The K-Property for Subadditive Equilibrium States
aa r X i v : . [ m a t h . D S ] J u l THE K -PROPERTY FOR SUBADDITIVE EQUILIBRIUM STATES BENJAMIN CALL, KIHO PARK
Abstract.
By generalizing Ledrappier’s criterion [Led77] for the K -property of equilibriumstates, we extend the criterion to subadditive potentials. We apply this result to the singularvalue potentials of matrix cocycles, and show that equilibrium states of large classes of singularvalue potentials have the K -property. Introduction
Given a continuous potential ϕ : X → R over a dynamical system ( X, f ) , its pressure may bedefined as P ( ϕ ) = sup µ ∈M ( f ) n h µ ( f ) + Z ϕ dµ o and we call the f -invariant measures achieving the supremum equilibrium states . These play animportant role in the study of the dynamical system ( X, f ) . Provided the entropy map µ h µ ( f ) is upper semicontinuous, these equilibrium states exist. However, without further assumptions onthe potential ϕ or the base dynamical system ( X, f ) , their uniqueness is not guaranteed.On the other hand, in his fundamental work, Bowen [Bow74] established the following result thatguarantees the existence and uniqueness of equilibrium states. Given a potential with a regularitycondition, later named the Bowen property, over an expansive dynamical system with specification,the system has a unique equilibrium state; see Proposition 2.13 and 2.14 for more details. Suchunique equilibrium states are now well-studied with various known constructions as well as strongergodic and statistical properties; see [Bow74, Bow75, Rat73, Rue76, PP90]. An important examplethat fits into this framework consists of Hölder potentials over uniformly hyperbolic systems.Since then, the theory has been extended in mainly two different directions. One direction aimsto relax the uniform hyperbolicity of the base dynamics; see for instance [Kni98, CT16, BCFT18,CKP20]. The other aims to relax and generalize the assumptions on the potential. In particular,much attention was recently brought to the subadditive generalization of thermodynamic formalismdue to its applications to the dimension theory of fractals; see for instance [Fal88, Zha97, CP10,BCH10, FS14, BHR19] and references therewithin. In this paper, we pursue the latter generalizationand study ergodic properties of the subadditive equilibrium states.Denoting a mixing subshift of finite type by (Σ T , σ ) and the full shift by (Σ , σ ) , consider asequence of continuous functions Φ = { log ϕ n } n ∈ N on Σ T . We say Φ is subadditive if log ϕ m + n ≤ log ϕ m + log ϕ n ◦ σ m (1.1)for all m, n ∈ N . Associated to the subadditive potential Φ is the subadditive pressure P (Φ) introduced by Cao, Feng, and Huang [CFH08] by generalizing the usual definition of the pressurein thermodynamic formalism; see Section 2. In particular, the subadditive pressure P (Φ) satisfiesthe subadditive variational principle [CFH08]: P (Φ) = sup µ ∈M ( σ ) n h µ ( σ ) + lim n →∞ n Z log ϕ n dµ o . (1.2) Date : July 7, 2020.2010
Mathematics Subject Classification.
Key words and phrases.
Equilibrium states, Thermodynamic Formalism, Kolmogorov property.B.C. is partially supported by NSF grant DMS- . Any σ -invariant measure µ ∈ M ( σ ) achieving the supremum in (1.2) is called an equilibrium state of Φ .Denoting by L the set of all admissible words of Σ T , we associate to a subadditive potential Φ = { log ϕ n } n ∈ N a function e Φ :
L → R defined by e Φ( I ) := sup x ∈ [ I ] ϕ n ( x ) . (1.3)We say a subadditive potential Φ = { log ϕ n } n ∈ N is quasi-multiplicative if there exists c > and k ∈ N such that for any I , J ∈ L , there exists K ∈ L with | K | ≤ k such that e Φ( IKJ ) ≥ c e Φ( I ) e Φ( J ) . Quasi-multiplicativity may be thought of as follows: given any two words I , J ∈ L of arbitrarylength, we obtain the inequality opposite to subadditivity (1.1) at a cost of inserting a connectingword K ∈ L of bounded length in between I and J . Quasi-multiplicativity is a rather mild assump-tion, and hence it is enjoyed by a large class of subadditive potentials; see Proposition 4.2 andProposition 4.11. A particularly important application is that, together with the bounded distor-tion property, quasi-multiplicativity serves as a sufficient condition to generalize Bowen’s theoremon the uniqueness of equilibrium states; see Proposition 2.17 for the precise statement.In this paper we study ergodic properties of unique equilibrium states guaranteed by quasi-multiplicativity and bounded distortion. Morris [Mor18] recently showed that if the unique equi-librium states associated to a class of matrix cocycles are totally ergodic, then they are actuallymixing. The main result of this paper is similar in flavor to Morris’s result. However, we workin a more general class of subadditive potentials, and show that under suitable assumptions, totalergodicity can be promoted to the K -property, which is stronger than mixing of all orders andweaker than Bernoulli. Theorem A.
Let
Φ = { log ϕ n } n ∈ N be a subadditive potential on Σ T , and suppose it is quasi-multiplicative and has bounded distortion. Suppose further that the unique equilibrium state µ ∈ M ( σ ) of Φ guaranteed by Proposition 2.17 is totally ergodic. Then, µ has the K -property.The key result used to prove this theorem is Theorem 3.5, which shows that for subadditiveequilibrium states, weak mixing is equivalent to the K -property under some suitable assumptions,similar to those used in [Bow74]. This holds even for non-symbolic systems, and we expect it to beof independent interest. The remaining results in this paper are obtained by applying Theorem Ato GL d ( R ) -cocycles, including locally constant cocycles and fiber-bunched cocycles. For any cocycle A : Σ T → GL d ( R ) , we define its norm potential Φ A = { log ϕ A ,n } n ∈ N as ϕ A ,n ( x ) := kA ( σ n − x ) . . . A ( x ) k . From the submultiplicativity of the operator norm k · k , it is clear that Φ A is subadditive. Thesingular value potentials are natural generalizations of the norm potentials; see Section 2 for theprecise definition.For irreducible locally constant cocyles A : Σ → M d × d ( R ) , it was shown by Feng [Fen09] thatnorm potentials Φ A have unique equilibrium states µ A ∈ M ( σ ) by establishing quasi-multiplicativity.Morris [Mor19] then obtained a characterization for µ A to be mixing under an extra assumptionthat at least one matrix in the image of A is invertible. This assumption is automatically met when A takes values in GL d ( R ) . We partly reformulate [Mor19, Corollary 3] for GL d ( R ) -cocycles. Proposition 1.1. [Mor19, Corollary 3]
Suppose A : Σ → GL d ( R ) is an irreducible locally constantcocycle. Then the following are equivalent:(1) The unique equilibrium state µ is mixing with respect to σ .(2) The measure µ is ergodic with respect to σ d .Remark . In both [Mor18] and [Mor19], Morris works with one-sided full shifts only, and so forbrevity, we limit the following result to full shifts, though we expect it to hold for shifts of finite HE K -PROPERTY FOR SUBADDITIVE EQUILIBRIUM STATES 3 type. The generalization to two-sided shifts follows from the theory of natural extensions, whichwe discuss in Subsection 3.3.By direct application of Theorem A to norm potentials of irreducible locally constant cocycles A : Σ → GL d ( R ) , we may add to the list of equivalent conditions in Proposition 1.1 that µ has the K -property. This improves the result of Morris. Theorem B.
Let A : Σ → GL d ( R ) be an irreducible locally constant cocycle. Suppose the uniqueequilibrium state µ A ∈ M ( σ ) of Φ A satisfies any one of the equivalent conditions from Proposition1.1. Then µ A has the K -property.In Conjecture 2 of [Mor18], Morris conjectured that the natural extension of every totally ergodicmatrix equilibrium state for a certain collection of potentials has the Bernoulli property. TheoremB establishes partial progress towards this conjecture for the class of norm potentials of GL d ( R ) locally constant cocycles, and in Remark 4.3, we discuss how Theorem A applies to all potentialsconsidered in [Mor18]. We note that Theorem B is related to results of Feng [Fen11] where heestablishes the K -property for some weighted equilibrium states.The remaining results are concerned with the thermodynamic formalism of α -Hölder and fiber-bunched cocycles; see Definition 2.8. We denote the space of α -Hölder and fiber-bunched GL d ( R ) -cocycles by C αb (Σ T , GL d ( R )) . In particular, every locally constant cocycle falls into these categories,while the converse is not true.Specific to fiber-bunched GL ( R ) -cocycles, in [BP19] Butler and the second-named author obtaina precise description of the equilibrium states for the norm potentials Φ A . In particular, they have acomplete characterization for when the norm potentials Φ A of fiber-bunched GL ( R ) -cocycles A failto have unique equilibrium states. By considering all cases depending on the number of equilibriumstates, we show that all equilibrium states of fiber-bunched GL ( R ) -cocycles have the K -property. Theorem C.
Let A : Σ T → GL ( R ) be a Hölder continuous and fiber-bunched cocycle. Everyergodic equilibrium state of A is K up to a period, that is, (Σ T , σ, µ A ) is isomorphic to a K -systemtimes a finite rotation. Indeed, the only setting when µ A is not K is when A can be conjugated toanother cocycle B ( x ) = (cid:18) a ( x ) b ( x ) 0 (cid:19) such that α ( x ) := log | a ( σx ) b ( x ) | and β ( x ) := log | b ( σx ) a ( x ) | viewed as potentials over (Σ T , σ ) have the same pressures, but distinct equilibrium states µ and µ .Theorem C may be thought of as follows: for fiber-bunched GL ( R ) -cocycles, the specified caseis the only case where the ergodic equilibrium states for Φ A fail to be K .The final result of this paper applies to a large subset of fiber-bunched cocycles and their sin-gular value potentials. More specifically, Bonatti and Viana [BV04] introduced a notion of typicalcocycles among C αb (Σ T , GL d ( R )) and established that typicality serves as a sufficient condition forthe simplicity of Lyapunov exponents with respect to any ergodic measures with continuous localproduct structure. Additionally, they showed that the set of typical cocycles is open and dense in C αb (Σ T , GL d ( R )) ; see Definition 2.11 for the precise formulation of the typicality assumption.In [Par20, Theorem B], the second-named author shows that for any s ∈ [0 , ∞ ) the singular valuepotentials Φ s A of typical cocycles A have unique equilibrium states µ A ,s ∈ M ( σ ) . By verifying theassumptions in Theorem A, we show that such equilibrium states µ A ,s have the K -property: Theorem D.
Let A : Σ T → GL d ( R ) be a Hölder continuous and fiber-bunched cocycle. If A istypical, then for any s ∈ [0 , ∞ ) the unique equilibrium state µ A ,s ∈ M ( σ ) of Φ s A has the K -property.We remark that unlike Theorem B and C where our results only apply to norm potentials Φ A ,the typicality assumption in Theorem D allows us to apply it to singular value potentials Φ s A forall s ∈ [0 , ∞ ) .In Section 2, we introduce and survey relevant preliminary results. In Section 3, we establishsufficient criteria for the K -property in the subadditive setting. Then we prove the main theoremsin Section 4. BENJAMIN CALL, KIHO PARK Preliminaries
Throughout, X is a compact metric space, f : X → X is a homeomorphism, and M ( f ) denotesthe set of all f -invariant probability measures.2.1. Mixing properties of invariant measures.
Although many of the results in this paper areconcerned with the K -property, we make use of various mixing properties along the way. We givea brief introduction to the ones that we will use, in increasing order of strength. In many cases,there are many equivalent formulations of these definitions which we lack the space to include. Fora more comprehensive discussion on various mixing properties, we refer the readers to [Pet83] Definition 2.1.
A measure-preserving transformation ( X, f, µ ) is totally ergodic if ( X, f n , µ ) isergodic for all n ∈ N . Definition 2.2.
We say µ ∈ M ( f ) is weakly mixing if for all measurable subsets A, B ⊆ X , thereexists E ⊂ N with upper density ¯ d ( E ) = 0 such that for n / ∈ E , lim n →∞ µ ( f n A ∩ B ) = µ ( A ) µ ( B ) . Observe from this definition that if µ is weak mixing, then it is totally ergodic. The following isa now classical result that we will also make use of periodically. Proposition 2.3. ( X, f, µ ) is weak mixing if and only if ( X × X, f × f, µ × µ ) is ergodic. Finally, we introduce the K -property. There are a myriad number of equivalent formulations,for details of which we refer to [CFS82]. Definition 2.4.
Let ( X, B , µ ) be a probability space, and let f be a measure-preserving invertibletransformation of ( X, B , µ ) The system has the
Kolmogorov property , or simply the
K-property , ifthere exists a sub- σ -algebra K ⊂ B satisfying
K ⊂ f K , ∞ W i =0 f i K = B , and ∞ T i =0 f − i K = {∅ , X } .An equivalent definition of independent interest is that of completely positive entropy . Proposition 2.5.
A measure µ ∈ M ( f ) has the K -property if and only if it has completely positiveentropy, that is, there are no non-trivial zero entropy factors.Remark . In particular, any K -system has positive measure-theoretic entropy.We say a system ( X, f, µ ) is Bernoulli if it is measurably isomorphic to a Bernoulli shift. Withouttoo much difficulty, one can use the above definition to show that every Bernoulli system is K .2.2. Base dynamical system.
Let T be a q × q square-matrix with entries in { , } . We define Σ T ⊂ { , · · · , q } Z to be the set of all bi-infinite sequences of q symbols such that ij is a wordfor ≤ i, j ≤ q if and only if T ij = 1 . An admissible word of length n is a word i . . . i n − with i j ∈ { , . . . , q } such that T i j ,i j +1 = 1 for all ≤ j ≤ n − . Let L be the collection of all admissiblewords. For I ∈ L , we denote its length by | I | . For each n ∈ N , let L ( n ) ⊂ L be the set of alladmissible words of length n . For any I = i . . . i n − ∈ L ( n ) , we define the associated cylinder by [ I ] = [ i . . . i n − ] := { y ∈ Σ T : y j = i j for all ≤ j ≤ n − } . We endow Σ T with the metric d defined as follows: for x = ( x i ) i ∈ Z , y = ( y i ) i ∈ Z ∈ Σ T , we have d ( x, y ) = 2 − k , where k is the largest integer such that x i = y i for all | i | < k . Equipped with such a metric, theleft shift operator σ becomes a hyperbolic homeomorphism of a compact metric space Σ T . Given x ∈ Σ T , the local stable set of x is W s loc ( x ) = { y ∈ Σ T | x i = y i for i ≥ } and analogously, the local unstable set of x is W u loc ( x ) = { y ∈ Σ T | x i = y i for i ≤ } . HE K -PROPERTY FOR SUBADDITIVE EQUILIBRIUM STATES 5 These local stable and unstable sets extend to define global stable and unstable sets W s/u ( x ) ,respectively, in the standard manner.Finally, we will always assume that the adjacency matrix T is primitive , meaning that thereexists N > such that all entries of T N are positive. The primitivity of T is equivalent to (Σ T , σ ) being topologically mixing.2.3. Linear cocycles.
To any A : Σ T → M d × d ( R ) and n ∈ N , we define A n ( x ) := A ( σ n − x ) . . . A ( x ) . It is clear from the definition that the following cocycle equation holds: A n + m ( x ) := A n ( σ m x ) A m ( x ) for all n, m ∈ N . When the image of A is a subset of GL d ( R ) , we define A ( · ) ≡ I and A − n ( x ) := (cid:0) A n ( σ − n x ) (cid:1) − for n ∈ N so that the cocycle equation holds for all n, m ∈ Z .We now introduce two classes of cocycles appearing in Theorems B, C, and D. First is the classof locally constant cocycles. A locally constant cocycle A is a cocycle whose generator A is locallyconstant. If A : Σ T → M d × d ( R ) is locally constant, then from the compactness of Σ T , there exists k ∈ N such that A ( x ) depends only on the word x − k . . . x k ∈ L (2 k + 1) for every x = ( x i ) i ∈ Z ∈ Σ T .For any locally constant GL d ( R ) -valued function A on Σ T , there exists a recoding of Σ T to anothersubshift of finite type Σ S such that A is carried to a GL d ( R ) -valued function on Σ S depending onlyon the 0-th entry x of x = ( x i ) i ∈ Z ∈ Σ S . Remark . For simplicity, we assume that all locally constant cocycles considered in this paperdepend only on the 0-th entry.The second class consists of fiber-bunched cocycles:
Definition 2.8. An α -Hölder cocycle A ∈ C α (Σ T , GL d ( R )) is fiber-bunched if for every x ∈ Σ T , kA ( x ) k · kA ( x ) − k < α . Clearly, conformal cocycles are fiber-bunched. Moreover, small perturbations of conformal cocy-cles are also fiber-bunched; in fact, fiber-bunched cocycles may be thought of as nearly conformalcocycles. We denote the set of α -Hölder and fiber-bunched cocycles by C αb (Σ T , GL d ( R )) . From thedefinition C αb (Σ T , GL d ( R )) , is an open subset of C α (Σ T , GL d ( R )) .The fiber-bunching assumption is mainly used for the convergence of the canonical stable/unstableholonomy H s/ux,y : for any y ∈ W s/u loc ( x ) , H sx,y := lim n →∞ A n ( y ) − A n ( x ) and H ux,y := lim n →−∞ A n ( y ) − A n ( x ) . (2.1)Moreover, the canonical holonomies vary Hölder continuously in the basepoints x, y ∈ Σ T with y ∈ W s/u loc ( x ) : there exists C > such that k H s/ux,y − I k ≤ C · d ( x, y ) α . (2.2)See [KS13] for further details.It can be easily checked that the canonical stable holonomies H sx,y satisfy the following properties:(1) H sx,x = I and H sy,z ◦ H sx,y = H sx,z for any y, z ∈ W s loc ( x ) ,(2) A ( x ) = H sσy,σx ◦ A ( y ) ◦ H sx,y ,(3) H s : ( x, y ) H sx,y is continuous.Likewise, the canonical unstable holonomies H ux,y satisfy the analogous properties. Using the secondproperty, the canonical holonomies H s/u can be defined for y ∈ W s/u ( x ) as well; i.e., for y notnecessarily in W s/u loc ( x ) but belonging to W s/u ( x ) .We now formulate the typicality assumption appearing in Theorem D. Consider any periodicpoint p ∈ Σ T and a homoclinic point z ∈ W s ( p ) ∩ W u ( p ) \ { p } . We define the holonomy loop ψ zp asthe composition of the unstable holonomy from p to z and the stable holonomy from z to p : ψ zp := H sz,p ◦ H up,z . BENJAMIN CALL, KIHO PARK
The following definition is a slight variation of typicality first introduced in [BV04]; this versionof typicality is identical to the definition which appeared in [Par20].
Definition 2.9.
Let
A ∈ C αb (Σ T , GL d ( R )) be a fiber-bunched cocycle and H s/u be its canonicalholonomies. We say that A is if it satisfies the following two extra conditions:(1) there exists a periodic point p such that P := A per ( p ) ( p ) has simple real eigenvalues ofdistinct norms. Let { v i } ≤ i ≤ d be the eigenvectors of P .(2) there exists a homoclinic point z of p such that ψ zp twists the eigendirections of P intogeneral position: for any ≤ i, j ≤ d , the image ψ zp ( v i ) does not lie in any hyperplane W j spanned by all eigenvectors of P other than v j . Equivalently, the coefficients c i,j in ψ zp ( v i ) = X ≤ j ≤ d c i,j v j , are nonzero for all ≤ i, j ≤ d .These two conditions in the above definition are often called pinching and twisting , respectively.For each ≤ t ≤ d , we denote by A ∧ t the action of A on the exterior product ( R d ) ∧ t . Then theexterior product cocycles A ∧ t , t ∈ { , . . . , d } , also admit stable and unstable holonomies, namely ( H s/u ) ∧ t . So, for a 1-typical function A , we consider similar conditions appearing in Definition 2.9on A ∧ t . Definition 2.10.
Let
A ∈ C αb (Σ T , GL d ( R )) be 1-typical. For ≤ t ≤ d − , we say A is t-typical if the same points p, z ∈ Σ T from Definition 2.9 satisfy(1) all the products of t distinct eigenvalues of P are distinct;(2) the induced map ( ψ zp ) ∧ t on ( R d ) ∧ t satisfies the corresponding twisting condition to thatgiven by Definition 2.9 with respect to the eigenvectors { v i ∧ . . . ∧ v i t } ≤ i <...
We say
A ∈ C αb (Σ T , GL d ( R )) is typical if A is t -typical for all ≤ t ≤ d − . Remark . Typicality was first introduced by Bonatti and Viana [BV04] for fiber-bunchedSL d ( R ) -cocycles as a sufficient condition to guarantee the simplicity of Lyapunov exponents withrespect to any ergodic invariant measures with continuous local product structure and full support.They also showed that the set of typical cocycles is open and dense in C αb (Σ T , SL d ( R )) and thisproperty easily generalizes to fiber-bunched GL d ( R ) -cocycles.The pinching and twisting assumptions of typicality are designed to replicate the effects ofproximality and strong irreducibility from Furstenberg’s theorem [Fur63] on positivity of the topLyapunov exponent.2.4. Thermodynamic Formalism.
In this section, we will introduce some of the key ideas ofboth additive and subadditive thermodynamic formalism that we will use. For shorthand, we willoften refer to the metric d n ( x, y ) := max ≤ i ≤ n − d ( f i x, f i y ) and to Bowen balls B n ( x, ε ) := { y ∈ X | d n ( x, y ) ≤ ε } . For any n ∈ N and ε > , E ⊂ X is ( n, ε ) -separated if given x, y ∈ E , d n ( x, y ) ≥ ε . Such a set is maximally ( n, ε ) -separated if for any z / ∈ E , d n ( x, z ) ≤ ε for some x ∈ E .Whenever we look at at a product space, we take the metric to be the maximum of the distancein each coordinate: d (( x , y ) , ( x , y )) := max { d ( x , x ) , d ( y , y ) } . (2.3)As mentioned in the introduction, we are interested in studying mixing properties of uniqueequilibrium states. One of the “ideal” results is the following, a proof of which can be found in[Bow75, Theorem 4.1]. Proposition 2.13.
Let f : X → X be a transitive Anosov homeomorphism, and let ϕ : X → R be aHölder continuous potential. Then ϕ has a unique equilibrium state µ ∈ M ( f ) , and µ is Bernoulli. HE K -PROPERTY FOR SUBADDITIVE EQUILIBRIUM STATES 7 The proof of this relies on the construction of a Markov coding, which in turn establishes theBernoulli property. In general, however, the existence, uniqueness, as well as the mixing proper-ties of equilibrium states are not well-known. While the existence of equilibrium states is oftenguaranteed under mild conditions such as entropy expansivity which implies upper semi-continuityof the entropy map [Bow72, Mis76] or C ∞ -smoothness of the system [New89], other properties ofequilibrium states are harder to come by.Even when uniqueness is guaranteed, the equilibrium states may not have strong mixing proper-ties as in Proposition 2.13. For instance, Bowen has shown the following theorem which guaranteesuniqueness of equilibrium states, but stops short of showing the Bernoulli property. Proposition 2.14. [Bow74]
Let f : X → X be expansive and have specification, and suppose that ϕ : X → R has the Bowen property, that is, for all ε > , there exists K such that for all n ∈ N and x ∈ X , sup ((cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n − X i =0 ϕ ( f i x ) − ϕ ( f i y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) : d n ( x, y ) ≤ ε ) ≤ K. Then there is a unique equilibrium state for ϕ . Ledrappier then showed that these equilibrium states have the K -property by means of thefollowing proposition. Proposition 2.15. [Led77, Proposition 1.4]
Let ( X, f ) be asymptotically entropy expansive andlet ϕ : X → R be continuous. Suppose that ( X × X, f × f ) has a unique equilibrium state for thepotential Φ( x, y ) = ϕ ( x ) + ϕ ( y ) . Then the unique equilibrium state for ϕ has the K -property. We now present some definitions and results in subadditive thermodynamic formalism that arealready known. Consider a sequence of continuous functions
Φ = { log ϕ n } n ∈ N on ( X, f ) . We say Φ is subadditive if log ϕ m + n ≤ log ϕ m + log ϕ n ◦ f m (2.4)for all m, n ∈ N .A subadditive potential is a natural generalization of the Birkhoff sum of an additive potentialin the following sense: given a continuous potential ϕ : X → R and denoting its n -th Birkhoff sumby S n ϕ , we obtain an equality in (2.4) if we replace each log ϕ n by S n ϕ .Then following the definition of [CFH08], we define the topological pressure of Φ as P (Φ) = lim ε → lim sup n →∞ n log sup n X x ∈ E ϕ n ( x ) | E ⊂ X is ( n, ε ) -separated o . The convergence of the limit is guaranteed by the subadditivity of Φ .There is another definition for the subadditive pressure introduced by Barreira [Bar96] usingopen covers. While it is not known whether two notions of the subadditive pressure coincide in themost general situations, they are shown to be equal when the entropy map µ h µ ( f ) is uppersemi-continuous [CFH08], which is true in our setting.As mentioned in the introduction, it was shown in [CFH08] that the subadditive pressure satisfiesthe following variational principle: if ( X, f ) has finite topological entropy, then P (Φ) = sup µ ∈M ( f ) P µ (Φ) where P µ (Φ) := h µ ( σ ) + lim n →∞ n Z log ϕ n dµ. Similar to additive potentials, the existence of equi-librium states for subadditive potentials can be guaranteed by mild conditions on the base such asthe upper semi-continuity of the entropy map. However, the questions on uniqueness and mixingproperties are more subtle.We also will make use of the subadditive versions of the Gibbs property as well as the boundeddistortion property. A probability measure µ on X has the subadditive Gibbs property with respect BENJAMIN CALL, KIHO PARK to Φ if for any ε > there exists C ≥ such that for all x ∈ X and n ≥ , C − ≤ µ ( B n ( x, ε )) e − nP (Φ) ϕ n ( x ) ≤ C. (2.5)If just the lower inequality holds, we say that Φ has the lower subadditive Gibbs property . Asubadditive potential Φ = { log ϕ n } on X has bounded distortion if there exists C ≥ such that forall ε > sufficiently small, x ∈ X , n ∈ N , and y, z ∈ B n ( x, ε ) , we have C − ≤ ϕ n ( y ) ϕ n ( z ) ≤ C. (2.6) Remark . This is a stronger property than that referred to in [CFH08], and should be thoughtof as analogous to the Bowen property for additive potentials in Proposition 2.14While Bowen’s theorem 2.14 does not extend directly to general subadditive potentials, it doesgeneralize to quasi-multiplicative subadditive potentials over uniformly hyperbolic base dynamics:
Proposition 2.17. [Fen11, Theorem 5.5]
Let
Φ = { log ϕ n } n ∈ N be a subadditive potential over (Σ T , σ ) with bounded distortion. If Φ is quasi-multiplicative, then Φ has a unique equilibrium state µ ∈ M ( σ ) . Moreover, µ has the subadditive Gibbs property: there exists C ≥ such that for any n ∈ N , I ∈ L ( n ) , and x ∈ [ I ] , C − ≤ µ ([ I ]) e − nP (Φ) ϕ n ( x ) ≤ C. (2.7)We note that the original setting of [Fen11, Theorem 5.5] deals with quasi-multiplicative func-tions on the set of admissible words L , while Proposition 2.17 considers more general subadditivepotentials. However, such a generalization is rather trivial using the bounded distortion property:we may treat Φ like a function on L via (1.3) and apply the result of [Fen11]. Another such instancecan be found in the proof of Lemma 4.1. Moreover, the subadditive Gibbs property of µ A fromProposition 2.17 will play a crucial role in establishing our results.2.5. Singular value potentials and previously known results.
In this subsection, we intro-duce a specific class of subadditive potentials known as the singular value potentials arising frommatrix cocycles. Since all of our results deal with the singular value potentials over subshifts offinite type (Σ T , σ ) , we assume that our base dynamic is (Σ T , σ ) throughout the subsection.The singular values of A ∈ M d × d ( R ) are eigenvalues of √ A ∗ A . We define the singular valuefunction ϕ s : M d × d ( R ) → R with parameter s ≥ as follows: ϕ s ( A ) = ( α ( A ) . . . α ⌊ s ⌋ ( A ) α ⌈ s ⌉ ( A ) { s } ≤ s ≤ d, | det( A ) | s/d s > d, where α ( A ) ≥ . . . ≥ α d ( A ) ≥ are the singular values of A . The function ( A, s ) ϕ s ( A ) isupper semi-continuous, and has a discontinuity at s = k ∈ N only if there is a jump in the singularvalues of the form α k − ( A ) > α k ( A ) = 0 . In particular, if A takes values in GL d ( R ) , then ϕ s ( A ) iscontinuous in both A and s .For each s ∈ R + , the singular value potential is defined by Φ s A := { log ϕ s A ,n } n ∈ N where ϕ s A ,n ( x ) := ϕ s ( A n ( x )) . As ϕ s is submultiplicative for all s , it follows that Φ s A is a subadditive potential on Σ T . In the casewhere s = 1 , Φ A coincides with the norm potential Φ A introduced in the introduction. We end thissection with a few remarks on the singular value potentials. Remark . The bounded distortion (2.6) holds for all norm potentials Φ A and singular valuepotentials Φ s A considered in our results. If A is locally constant, then we may take C = 1 from(2.6). For fiber-bunched cocycles A ∈ C αb (Σ T , GL d ( R )) , the bounded distortion follows from theHölder continuity of the canonical holonomies (2.2). HE K -PROPERTY FOR SUBADDITIVE EQUILIBRIUM STATES 9 Remark . We observe that equilibrium states are preserved under a continuous conjugacy. Morespecifically, we say
A ∈ C (Σ T , GL d ( R )) is continuously conjugated to another GL d ( R ) -cocycle B if there exists C ∈ C (Σ T , GL d ( R )) such that B ( x ) = C − ( σx ) A ( x ) C ( x ) . This follows from thesubadditive variational principle (1.2) and the fact that the norm kC ( x ) k is uniformly boundedfrom the compactness of Σ T .3. Subadditive Thermodynamic Formalism and the K -property Many of the techniques and results in thermodynamic formalism necessary for our proof ofTheorem A hold in general settings; as such, we set them apart here. We first show that under somegeneral conditions, ergodicity and the Gibbs property of an equilibrium state implies uniqueness. Wethen establish Ledrappier’s criterion in the subadditive setting. These combine to prove a generalresult which shows that in our setting, weak mixing is equivalent to the K -property. Throughoutthis section, we will consider ( X, f ) to be an expansive homeomorphism on a compact metric space.3.1. Uniqueness of Equilibrium States.
In this subsection, we establish sufficient conditionsfor subadditive equilibrium states to be unique, based on [Bow74]. In doing so, we will need tomake use of the Kolmogorov-Sinai entropy of a transformation. For any measure ν on X and anyfinite partition ξ of X , define H ν ( ξ ) = − X A ∈ ξ ν ( A ) log ν ( A ) and h ν ( f, ξ ) := lim n →∞ n H ν ( n − _ i =0 f − i ξ ) = inf n →∞ n H ν ( n − _ i =0 f − i ξ ) (3.1)where the infimum is due to subadditivity. Then the Kolmogorov-Sinai entropy of ν is defined by h ν ( f ) = sup finite partitions ξ h ν ( f, ξ ) . By Sinai’s theorem, if a partition ξ generates the Borel σ -algebra, then h ν ( f ) = h ν ( f, ξ ) . Lemma 3.1.
Let
Φ = { log ϕ n } n ∈ N be a subadditive potential on X with bounded distortion as in (2.6) and suppose η ∈ M ( f ) is an ergodic equilibrium state of Φ with the subadditive lower Gibbsproperty (2.5) . Then η is the unique equilibrium state of Φ .Proof. We follow the proof of [Bow74, Lemma 8] closely. Assume for the sake of contradiction that ν ∈ M ( f ) is an ergodic equilibrium state not equal to η . Then ν and η are mutually singular, andso there exists a ( ν + η ) -measurable set B ⊂ X such that f ( B ) = B , η ( B ) = 0 and ν ( B ) = 1 . Forinstance, we could take B to be the set of generic points for ν .Let ε > be smaller than the expansivity constant of ( X, f ) and small enough for boundeddistortion (2.6) to hold. For each n ∈ N we fix a maximal ( n, ε ) -separated set E n ⊂ X . Then wefix an adapted partition ξ n := { A x : x ∈ E n } of X such that B n ( x, ε ) ⊆ A x ⊆ B n ( x, ε ) for each x ∈ E n .In order to make use of the expansivity assumption, define for all n , the partition Ω n := f [ n/ ξ n and denote the element of Ω n containing y ∈ X by ω n ( y ) . From the construction of Ω n , for any y ∈ X there exists some x ∈ E n such that B n ( x, ε ) ⊆ f − [ n/ ω n ( y ) ⊆ B n ( x, ε ) . It then followsthat f − [ n/ ω n ( y ) ⊆ B n ( y, ε ) . Therefore expansivity gives T n ∈ N ω n ( y ) = { y } for all y ∈ X , and by[CT16, Lemma 5.10] there exists a sequence { C n } n ∈ N where C n is a union of elements of Ω n suchthat lim n →∞ ( ν + η )( C n △ B ) → . Since B is f -invariant, setting U n := f − [ n/ C n ⊆ ξ n , we have ( ν + η )( U n △ B ) → . From the assumptions on B , this is equivalent to η ( U n ) → and ν ( U n ) → . As ( X, f ) is expansive, ξ n is a generator under f n by observing that given y, z ∈ T k ∈ Z f kn B n ( x k , ε ) for some { x k } k ∈ Z ⊂ X , we have that d ( f k y, f k z ) ≤ ε for all k ∈ Z . Consequently, nh ν ( f ) = h ν ( f n ) = h ν ( f n , ξ n ) ≤ H ν ( ξ n ) where the last inequality is from (3.1). Moreover, from the subadditivity of Φ , we have lim k →∞ k Z log ϕ k dν = inf k →∞ k Z log ϕ n dν ≤ n Z log ϕ n dν for each n ∈ N . Hence, nP (Φ) = n (cid:16) h ν ( f ) + lim k →∞ k Z log ϕ k dν (cid:17) ≤ H ν ( ξ n ) + Z log ϕ n dν = X A x ∈ ξ n (cid:16) − ν ( A x ) log ν ( A x ) + Z log ϕ n · χ A x dν (cid:17) . Let C be the constant given by the bounded distortion (2.6) on Φ . Then Z log ϕ n · χ A x dν ≤ ν ( A x ) (cid:0) C + log ϕ n ( x ) (cid:1) for all n sufficiently large. In particular, we have nP (Φ) ≤ C + X A x ∈U n ν ( A x ) (cid:16) − log ν ( A x ) + log ϕ n ( x ) (cid:17) + X A x ∩U n = ∅ ν ( A x ) (cid:16) − log ν ( A x ) + log ϕ n ( x ) (cid:17) . Applying a Jensen-type inequality (see [Bow74, Lemma 7]) to each sum, we have nP (Φ) − C ≤ C ∗ + ν ( U n ) log X A x ∈U n ϕ n ( x ) ! + ν ( U cn ) log X A x ∩U n = ∅ ϕ n ( x ) , where C ∗ := max t ∈ [0 , − t log t .Let C be the constant from the subadditive lower Gibbs property (2.5) of η . Then after rear-ranging the terms, we have − C ∗ − C ≤ ν ( U n ) log X A x ∈U n ϕ n ( x ) e − nP (Φ) ! + ν ( U cn ) log X A x ∩U n = ∅ ϕ n ( x ) e − nP (Φ) ≤ ν ( U n ) log( C η ( U n )) + ν ( U cn ) log( C η ( U cn ))= log C + ν ( U n ) log η ( U n ) + ν ( U cn ) log η ( U cn ) . This, however, is a contradiction because as we send n → ∞ , the lower bound − C ∗ − C isindependent of n ∈ N while ν ( U n ) log η ( U n ) → −∞ and ν ( U cn ) log η ( U cn ) → . Hence, ν cannot bean equilibrium state of Φ . (cid:3) Subadditive generalization of Ledrappier’s criterion.Lemma 3.2.
For any subadditive potential
Φ = { log ϕ n } n ∈ N on ( X, f ) , consider a sequence ofcontinuous functions Ψ = { log ψ n } n ∈ N on ( X × X, f × f ) defined by ψ n ( x, y ) := ϕ n ( x ) · ϕ n ( y ) . (3.2) Then Ψ is subadditive and P (Ψ) = 2 P (Φ) . HE K -PROPERTY FOR SUBADDITIVE EQUILIBRIUM STATES 11 Proof.
Subadditivity of Ψ follows immediately: as for all n, m ∈ N , log ψ n + m ( x, y ) = log ϕ n + m ( x ) + log ϕ n + m ( y ) ≤ log ϕ n ( x ) + log ϕ n ◦ f m ( x ) + log ϕ n ( y ) + log ϕ n ◦ f m ( y )= log ψ n ( x, y ) + log ψ n ( f m x, f m y ) . For the second statement, let µ be an equilibrium state for Φ . Then µ × µ ∈ M ( f × f ) , and wealso have P µ × µ (Ψ) = h µ × µ ( f × f ) + lim n →∞ n Z log ψ n dµ × µ = 2 h µ ( f ) + 2 lim n →∞ n Z log ϕ n dµ = 2 P µ (Φ) . Therefore, by the variational principle (1.2), we see that P (Ψ) ≥ P (Φ) .For the reverse direction, we again proceed by the variational principle. Let ν ∈ M ( f × f ) be arbitrary, and write ν and ν to be the projections of ν onto the first and second coordinate,respectively. Each ν i is a f -invariant measure on X . An elementary calculation shows that h ν ( f × f ) ≤ h ν ( f ) + h ν ( f ) (see for instance, [Dow11, Fact 4.4.3]), and lim n →∞ n Z log ψ n dν = lim n →∞ n (cid:16) Z log ϕ n dν + Z log ϕ n dν (cid:17) . Therefore, P ν (Ψ) ≤ h ν ( f ) + lim n →∞ n Z log ϕ n dν + h ν ( f ) + lim n →∞ n Z log ϕ n dν ≤ P (Φ) . (cid:3) We immediately have the following corollary:
Corollary 3.3. If µ ∈ M ( f ) is an equilibrium state for Φ , then µ × µ ∈ M ( f × f ) is an equilibriumstate for Ψ . We can now state the subadditive generalization of Proposition 2.15 for establishing the K -property. Recall that we call a measure µ is K if and only if it has no nontrivial zero entropyfactors. Equivalently, the maximal zero entropy factor, called the Pinsker factor , is trivial.
Proposition 3.4.
Let
Φ = { log ϕ n } n ∈ N be a subadditive potential on X with unique equilibrumstate µ ∈ M ( f ) . If µ × µ ∈ M ( f × f ) is the unique equilibrium state for Ψ , then µ has the K -property.Proof. We follow the original proof of Ledrappier, and prove the contrapositive. Let µ ∈ M ( f ) bethe unique equilibrium state for Φ , and suppose it is not K . Then the Pinsker factor Π for µ isnon-trivial. We therefore can define m ∈ M ( f × f ) different from µ × µ to be m ( A × A ′ ) = Z A E [ χ A ′ | Π] dµ for all measurable A, A ′ ⊂ X . To see this is different from µ × µ , take A to be Π -measurable, andobserve that m ( A × A ) = µ ( A ) = µ ( A ) = ( µ × µ )( A × A ) . For those familiar with joinings, this isthe relatively independent self-joining of µ over Π .The entropy calculation from [Led77] is purely dependent on the measure, and so is unaffectedby the subadditive setting. For a reference where this calculation is carried out in full, see [Cal20].Hence, h m ( f × f ) = 2 h µ ( f ) . Now because m ( A × X ) = m ( X × A ) = µ ( A ) , and ψ is is definedindependently in each coordinate, we observe that for all n ∈ N , Z log ψ n dm = 2 Z log ϕ n dµ .Therefore, P m (Ψ) = h m ( f × f ) + lim n →∞ n Z log ψ n dm = 2 h µ ( f ) + 2 lim n →∞ n Z log ϕ n dµ = 2 P µ (Φ) = 2 P (Φ) . Hence, m is an equilibrium state for Ψ in M ( f × f ) , as is µ × µ . So there exist multiple equilibriumstates for the product system. (cid:3) Now, recall from Proposition 2.3 that a measure µ ∈ M ( f ) is weak mixing if and only if µ × µ ∈M ( f × f ) is ergodic. Using this fact, we obtain the following theorem: Theorem 3.5.
Let ( X, f ) be an expansive homeomorphism on a compact metric space and Φ = { log ϕ n } n ∈ N be a subadditive potential on X with bounded distortion (2.6) . Suppose η ∈ M ( f ) is aweak mixing equilibrium state of Φ with the lower subadditive Gibbs property (2.5) . Then η has the K -property.Proof. First, as η is a weak mixing equilibrium state, η × η is an ergodic equilibrium state. Therefore,if we can show Lemma 3.1 holds for the system ( X × X, f × f ) with potential Ψ defined as (3.2),then it follows that η × η is the unique equilibrium state. Therefore, by the subadditive version ofLedrappier’s criterion, it immediately follows that η is K .We now verify the assumptions in Lemma 3.1. First, ( X × X, f × f ) is still an expansivehomeomorphism on a compact metric space. Thus, we only need to check that Ψ has the boundeddistortion and the subadditive Gibbs property.Since the metric on our product space is the maximum of the distance in each coordinate (2.3),it follows that B n (( x, y ) , ε ) = B n ( x, ε ) × B n ( y, ε ) . From this, it follows that the subadditive Gibbs property on η and the bounded distortion of Φ induce the corresponding properties on η × η and Ψ . (cid:3) We note that weak mixing is a natural assumption to impose in this theorem, as one can easilydefine a system which is not weak mixing and satisfies all other conditions of this theorem.3.3.
Relationship between one and two-sided results.
Many of the results that we cite (seefor instance, [Mor18, Mor19, Fen09]) are written in the case where the base dynamic is a one-sidedshift. While it is not difficult to see that those that we cite hold in the invertible setting as well, forcompleteness, we sketch some of the arguments here. Throughout, let (Σ + T , σ ) be a mixing one-sidedshift of finite type, and let (Σ T , σ ) be its natural extension, which is a two-sided subshift of finitetype with the same list of forbidden words. Finally, let π : Σ T → Σ + T be the standard projectionmap, taking ( x i ) i ∈ Z to ( x i ) i ∈ N .The following proposition is a consequence of the definition for the subadditive pressure. Proposition 3.6.
Let Φ be a subadditive potential on (Σ + T , σ ) , and consider the subadditive potential Ψ := Φ ◦ π on (Σ T , σ ) . Then P (Φ) = P (Ψ) . Since Ψ is defined as Φ ◦ π and the entropy is preserved under the natural extension, the followingcorollary is immediate from the above proposition. Corollary 3.7.
Let µ be an equilibrium state for (Σ + T , σ ) and Φ . Then the natural extension of µ is an equilibrium state for (Σ T , σ ) and Ψ . Proposition 3.8.
There is a unique equilibrium state for a subadditive potential Φ on (Σ + T , σ ) , ifand only if its natural extension is the unique equilibrium state for Ψ on (Σ T , σ ) .Proof. It suffices to show that different equilibrium states for the natural extension project todifferent equilibrium states for the one-sided system. That they project to different measuresfollows from shift-invariance and the fact that they must differ on some cylinder set. That theyproject to equilibrium states follows because Ψ is defined as Φ ◦ π and the entropy is preservedunder the natural extension. (cid:3) Corollary 3.9.
For any locally constant cocycle A and s > , the natural extension of any equi-librium state for Φ A over (Σ + T , σ ) is an equilibrium state for the invertible setting with the samepotential. Using the classical fact that mixing and ergodicity of natural extensions are equivalent to therespective properties for the one-sided systems, we see that Proposition 1.1 holds for two-sidedshifts as well. HE K -PROPERTY FOR SUBADDITIVE EQUILIBRIUM STATES 13 Proof of main theorems
Proof of Theorem A.
We recall the setting of Theorem A. Let (Σ T , σ ) be a mixing subshiftof finite type and Φ = { log ϕ n } n ∈ N be a quasi-multiplicative subadditive potential with boundeddistortion. Let µ ∈ M ( σ ) be the unique equilibrium state for Φ with the Gibbs property fromProposition 2.17, and suppose that µ is totally ergodic. We wish to show that µ is K . By Theorem3.5, it suffices to show that µ is weak mixing.The following proposition is essentially a reformulation of [Mor18, Theorem 5 (ii)]. The settingthere is for norm potentials of irreducible locally constant cocycles; however, the proof generalizeseasily to any quasi-multiplicative subadditive potentials with bounded distortion. Lemma 4.1.
Let
Φ = { log ϕ n } n ∈ N be a quasi-multiplicative subadditive potential on Σ T withbounded distortion. Suppose the unique equilibrium state µ ∈ M ( σ ) of Φ from Proposition 2.17is totally ergodic. Then µ is mixing.Proof. The proof of [Mor18, Theorem 5 (ii)] extends without much modification; we only point outminor modifications required to extend the proof.From Proposition 2.17, it follows that µ has the Gibbs property (2.7) with constant C . Recallingthe notation e Φ from (1.3) and denoting the constant from the bounded distortion (2.6) of Φ by C ,for any n ∈ N , I ∈ L ( n ) , and x ∈ [ I ] we have the following bounds on µ ([ I ]) / (cid:0) e − nP (Φ) e Φ( I ) (cid:1) : ( C C ) − ≤ C − · µ ([ I ]) e − nP (Φ) ϕ n ( x ) ≤ µ ([ I ]) e − nP (Φ) e Φ( I ) ≤ C Then for any cylinders I , J ∈ L of length n and m , we have for any k > n , µ ([ I ] ∩ f − k [ J ]) = X | K | = k − n IKJ ∈L µ ([ IKJ ]) ≤ C X | K | = k − n IKJ ∈L e − ( k + m ) P (Φ) e Φ( IKJ ) ≤ C X | K | = k − n IKJ ∈L e − ( k + m ) P (Φ) e Φ( I ) e Φ( K ) e Φ( J ) ≤ C C µ ([ I ]) µ ([ J ]) (cid:16) X | K | = k − n IKJ ∈L µ ([ K ]) (cid:17) ≤ C C µ ([ I ]) µ ([ J ]) . This gives lim sup k →∞ µ ([ I ] ∩ f − k [ J ]) ≤ Cµ ([ I ]) µ ([ J ]) where C = C C . Using this property togetherwith total ergodicity of µ , the rest of the proof from here on (i.e., promoting total ergodicity toweak mixing, and then to mixing) follows that of [Mor18, Theorem 5 (ii)] verbatim, following themethod of Ornstein [Orn72]. (cid:3) Theorem A now follows as an easy consequence of Proposition 2.17 which gives an equilibriumstate with the Gibbs property, Lemma 4.1 which shows that total ergodicity is enough to get weakmixing, and Theorem 3.5, which lifts these together to K .4.2. Proof of Theorem B.
Suppose A : Σ → GL d ( R ) is a locally constant cocycle, and its imageconsists of matrices { A , . . . , A q } . We say A is irreducible if there does not exist a proper subspace V ⊂ R d with A i V = V for every ≤ i ≤ q .The following result of [Fen09] guarantees the quasi-multiplicativity of the unique equilibriumstate of Φ A under irreducibility: Proposition 4.2. [Fen09, Proposition 2.8]
Let A : Σ → GL d ( R ) be a locally constant cocycle. If A is irreducible, then the norm potential Φ A is quasi-multiplicative and has a unique equilibrium state µ A ∈ M ( σ ) . Then Theorem B is a direct consequence of Theorem A and Proposition 4.2:
Proof of Theorem B.
The norm potentials Φ A of any locally constant cocycles A : Σ → GL d ( R ) immediately have bounded distortion with constant C = 1 . Irreducibility of A then gives quasi-multiplicativity of Φ A by Proposition 4.2. Hence, if µ A ∈ M ( σ ) satisfies either of the equivalentconditions in Proposition 1.1, then µ A is mixing which is stronger than total ergodicity, and it thenfollows from Theorem A that µ A has the K -property. (cid:3) We end this subsection with a few related remarks. First, we comment on the difference betweennorm potentials considered in Theorem B with similar subadditive potentials considered by Morris.
Remark . Morris [Mor18, Mor19] works with similar subadditive potentials. He considers locallyconstant cocycles A over one-sided full shifts (Σ + , σ ) , and defines the subadditive pressure of ( A , s ) as P ( A , s ) = lim n →∞ n log (cid:16) q X i ,...,i n =1 k A i n . . . A i k s (cid:17) . Making this explicit, for all s > , the corresponding subadditive potential is given by Ψ s A = { log( ψ n ) s } , where ψ n ( x ) = kA n ( x ) k . Thus, we see that the subadditive potential ( A , in his papersagrees with the norm potential Φ A , and in fact, ( A , s ) agrees with the singular value potentials Φ s A when s ≤ . This is not necessarily so when s > . Nevertheless, Theorem A applies toMorris’ subadditive potentials ( A , s ) for all s . If A is an irreducible locally constant cocycle, thenby Proposition 4.2, Ψ s A is quasi-multiplicative, and the bounded distortion condition is satisfiedtrivially because it is locally constant. Therefore, if the unique equilibrium state µ of ( A , s ) istotally ergodic, then it is K , establishing partial progress towards Conjecture 2 of [Mor18]. Remark . For irreducible locally constant cocycles, their unique equilibrium states are shown topossess various ergodic properties depending on the assumptions. Theorem B is one result in thisdirection. Another such instance can be found in [Pir18]; Piraino established that under primivityor strong irreduciblity and proximality, the unique equilibrium states are weakly Bernoulli, and soare isomorphic to a Bernoulli shifts. On the other hand, such equilibrium states are rarely Bernoullimeasures. Morris and Sert [MS19] established a mild assumption that prevents such equilibriumstates from being Bernoulli measures on full shifts Σ . We stress that both of these results lie in thesetting of Theorem B, and do not apply to the setting of the following results.4.3. Proof of Theorem C.
Let A : Σ T → GL ( R ) be a Hölder continuous and fiber-bunchedcocycle. The proof of Theorem C relies on the results of [BP19]. We begin by introducing thenotion of irreducibility for fiber-bunched GL d ( R ) -cocycles. Definition 4.5.
We say a fiber-bunched cocycle
A ∈ C αb (Σ T , GL d ( R )) is reducible if there exists aproper A -invariant and H s/u -invariant sub-bundle. We say A is irreducible if it is not reducible. Remark . For fiber-bunched cocycles, irreducibility is a weaker assumption than typicality be-cause typical cocycles are necessarily irreducible. Additionally, whenever a cocycle is both locallyconstant and fiber-bunched, this definition of irreducibility coincides with that of the previoussection.The following proposition summarizes the results in [BP19]:
Proposition 4.7. [BP19]
Let
A ∈ C αb (Σ T , GL ( R )) . If A is irreducible, then Φ A is quasi-multiplicative, and hence, has a unique equilibrium state µ A ∈ M ( σ ) .If A is reducible, then A is Hölder continuously conjugated to another GL ( R ) -cocycle B takingvalues in the group of upper triangular matrices: B ( x ) := (cid:18) a ( x ) b ( x )0 c ( x ) (cid:19) . (4.1) HE K -PROPERTY FOR SUBADDITIVE EQUILIBRIUM STATES 15 The set of ergodic equilibrium states of Φ A is a subset of { µ log | a | , µ log | c | } where µ log | τ | , τ ∈ { a, c } ,is the unique equilibrium state for log | τ | .Moreover, Φ A has two distinct ergodic equilibrium states if and only if(1) log | a | is not cohomologous to log | c | , and(2) P (log | a | ) = P (log | c | ) .Otherwise, Φ A has a unique equilibrium state.Remark . In [BP19], it was shown that Φ A has a unique equilibrium state if A is irreducible.It was done via the dichotomy that either Φ A is quasi-multiplicative or A is Hölder continuouslyconjugated into the group of conformal linear transformations. The latter case is clearly also quasi-multiplicative. Hence, we stated in Proposition 4.7 that Φ A is quasi-multiplicative when A isirreducible. Lemma 4.9. If A ∈ C αb (Σ T , GL ( R )) is reducible, then all ergodic equilibrium states of Φ A areBernoulli.Proof. Since the set of equilibrium states of Φ A is a subset of { µ log | a | , µ log | c | } from Proposition 4.7and both µ log | a | and µ log | c | are Bernoulli from Proposition 2.13, our claim follows. (cid:3) For each n ∈ N , consider A n as a cocycle over (Σ T , σ n ) and denote the corresponding normpotential by Φ A n . It can be easily checked from the definition that if A is fiber-bunched over (Σ T , σ ) , then so is A n over (Σ T , σ n ) . Lemma 4.10.
For any n ∈ N , we have P (Φ A n ) = nP (Φ A ) . Moreover, any equilibrium state µ ∈ M ( σ ) of Φ A is an equilibrium state of Φ A n .Proof. We proceed via the variational principle (1.2). Observe that ϕ A n ,m = ϕ A ,mn . Then, we seethat if µ ∈ M ( σ ) , h µ ( σ n ) + lim m →∞ m Z log ϕ A n ,m dµ = nh µ ( σ ) + n lim m →∞ mn Z log ϕ A ,mn dµ. Considering µ as a σ n -invariant measure, we have just shown that P µ (Φ A n ) = nP µ (Φ A ) and thevariational principle implies that nP (Φ A ) ≤ P (Φ A n ) .For the reverse inequality, take µ ∈ M ( σ n ) and define ν = n − X i =0 ( σ i ) ∗ µn . Then ν is σ -invariant,and furthermore, h µ ( σ n ) = h ν ( σ n ) = nh ν ( σ ) . Since A is continuous and Σ T is compact, for any ≤ i ≤ n − , two functions log ϕ A n ,m ◦ σ i and log ϕ A n ,m are uniformly comparable. Hence, lim m →∞ m Z log ϕ A n ,m ◦ σ i dµ = lim m →∞ m Z log ϕ A n ,m dµ. Then it follows that lim m →∞ mn Z log ϕ A ,mn dν = lim m →∞ mn n − X i =0 n Z log ϕ A n ,m ◦ σ i dµ = 1 n lim m →∞ m Z log ϕ A n ,m dµ. Therefore, P µ (Φ A n ) = nP ν (Φ A ) . This gives the reverse inequality, and the result follows. That anyequilibrium state of Φ A is an equilibrium state of Φ A n is now a direct consequence. (cid:3) We are now ready to prove Theorem C.
Proof of Theorem C.
In view of Lemma 4.9 it suffices to focus on irreducible GL ( R ) -cocycles. Let A ∈ C αb (Σ T , GL ( R )) be irreducible, and µ A ∈ M ( σ ) be the unique equilibrium state for Φ A fromProposition 4.7. We then consider the cocycle A over (Σ T , σ ) . Noting that Proposition 4.7 also applies to A ,we divide into two cases depending on the number of equilibrium states of Φ A . Case 1: Φ A has a unique equilibrium state. Such a unique equilibrium state must be µ A by Lemma 4.10. Uniqueness then implies that (Σ T , σ , µ A ) is ergodic. We claim that in fact, µ A is totally ergodic, which by Theorem A, wouldimply that µ A is K .Assume not for contradiction. Let n ∈ N be the least integer such that µ A is not ergodic withrespect to (Σ T , σ n ) . As (Σ T , σ, µ A ) and (Σ T , σ , µ A ) are ergodic, we know that n ≥ . Since A n isstill a fiber-bunched GL ( R ) -cocycle over (Σ T , σ n ) , Φ A n has at most two distinct ergodic equilib-rium states by Proposition 4.7. Furthermore, in the proof of [Mor19, Theorem 2], Morris showedthat the number of distinct ergodic equilibrium states for Φ A n bounds n . Therefore, ≥ n ≥ , acontradiction. Case 2: Φ A has multiple equilibrium states. From Proposition 4.7, A over (Σ T , σ ) mustbe reducible and Φ A must have two distinct ergodic equilibrium states µ , µ ∈ M ( σ ) . In fact,denoting the A -invariant and H s/u -invariant line bundle by L , consider another line bundle L defined by L ( σx ) := A ( x ) L ( x ) . Since A is irreducible, L is different from L . We then have L ( σx ) = A ( x ) L ( x ) from the A -invariance of L , and L is also A -invariant.For each x ∈ Σ T , let C ( x ) ∈ GL ( R ) be the unique linear map that takes the standard basis of R into { L ( x ) , L ( x ) } . Then B ( x ) := C ( σx ) − A ( x ) C ( x ) exchanges the coordinate axes of R , andhence must be of the form specified in Theorem C: B ( x ) = (cid:18) a ( x ) b ( x ) 0 (cid:19) . Then B ( x ) is the diagonal matrix given by diag ( a ( σx ) b ( x ) , a ( x ) b ( σx )) . Moreover, two potentials α ( x ) := log | a ( σx ) b ( x ) | and β ( x ) := log | b ( σx ) a ( x ) | have the same pressure (with respect to σ ),and their σ -ergodic equilibrium states are µ and µ , respectively, each of which is Bernoulli byLemma 4.9. From the assumption that µ and µ are distinct, we must have that σ ∗ µ = µ and σ ∗ µ = µ . This is because σ ∗ µ is an σ -ergodic invariant measure and an equilibrium statefor Φ A , and hence it must be either µ itself or µ . However, it cannot be equal to µ as thiswould imply the σ -invariance of µ , and by the uniqueness of the equilibrium state µ A for Φ A , thiswould imply µ = µ A . But applying the same argument to µ contradicts µ and µ being distinctmeasures.We will now show that µ A is the average of µ and µ . Indeed,
12 ( µ + σ ∗ µ ) = 12 ( µ + µ ) is σ -invariant from the σ -invariance of µ , and is an equilibrium state for Φ A . Since µ A is the uniqueequilibrium state for Φ A , we must have that µ A = 12 ( µ + µ ) . (cid:3) Proof of Theorem D.
For any GL d ( R ) -cocycle A , n ∈ N , and s > , similar to the notationfrom the previous subsection we denote by Φ s A n the s -singular value potential of A n with respectto (Σ T , σ n ) .The following proposition gives us that the singular value potentials of typical cocycles haveunique equilibrium states: Proposition 4.11. [Par20, Theorem B]
Let
A ∈ C αb (Σ T , GL d ( R )) be an α -Hölder and fiber-bunchedcocycle. If A is typical, then Φ s A is quasi-multiplicative, and hence, has a unique equilibrium state µ A ,s ∈ M ( σ ) for every s ∈ [0 , ∞ ) . In view of Theorem A and Proposition 4.11, the only missing ingredient in proving Theorem Dis the total ergodicity of µ A ,s which we establish below. The idea of the proof is similar to that of[Mor19, Theorem 5 (i)]. Proposition 4.12.
Let
A ∈ C αb (Σ T , GL d ( R )) be typical and s ∈ [0 , ∞ ) . Then the unique equilib-rium state µ A ,s ∈ M ( σ ) of Φ s A is totally ergodic. HE K -PROPERTY FOR SUBADDITIVE EQUILIBRIUM STATES 17 Proof.
For any n ∈ N , A n is typical with respect to (Σ T , σ n ) via the same periodic and thehomoclinic points p and z from the definition of typical cocycles. Applying Proposition 4.11 to A n and (Σ T , σ n ) , Φ s A n has a unique equilibrium state µ A n ,s ∈ M ( σ n ) . In particular, µ A n ,s isergodic with respect to (Σ T , σ n ) . From Lemma 4.10 which also applies to Φ s A n , it follows that µ A n ,s coincides with µ A ,s . Hence, µ A ,s is totally ergodic. (cid:3) Acknowledgments.
We would like to thank Ian Morris, Mark Piraino, Anthony Quas, and CagriSert for helpful comments, as well as for pointing out an incorrect statement about Bernoulli systemsin an earlier version of this paper.
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