Weak Gibbs measures as Gibbs measures for asymptotically additive sequences

Abstract

In this note we prove that every weak Gibbs measure for an asymptotically additive sequences is a Gibbs measure for another asymptotically additive sequence. In particular, a weak Gibbs measure for a continuous potential is a Gibbs measure for an asymptotically additive sequence. This allows, for example, to apply recent results on dimension theory of asymptotically additive sequences to study multifractal analysis for weak Gibbs measure for continuous potentials.

Additive, almost additive and asymptotically additive potential sequences are equivalent

“…, and thus F is asymptotically additive (this was also observed in [33]). However, F admits µ as a Gibbs measure, and also µ as a weak Gibbs measure; in particular, both µ and µ are equilibrium states for F.…”

“…where C n (x) is the cylinder set of rank n containing x, and we say that µ is a weak Gibbs measure if (3.6) holds with K replaced by K n , where lim n→∞ n −1 log K n = 0. See for example [33].…”

“…In fact, all that can be said a priori is that any measure that is Gibbs with respect to F is weak Gibbs with respect to f . One can argue that this is enough for many applications, and that when working with asymptotically additive sequences of potentials, the notions of weak Gibbs and Gibbs measures are no different; in fact, on Markov shifts, any measure that is weak Gibbs with respect to an asymptotically additive sequence is also Gibbs for another, well-chosen, asymptotically additive sequence, see [33]. However, in the special case where F is almost additive, the Gibbs property is genuinely stronger than the weak Gibbs property.…”

“…Remark 4.3. Note that the function f in (4.4) has vanishing topological pressure, since the topological pressure of the sequence (4.2) is easily seen to be zero (see also [33]).…”

Domination, almost additivity, and thermodynamic formalism for planar matrix cocycles

“…Such sequences of functions are referred to in the literature as almost additive and have been investigated in [4,6,10,21,33]. The condition of almost additivity implies trivially a further property, asymptotic additivity (see for example Feng and Huang [16,Proposition A.5]), which has been applied in [13,16,22]. In another category of works, positivity is replaced by the more general hypothesis of domination: under this hypothesis there exists a continuous splitting R d = U(x) ⊕ V(x), which is preserved by the cocycle, such that A n T (x)u Ce nε A n T (x)v for all unit vectors u ∈ U(x) and v ∈ V(x), for some constants C, ε > 0 (see [7] and references therein).…”