Sub-additive setting Clause Samples

Sub-additive setting. In this section we present the thermodynamic formalism of the sub-additive poten- tials that will be considered in Chapter 6. Sub-additive thermodynamic formalism was developed as an extension of the standard thermodynamic formalism for addi- tive potentials, in part due to its applications to the study of measures supported on self-affine sets, see for instance [F3], [K]. Essentially, this theory is concerned with generalising the classical results of ▇▇▇▇▇▇, ▇▇▇▇▇ and ▇▇▇▇▇▇▇ which connects the topological pressure with the measure theoretic entropy and Lyapunov exponents via a ‘variational principle’, to the setting where additive potentials are replaced by sub-additive potentials on Σ = {1, . . . , l}∞ (or more generally, a compact metric space X, equipped with a continuous mapping T ). We say that a sequence F = {log fn}∞n=1 of functions on Σ is sub-additive if each fn is a continuous non-negative function on Σ such that 0 ™ fn+m(i) ™ fn(i)fm(σni) for each i ∈ Σ and n, m ∈ N. In this setting, the pressure of the sub-additive sequence F can be defined as P (F) = lim log  f (i) 1 Σ i∈{1,...,l}n = lim log  exp(log f 1 Σ i∈{1,...,l}n (i)  where the limit exists by sub-additivity of the sequence F. Thus, it should be clear from the second displayed equation above, that in the sub-additive setting, the sequence log fn plays the role of Snf in the classical setting. In particular, if we put fn = exp Snf for all n ∈ N then P (F) = P (f ) and indeed we are in the additive case since fn+m(i) = exp(Sn+mf (i)) = exp(Snf (i)) exp(Smf (σni)) = fn(i)fm(σni) for each i ∈ Σ and n, m ∈ N. Once P (F) is defined for a sub-additive potential, the goal is then to prove a variational principle, analogous to (2.5). We will be dealing with a very specific case of this theory which applies to the singular value function related to a family of matrices. Therefore, our presentation of the theory will differ from the general formulation provided above, with the hope of improving on the clarity of the expo- sition. For a more general treatment of sub-additive thermodynamic formalism, the reader is directed to [CFH]. i=1 Let {Ai}l be a family of invertible d × d matrices. We denote Σ to be the full shift on l symbols. Recall that the singular value function φs : Σ∗ → R+ was defined for each 0 ™ s ™ d by |s| (i) = α1(i) · · · α|s|−1 (i) αs−|s|