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

Abstract

Let {{\displaystyle \{{{\displaystyle S}}_i\}}}_{i=\mathrm{1}}^l be an iterated function system (IFS) on {ℝ}^d with an attractor K. Let (Σ, σ) denote the one-sided full shift over the finite alphabet {1, 2, . . . , l}, and let π: Σ → K be the coding map. Given an asymptotically (sub)-additive sequence of continuous functions F={{\displaystyle \{{{\displaystyle f}}_n\}}}_{n\ge \mathrm{1}}, we define the asymptotically additive projection pressure P_π(F) and show the variational principle for P_π(F) under certain affine 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 P_π(βF) with positive parameter β.

Key words： Projection pressure; asymptotically (sub)-additive potentials; variational principle; zero temperature limits

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