The<i>K</i>-property for subadditive equilibrium states

Let $$(\Sigma _T,\sigma )$$ ( Σ T , σ ) be a subshift of finite type with primitive adjacency matrix $$T$$ T , $$\psi :\Sigma _T \rightarrow \mathbb {R}$$ ψ : Σ T → R a Hölder continuous potential, and $$\mathcal {A}:\Sigma _T \rightarrow \textrm{GL}_d(\mathbb {R})$$ A : Σ T → GL d ( R ) a 1-typical, one-step cocycle. For $$t \in \mathbb {R}$$ t ∈ R consider the sequences of potentials $$\Phi _t=(\varphi _{t,n})_{n \in \mathbb {N}}$$ Φ t = ( φ t , n ) n ∈ N defined by $$\begin{aligned}\varphi _{t,n}(x):=S_n \psi (x) + t\log \Vert \mathcal {A}^n(x)\Vert , \, \forall n \in \mathbb {N}.\end{aligned}$$ φ t , n ( x ) : = S n ψ ( x ) + t log ‖ A n ( x ) ‖ , ∀ n ∈ N . Using the family of transfer operators defined in this setting by Park and Piraino, for all $$t<0$$ t < 0 sufficiently close to 0 we prove the existence of Gibbs-type measures for the superadditive sequences of potentials $$\Phi _t$$ Φ t . This extends the results of the well-understood subadditive case where $$t \ge 0$$ t ≥ 0 . Prior to this, Gibbs-type measures were only known to exist for $$t<0$$ t < 0 in the conformal, the reducible, the positive, or the dominated, planar settings, in which case they are Gibbs measures in the classical sense. We further prove that the topological pressure function $$t \mapsto P_{\textrm{top}}(\Phi _t,\sigma )$$ t ↦ P top ( Φ t , σ ) is analytic in an open neighbourhood of 0 and has derivative given by the Lyapunov exponents of these Gibbs-type measures.

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Uniformity of singular value exponents for typical cocycles

Chen

Cao

Zou

2023

Stoch. Dyn.

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Let [Formula: see text] be a [Formula: see text]-valued cocycle over a subshift of finite type. Under a certain twisting assumption, we prove that [Formula: see text] has a uniform Lyapunov exponent if and only if the largest Lyapunov exponent of [Formula: see text] at all periodic points equals. Under the typicality assumption, we give two checkable criteria for deciding whether [Formula: see text] has uniform singular value exponents.

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TheK-property for subadditive equilibrium states

Cited by 3 publications

References 57 publications

Bernoulli property of subadditive equilibrium states

Bernoulli property of subadditive equilibrium states

On the Superadditive Pressure for 1-Typical, One-Step, Matrix-Cocycle Potentials

Uniformity of singular value exponents for typical cocycles

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