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Beyond Bowen’s Specification Property

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Thermodynamic Formalism

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2290))

Abstract

A classical result in thermodynamic formalism is that for uniformly hyperbolic systems, every Hölder continuous potential has a unique equilibrium state. One proof of this fact is due to Rufus Bowen and uses the fact that such systems satisfy expansivity and specification properties. In these notes, we survey recent progress that uses generalizations of these properties to extend Bowen’s arguments beyond uniform hyperbolicity, including applications to partially hyperbolic systems and geodesic flows beyond negative curvature. We include a new criterion for uniqueness of equilibrium states for partially hyperbolic systems with 1-dimensional center.

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Notes

  1. 1.

    In particular, this holds if X = M is compact and f is a transitive Anosov diffeomorphism.

  2. 2.

    The terminology in the literature for these different variants (weak specification, almost specification, almost weak specification, transitive orbit gluing, etc.) is not always consistent, and we make no attempt to survey or standardize it here. To keep our terminology as simple as possible, we just use the word specification for the version of the definition which is our main focus. In places where a different variant is considered, we take care to emphasize this.

  3. 3.

    For other purposes, and especially in the absence of any expansivity property, the difference between ≤ τ and = τ can be quite substantial, see for example [54, 55].

  4. 4.

    The notes at https://vaughnclimenhaga.wordpress.com/2020/06/23/specification-and-the-measure-of-maximal-entropy/ give a slightly more detailed version of this proof.

  5. 5.

    We will encounter this general principle multiple times: many of our proofs rely on obtaining uniform bounds (away from 0 and ∞) for quantities that a priori can grow or decay subexponentially.

  6. 6.

    This requires ergodicity of μ; one can also give a short argument directly from the definition of h μ(σ) that does not need ergodicity.

  7. 7.

    Since \(\mathcal {G}^M\) corresponds to a collection of orbit segments rather than a subset of the space, the most accurate analogy might be to think of \(\mathcal {G}^M\) as corresponding to orbit segments that start and end in a given regular level set.

  8. 8.

    The constant K M increases exponentially with the transition time in the specification property for \(\mathcal {G}^M\), so we do not expect any explicit relationship between M and K M in general. Examples of S-gap shifts (see Remark 1.9) can be easily constructed to make the constants \(K_M^{-1}\) decay fast.

  9. 9.

    Formally, I a = {x ∈ [0,  1) : ⌊βx⌋ = a}, so \(I_a = [\frac a\beta , 1)\) if a = ⌈β⌉− 1, and \([\frac a\beta , \frac {a+1}\beta )\) otherwise.

  10. 10.

    In the symbolic setting, this corresponds to X being a subshift of finite type.

  11. 11.

    The statements in [68] used 𝜖 ≥ 28δ but this must be corrected to 𝜖 > 40δ; see [2, §5.7].

  12. 12.

    See [68, Proposition 2.7] for a detailed proof that h top(X, f, ρ) = h top(X, f) in this case.

  13. 13.

    Use specification to get \(y_n \in f^n(B_n(x,\delta )) \cap f^{-k_n}(B_n(q,\delta ))\) for 0 ≤ k n ≤ τ, choose k such that k n = k for infinitely many values of n, and let y be a limit point of the corresponding y n.

  14. 14.

    Note that \(f^{-\tau }(W^{ss}_\delta (q))\) intersects a local leaf of W cu in at most finitely many points, and thus intersects at most finitely many of the corresponding local leaves of W u; however, there are uncountably many of these corresponding to points that never enter B(q, ρ).

  15. 15.

    There is a clear analogy between what we are doing here and the notion of hyperbolic time introduced by Alves [81], and developed by Alves, Bonatti and Viana [82].

  16. 16.

    Observe that this is impossible if φ does not satisfy the Bowen property.

  17. 17.

    Another specification-based proof of uniqueness of the MME on surfaces without focal points was given by Gelfert and Ruggiero [89].

  18. 18.

    We note that the original formulation of expansivity for flows by Bowen and Walters [100] allows reparametrizations, which suggests that one might consider a potentially larger set in place of Γ𝜖 for expansive flows. The main motivation for allowing reparametrizations is to give a definition that is preserved under orbit equivalence. However, this is not relevant for our purposes. In our setup, the natural notion of expansivity would be to ask that there exists 𝜖 so that \(\mathrm {NE}(\epsilon , \mathcal {F})= \emptyset \). This definition is sufficient for the uniqueness results, and strictly weaker than Bowen–Walters expansivity, although it is not an invariant under orbit equivalence. See the discussion of kinematic expansivity in [37].

  19. 19.

    We mention that μ L(Reg) > 0 and that μ L|Reg is known to be ergodic. Ergodicity of μ L, which is a major open problem, is thus equivalent to the question of whether μ L(Sing) = 0.

  20. 20.

    Here, we are following a notation convention of Katok: when we say a geodesic, we mean oriented geodesic, and we are considering γ as a periodic orbit living in T 1 M.

  21. 21.

    For manifolds M with Dim(M) ≥ 2, we define λ: T 1 M → [0, ∞) as follows. Let H s, H u be the stable and unstable horospheres for v. Let \(\mathcal {U}^s_v \colon T_{\pi v} H^s \to T_{\pi v} H^s\) be the symmetric linear operator defined by \(\mathcal {U}(v)=\nabla _vN\), where N is the field of unit vectors normal to H on the same side as v. This determines the second fundamental form of the stable horosphere H s. We define \(\mathcal {U}^u_v \colon T_{\pi v} H^u \to T_{\pi v} H^u\) analogously. Then \(\mathcal {U}_v^u\) and \(\mathcal {U}_v^s\) depend continuously on v, \(\mathcal {U}^u\) is positive semidefinite, \(\mathcal {U}^s\) is negative semidefinite, and \(\mathcal {U}^u_{-v}=-\mathcal {U}^s_v\). For v ∈ T 1 M, let λ u(v) be the minimum eigenvalue of \(\mathcal {U}^u_v\) and let λ s(v) = λ u(−v). Let \(\lambda (v) = \min ( \lambda ^u(v), \lambda ^s(v))\).

    The functions λ u, λ s, and λ are continuous since the map \(v\mapsto \mathcal {U}^{u,s}_v\) is continuous, and we have λ u, s ≥ 0. When M is a surface, the quantities λ u, s(v) are just the curvatures at πv of the stable and unstable horocycles, and we recover the definition of λ stated above.

  22. 22.

    This allows us to use indicator functions of open sets, which is helpful in some applications.

  23. 23.

    We could also define the class of one-sided λ-decompositions by taking the longest initial segment in \(\mathcal {B}(\eta )\), declaring what is left over to be good, and setting \(\mathcal {S}=\emptyset \), or conversely by putting \(\mathcal {S} = \mathcal {B}(\eta )\) and \(\mathcal {P}=\emptyset \). This formalism is defined in [109]: the decompositions in Sect. 1.3.4.2 are examples of one-sided λ-decompositions.

  24. 24.

    In fact one can improve this estimate, but the formula is more complicated [20].

  25. 25.

    Formally, one needs to take a finite index subgroup of π 1(M) that avoids all non-identity elements corresponding to a large ball in \(\widetilde {M}\); this is possible because π 1(M) is residually finite.

  26. 26.

    This can also be formulated in terms of the Pinsker σ-algebra for μ, which can be thought of as the biggest σ-algebra with entropy 0: the measure μ has the K-property if and only if the Pinsker σ-algebra for μ is trivial.

  27. 27.

    In dimension 2, it is in fact an open problem whether Sing can contain non-periodic orbits [124], but this does not affect the argument that h(Sing) = 0.

  28. 28.

    The idea is that we want to split a word y [1,nN] into αN subwords and perform surgeries near the points where it was split; these are the “on” points in A.

  29. 29.

    Each such window determined by the set J has length some multiple of n. The surgery procedure is to remove the last t + 2τ symbols from each window and replace with a word of the form v 1 wv 2 where the words v j are provided by the specification property to ensure that this procedure creates a word in \(\mathcal {L}_{nN}(X)\).

  30. 30.

    https://vaughnclimenhaga.wordpress.com/2017/01/26/entropy-bounds-for-equilibrium-states/.

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Acknowledgements

Vaughn Climenhaga is partially supported by NSF DMS-1554794. D.T. is partially supported by NSF DMS-1461163 and DMS-1954463.

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  1. Department of Mathematics, University of Houston, Houston, TX, USA

    Vaughn Climenhaga

  2. Department of Mathematics, Ohio State University, Columbus, OH, USA

    Daniel J. Thompson

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  1. Department of Mathematics, Warwick University, Coventry, UK

    Mark Pollicott

  2. Institute de Mathématiques de Toulon, Toulon, France;, Centre de Physique Théorique de Luminy, Marseille, France

    Sandro Vaienti

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Climenhaga, V., Thompson, D.J. (2021). Beyond Bowen’s Specification Property. In: Pollicott, M., Vaienti, S. (eds) Thermodynamic Formalism. Lecture Notes in Mathematics, vol 2290. Springer, Cham. https://doi.org/10.1007/978-3-030-74863-0_1

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