Abstract
In recent papers, Kenyon et al. (Ergod Theory Dyn Syst 32:1567–1584 2012), and Fan et al. (C R Math Acad Sci Paris 349:961–964 2011, Adv Math 295:271–333 2016) introduced a form of non-linear thermodynamic formalism based on solutions to a non-linear equation using matrices. In this note we consider the more general setting of Hölder continuous functions.
Highlights
We first recall a classical result for matrices dating back to work of Perron (1907) and Frobenius (1912)
A k × k matrix A is called non-negative if all the entries are non-negative real numbers and aperiodic if there exists n > 0 such that all entries of the nth power An are strictly positive
We recall a generalisation of the Perron–Frobenius Theorem to Banach spaces of functions
Summary
We first recall a classical result for matrices dating back to work of Perron (1907) and Frobenius (1912) (cf. [5], p. 53). Remark 1.7 If φ ∈ Fθ as usual, by replacing φ by φ1 = φ + log ψφ − log ψφ ◦ σ ∈ Fθ , where ψ is the positive eigenfunction in Theorem 1.3, we can assume without loss of generality that ψφ (x) = 1 is the constant function taking the value 1, i.e., Lφ1 1 = λ1, where λ = λφ = λφ. Remark 1.7 If φ ∈ Fθ as usual, by replacing φ by φ1 = φ + log ψφ − log ψφ ◦ σ ∈ Fθ , where ψ is the positive eigenfunction in Theorem 1.3, we can assume without loss of generality that ψφ (x) = 1 is the constant function taking the value 1, i.e., Lφ1 1 = λ1, where λ = λφ = λφ1 For such special normalized functions φ1 the function ψ Theorem 1.5 can be identified as ψ = ψφ1 = λ1, we see that Lφ1 ψ = ψ2. The uniqueness and analyticity are relatively easy to establish
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