble des états d’équilibre de v.f. Soit ∂p(v) la sous-dérivée de p au point v (voir [15]).

Théorème 1. Soit (X, T) un système dynamique pour lequel l’application d’entropie h : M → ℝ est semi-continue supérieurement, et soit f : X → ℝ^{d} continue. Alors (a) f_{∗}(M) = \overline{f_{∗}(M_{f}(ℝ^{d}))} = \overline{∂p(ℝ^{d})}.

Si ϱ appartient à l’intérieur rélatif ri(f_{∗}(M)), alors

(b) il existe v ∈ ℝ^{d} tel que f_{∗}^{−1}(ϱ) ∩ M_{f}(ℝ^{d}) ⊂ M_{f}(v), et

(c) m ∈ f_{∗}^{−1}(ϱ) satisfait h(m) = H(ϱ) si et seulement si m ∈ f_{∗}^{−1}(ϱ) ∩ M_{f}(ℝ^{d}).

Notez que h est semi-continue supérieurement si, par exemple, T est une application C^{∞} d’une variété compacte, ou T est n’importe quel sous-shift symbolique.

Théorème 2. Soit (X, T) un sous-shift de type fini mélangeant (unidirectionnel ou bidirectionnel). Soit f de variation sommable, où les fonctions coordonnées f_{i} sont cohomologiquement indépendantes. Les fonctions H et H coïncident dans l’intérieur de f_{∗}(M), et sur l’ensemble des points exposés. Pour tout point ϱ de l’intérieur, H(ϱ) = H(ϱ) est atteint par une seule mesure dans f_{∗}^{−1}(ϱ), et cette mesure est perdue. Pour les points ϱ exposés, H(ϱ) = H(ϱ) est atteint par au moins une mesure directionnelle dans f_{∗}^{−1}(ϱ), et par aucune mesure perdue.

La partie la plus importante du théorème 2, ainsi que le résultat principal de cette Note, est le fait que les deux entropies H et H sont (essentiellement) égales.

On se réfèra à [14] pour la définition d’un sous-shift de type fini mélangeant. On se rappelle que variation sommable veut dire que ∑_{n=1}^{∞} var_{n}(f) < ∞, où

var_{n}(f) := max{|f(x)−f(y)| : x, y appartiennent au même cylindre de longueur n}.

Indépendance cohomologique signifie que toute combinaison linéaire non-triviale des f_{i} n’est pas égale à ψ ∘ T − ψ + c pour ψ Borel-mesurable, et c ∈ ℝ. Un vecteur de rotation ϱ est dit exposé si {ϱ} est une face de f_{∗}(M). Un point exposé est en particulier un point extrême (voir [15]). Pour un ensemble strictement convexe, tout point sur le bord est un point exposé, donc on a

Corollaire. Soit (X, T) un sous-shift de type fini mélangeant (unidirectionnel ou bidirectionnel). Soit f de variation sommable, où les fonctions coordonnées f_{i} sont cohomologiquement indépendantes. Si l’ensemble de rotation f_{∗}(M) est strictement convexe, alors les fonctions H et H coïncident.

On trouvera plus de détails sur les résultats de cette Note, ainsi que des références plus larges, dans [8]. Je remercie Thierry Bousch, Ai Hua Fan, et Evgeny Verbitski pour leurs remarques pertinentes sur ce travail.

1. Introduction

Consider a dynamical system (X, T), given by a continuous surjection T : X → X of a compact metrizable space X. Suppose the continuous function f : X → ℝ^{d} has coordinate functions f_{1}, …, f_{d}. If M denotes the set of T -invariant Borel probability measures, then f induces the map f_{∗} : M → ℝ^{d} given by

f_{∗}(μ) = (∫ f_{1} dμ, …, ∫ f_{d} dμ) .

We call f_{∗}(μ) the rotation vector of the measure μ ∈ M. The image f_{∗}(M), called the rotation set, is compact and convex.

Rotation sets arise in various contexts: rotation sets of torus maps [9], [13]; norm balls for the stable norm on the real homology space of various surfaces [11], [12]; the set of barycentres ∫_{S^{1}} z dμ(z) of invariant probability measures for a circle map [1], [5], [6], [7]; the spectrum of Lyapunov exponents for one-dimensional maps [2].

Geller & Misiurewicz [4] call an ergodic measure μ directional if all invariant measures whose support is contained in the support of μ have the same rotation vector as μ. Otherwise an ergodic μ is called lost. The directional entropy H(ϱ) of a rotation vector ϱ is then defined as the supremum of the entropies of those directional measures in f_{∗}^{−1}(ϱ), (where the supremum of the empty set is defined to be zero). Define also H(ϱ) to be the supremum of the entropies over all measures in f_{∗}^{−1}(ϱ). Clearly H(ϱ) ≤ H(ϱ) for all ϱ.

Geller & Misiurewicz [4] consider in detail the case of a mixing subshift of finite type, where the function f is constant on length-one cylinder sets. Fan & Feng [3] consider the more general case where f is any locally constant function.

In order to state our results we first recall some thermodynamic ideas. The pressure of a continuous function g : X → ℝ is given by P (g) = sup_{μ∈M} (h(μ) + ∫ g dμ), and any m ∈ M realising this supremum is called an equilibrium state for g. For v ∈ ℝ^{d}, let v.f denote the function v_{1}f_{1} + … + v_{d}f_{d}, and define p : ℝ^{d} → ℝ by p(v) = P (v.f). Let M_{f}(v) denote the set of equilibrium states of v.f. Let ∂p(v) denote the subdifferential of p at v (see [15]).

Theorem 1. Suppose (X, T) is a dynamical system with upper semi-continuous entropy map h : M → ℝ, and f : X → ℝ^{d} is continuous. Then

(a) f_{∗}(M) = \overline{f_{∗}(M_{f}(ℝ^{d}))} = \overline{∂p(ℝ^{d})}.

If the rotation vector ϱ lies in the relative interior ri(f_{∗}(M)), then

(b) there exists v ∈ ℝ^{d} with f_{∗}^{−1}(ϱ) ∩ M_{f}(ℝ^{d}) ⊂ M_{f}(v), and

(c) m ∈ f_{∗}^{−1}(ϱ) satisfies h(m) = H(ϱ) if and only if m ∈ f_{∗}^{−1}(ϱ) ∩ M_{f}(ℝ^{d}).

Note that upper semi-continuity of h is guaranteed, for example, if T is a C^{∞} map of a compact manifold, or if T is any symbolic subshift.

Theorem 2. Let (X, T) be a (one-sided or two-sided) mixing subshift of finite type. Let f : X → ℝ^{d} be of summable variation, where its coordinate functions f_{i} are cohomologically independent. Then H(ϱ) = H(ϱ) at all interior and exposed points ϱ of the rotation set f_{∗}(M). If ϱ lies in the interior of f_{∗}(M), then H(ϱ) = H(ϱ) is attained by a unique measure in f_{∗}^{−1}(ϱ), and this measure is lost. If ϱ is exposed, then H(ϱ) = H(ϱ) is attained by at least one directional measure in f_{∗}^{−1}(ϱ), and is not attained by any lost measures.

See [14] for the definition of a mixing subshift of finite type. Recall that summable variation means ∑_{n=1}^{∞} var_{n}(f) < ∞, where

var_{n}(f) := max{|f(x) − f(y)| : x, y belong to the same length-n cylinder}.

Cohomological independence means that each non-trivial linear combination of the f_{i} cannot equal ψ ∘ T − ψ + c (an essential coboundary) for Borel measurable ψ and c ∈ ℝ. It ensures f_{∗}(M) has interior. A rotation vector ϱ is called exposed if {ϱ} is a face of f_{∗}(M). Exposed points are in particular extremal points (see [15]). For a strictly convex set, every point on the boundary is an exposed point, so we have

Corollary. Let (X, T) and f : X → ℝ^{d} be as in Theorem 2. If the rotation set f_{∗}(M) is strictly convex, then the functions H and H coincide.

2. Proof of Theorem 1

(a) Clearly f_{∗}(M_{f}(ℝ^{d})) ⊂ f_{∗}(M), since M_{f}(ℝ^{d}) ⊂ M. Compactness of f_{∗}(M) means in fact \overline{f_{∗}(M_{f}(ℝ^{d}))} ⊂ f_{∗}(M).

We can show (see Prop. 1, [8]) that ∂p(v) = f_{∗}(M_{f}(v)) for all v ∈ ℝ^{d}. The ingredients for this are the representation of an equilibrium state as a tangent functional to pressure, the upper semi-continuity of h, and a little convex geometry. We therefore have ∂p(ℝ^{d}) = f_{∗}(M_{f}(ℝ^{d})), and hence \overline{∂p(ℝ^{d})} = \overline{f_{∗}(M_{f}(ℝ^{d}))}.

To show that f_{∗}(M) ⊂ \overline{∂p(ℝ^{d})}, let μ ∈ M be arbitrary. Defining

A_{μ}(v_{1}, …, v_{d}) = h(μ) + v_{1} ∫ f_{1} dμ + … + v_{d} ∫ f_{d} dμ = h(μ) + v.f_{∗}(μ),

the definition of pressure implies that p(v) ≥ A_{μ}(v) for all v ∈ ℝ^{d}. Thus Graph(p) lies above the hyperplane Graph(A_{μ}) in ℝ^{d+1}, possibly touching it tangentially.

This implies that the gradient of the hyperplane is contained in the closure of all subdifferentials of p, since if not then the graphs would intersect transversally. That is, the vector f_{∗}(μ) belongs to the closure of ∂p(ℝ^{d}). Since μ ∈ M was arbitrary then f_{∗}(M) ⊂ \overline{∂p(ℝ^{d})}.

(b) Note that f_{∗}^{−1}(ϱ) ∩ M_{f}(ℝ^{d}) is non-empty by the above. Suppose (b) is false. Then there exists v ≠ v′, and a measure m which belongs to M_{f}(v) but not to M_{f}(v′), and a measure m′ which belongs to M_{f}(v′) but not to M_{f}(v), but such that m, m′ ∈ f_{∗}^{−1}(ϱ). This implies that ∫ v.f dm = ∫ v.f dm′, but the fact that m is an equilibrium state for v.f, yet m′ is not, gives h(m) > h(m′). Similarly we have ∫ v′.f dm = ∫ v′.f dm′, and now the fact that m′ is an equilibrium state for v′.f, yet m is not, gives h(m′) > h(m). This is a contradiction.

(c) Let m ∈ f_{∗}^{−1}(ϱ)∩M_{f}(ℝ^{d}), and let v ∈ ℝ^{d} be such that m ∈ M_{f}(v). Now let μ be some measure in f_{∗}^{−1}(ϱ) which does not belong to the family M_{f}(ℝ^{d}). Then ∫ v.f dm = ∫ v.f dμ, but the fact that m is an equilibrium state for v.f, whereas μ is not, gives h(m) > h(μ).

3. Proof of Theorem 2

For (X, T) and f as in the statement of Theorem 2, the function p is strictly convex, so that ∂p(ℝ^{d}) = f_{∗}(M_{f}(ℝ^{d})) is both open in ℝ^{d} and convex. Taking interiors in Theorem 1 gives that int(f_{∗}(M)) = f_{∗}(M_{f}(ℝ^{d})).

First let ϱ ∈ int(f_{∗}(M)). We claim there exists m ∈ f_{∗}^{−1}(ϱ) ∩ M_{f}(ℝ^{d}) with h(m) = H(ϱ), and that m is the unique measure in f_{∗}^{−1}(ϱ) with this property. By Theorem 1 we know there exists v ∈ ℝ^{d} with f_{∗}^{−1}(ϱ)∩M_{f}(ℝ^{d}) ⊂ M_{f}(v), Now suppose v′ ≠ v satisfies f_{∗}^{−1}(ϱ) ∩ M_{f}(ℝ^{d}) ⊂ M_{f}(v′). Since the f_{i} are cohomologically independent then v.f and v′.f are not essentially cohomologous, so the equilibrium states m_{v.f}, m_{v′.f} are distinct [16], Cor. 3.3(ii). So the argument used to prove Theorem 1 (b) again provides a contradiction. Thus there exists a unique v ∈ ℝ^{d} for which f_{∗}^{−1}(ϱ) ∩ M_{f}(ℝ^{d}) ⊂ M_{f}(v). But v.f is of summable variation, so it has a unique equilibrium state m, [16], Cor. 3.3(i). That is, f_{∗}^{−1}(ϱ) ∩ M_{f}(ℝ^{d}) = {m}. The fact that m is the unique measure in f_{∗}^{−1}(ϱ) satisfying h(m) = H(ϱ) then follows immediately from Theorem 1 (c).

The equilibrium state m is fully supported ([16], Cor. 3.3(i)). It follows that m is lost, since if not then f_{∗}(M) would be the singleton {ϱ}. But this cannot be, since a theorem of Livsic [10] (the proof in [14], Prop. 3.7 is easily adapted to the case of functions of summable variation) would imply that each coordinate function f_{i} is an essential coboundary, a contradiction.

It remains to show that H(ϱ) = H(ϱ). For any 0 < α < H(ϱ) we will construct a subshift Y_{α} ⊂ X, with topological entropy h_{top}(Y_{α}) > H(ϱ) − α, and such that

sup_{y∈Y_{α}} | ∑_{i=0}^{n−1} f(T^{i}y) − nϱ| = O(1) as n → ∞.  (1)

Consequently any ergodic measure supported on Y_{α} will be directional, and in the fibre f_{∗}^{−1}(ϱ). Moreover, the variational principle (see Walters [17], [18]) means we can find an ergodic measure μ_{α}, with supp(μ_{α}) ⊂ Y_{α}, such that h(μ_{α}) > H(ϱ) − α.

By a periodic block of length n we mean an n-tuple of symbols b = (b^{(1)}, …, b^{(n)}), such that A(b^{(i)}, b^{(i+1)}) = 1 for i = 1, …, n − 1, and A(b^{(n)}, b^{(1)}) = 1, where A is the transition matrix defining our subshift of finite type X. Each periodic block naturally corresponds to a unique measure μ ∈ M, so we define f_{∗}(b) = f_{∗}(μ), and call this the rotation vector of the block b.

Fix some q ∈ ℕ such that q > \frac{2(H(ϱ)−α)}{α}. Choose r > 0 such that B_{3rq}(ϱ) ⊂ f_{∗}(M), (where B_{t}(ϱ) denotes the open Euclidean ball of radius t around the point ϱ).

Fix ε > 0 such that ε < rq. We can also choose a natural number N satisfying the following four conditions:

(a) N is an integer multiple of q,

(b) V (f)/N < r − ε/q,

(c) There is a collection C of periodic blocks, each of length M := N/q, each beginning with the same symbol a, whose rotation vectors are ε-dense in f_{∗}(M),

(d) There is a collection B of length-N periodic blocks, of cardinality |B| ≥ e^{(H(ϱ)−α/2)N}, such that each b ∈ B begins with the symbol a and satisfies f_{∗}(b) ∈ B_{r}(ϱ).

To each \underline{b} = (b_{i}) ∈ B^{ℕ} we will identify a collection Z(\underline{b}) of points x ∈ X of the form x = b_{1}c_{1}b_{2}c_{2} …, where c_{i} ∈ C. Given \underline{b}, the j^{th} block c_{j} will depend on the b_{i}, i ≤ j, and c_{i}, i < j, and will be chosen to maintain |f^{n}(x) − nϱ| = O(1) (rather than the |f^{n}(x) − nϱ| = O(n), which would occur if x were constructed solely from blocks in B). We think of the blocks c ∈ C as being corrective blocks, since the c_{i} are chosen so as to reduce the distances |f^{n}(x) − nϱ|. For any \underline{b} ∈ B^{ℕ} we can show there exists at least one x of the above form, so that Z(\underline{b}) is non-empty. We then define Z to be the union of the Z(\underline{b}), and define Y_{α} = \overline{⋃_{i≥0} T^{i}(Z)}. In [8] we show that (1) holds, and that h_{top}(Y_{α}) > H(ϱ) − α.

Now let ϱ be an exposed point of f_{∗}(M). So there exists some normal v ∈ ℝ^{d} such that the supporting hyperplane {w ∈ ℝ^{d} : w.v = w.ϱ} intersects f_{∗}(M) at the single point ϱ. Thus μ ∈ f_{∗}^{−1}(ϱ) if and only if μ is v.f-optimal (for a continuous g : X → ℝ, we say μ ∈ M is g-optimal if ∫ g dμ = sup_{m∈M} ∫ g dm). But then a theorem of Bousch (see [1], [8]) implies that any m ∈ M with supp(m) ⊂ supp(μ) is also v.f-optimal, and hence that f_{∗}(m) = f_{∗}(μ). So if our μ ∈ f_{∗}^{−1}(ϱ) is ergodic then it is directional.

In fact f_{∗}^{−1}(ϱ) must contain at least one ergodic (and hence directional) measure, by the following argument. Since ϱ is exposed then it is also extremal in f_{∗}(M). Let ν ∈ M be an extremal point of the convex set f_{∗}^{−1}(ϱ). Suppose ν is not ergodic. Then there exist distinct measures ν_{1}, ν_{2} ∈ M and α ∈ (0, 1) such that

ν = αν_{1} + (1 − α)ν_{2}.  (2)

Thus f_{∗}(ν) = αf_{∗}(ν_{1})+(1−α)f_{∗}(ν_{2}), and extremality of ϱ = f_{∗}(ν) implies f_{∗}(ν_{1}) = ϱ = f_{∗}(ν_{2}). But now since ν_{1}, ν_{2} ∈ f_{∗}^{−1}(ϱ), (2) implies ν is not extremal in f_{∗}^{−1}(ϱ), a contradiction. Thus ν must be ergodic.

Now f_{∗}^{−1}(ϱ) is closed, and hence compact, so that there exists m ∈ f_{∗}^{−1}(ϱ) with h(m) = H(ϱ). But extremality of ϱ means that almost every measure in the ergodic decomposition of m must belong to f_{∗}^{−1}(ϱ), so since entropy h is affine then there is some ergodic (and hence directional) μ ∈ f_{∗}^{−1}(ϱ) with h(μ) ≥ h(m) = H(ϱ). Thus H(ϱ) ≥ H(ϱ). □.

References

[1] T. Bousch, Le poisson n’a pas d’arêtes, Ann. Inst. Henri Poincaré, Proba. et Stat., 36 (2000), 489–508.

[2] G. Contreras, A. Lopes and Ph. Thieullen, Lyapunov minimizing measures for expanding maps of the circle, preprint, Orsay, 1999.

[3] A. H. Fan and D. J. Feng, Analyse multifractale de la récurrence sur l’espace symbolique, C. R. Acad. Sci. Paris Sr. I Math., 327 (1998), 629–632.

[4] W. Geller and M. Misiurewicz, Rotation and entropy, Trans. Amer. Math. Soc., 351 (1999), 2927–2948.

[5] O. Jenkinson, Conjugacy rigidity, cohomological triviality, and barycentres of invariant measures, Ph.D. thesis, Warwick University, available at http://maths.qmw.ac.uk/∼omj

[6] O. Jenkinson, Geometric barycentres of invariant measures for circle maps, Ergod. Th. Dynam. Sys., to appear.

[7] O. Jenkinson, Frequency-locking on the boundary of the barycentre set, Experimental Mathematics, 9 (2000), 309–317.

[8] O. Jenkinson, Rotation, entropy, and equilibrium states, Trans. Amer. Math. Soc., to appear.

[9] S. Kim, R. S. MacKay, J. Guckenheimer, Resonance regions for families of torus maps, Nonlinearity, 2 (1989), 391–404.

[10] A. Livsic, Homology properties of Y -systems, Math. Zametki, 10 (1971), 758–763.

[11] G. McShane and I. Rivin, Simple curves on hyperbolic tori, C. R. Acad. Sci. Paris Sr. I Math., 320 (1995), 1523–1528.

[12] D. Massart, Normes stables des surfaces, Ph.D. Thesis, ENS Lyon, 1996.

[13] H. Poincaré, Sur les courbes définies par les équations différentielles, Œuvres Complètes, tome 1, Gauthier-Villars, Paris, 1952, 137–158.

[14] W. Parry and M. Pollicott, Zeta functions and the Periodic Orbit Structure of Hyperbolic Dynamics, Astérisque, 187–188, 1990.

[15] R. T. Rockafeller, Convex Analysis, Princeton University Press, 1970.

[16] P. Walters, Ruelle’s operator theorem and g-measures, Trans. Amer. Math. Soc., 214 (1975), 375–387.

[17] P. Walters, A variational principle for the pressure of continuous transformations, Amer. J. Math., 97 (1976), 937–971.

[18] P. Walters, An Introduction to Ergodic Theory, Springer-Verlag, 1982.

Oliver Jenkinson, School of Mathematical Sciences, Queen Mary & Westfield College, Mile End Road, London E1 4NS, UK.

Email: omj@maths.qmw.ac.uk  Web: http://maths.qmw.ac.uk/∼omj