Variational principle and zero temperature limits of asymptotically (sub)-additive projection pressure

Abstract

Let $$\{S_i\}_{i=1}^l$$ be an iterated function system (IFS) on ℝd with an attractor K. Let (∑, σ) denote the one-sided full shift over the finite alphabet {1, 2,...,l}, and let π : ∑ → K be the coding map. Given an asymptotically (sub)-additive sequence of continuous functions ℱ = {fn}n⩾1; we define the asymptotically additive projection pressure Pπ(ℱ) and show the variational principle for Pπ(ℱ) under certain affne IFS. We also obtain variational principle for the asymptotically sub-additive projection pressure if the IFS satisfies asymptotically weak separation condition (AWSC). Furthermore, when the IFS satisfies AWSC, we investigate the zero temperature limits of the asymptotically sub-additive projection pressure Pπ(βℱ) with positive parameter β.