Sub-additive setting Sample Clauses

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 Xxxxxx, Xxxxx and Xxxxxxx 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 n→∞ n n i∈{1,...,l}n = lim log  exp(log f 1 Σ n→∞ n i∈{1,...,l}n n
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