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

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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.

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Acknowledgements

The authors would like to thank the referees for their careful reading and valuable comments. Part of this work was done when the second author visited ICERM, the financial support are greatly appreciated. This work was partially supported by the National Natural Science Foundation of China (Grant Nos. 11790274, 11871361).

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  1. School of Mathematical Sciences, Soochow University, Suzhou, 215006, China

    Qiuhong Wang & Yun Zhao

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  1. Qiuhong Wang
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  2. Yun Zhao
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Correspondence to Yun Zhao.

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Wang, Q., Zhao, Y. Variational principle and zero temperature limits of asymptotically (sub)-additive projection pressure. Front. Math. China 13, 1099–1120 (2018). https://doi.org/10.1007/s11464-018-0720-1

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  • DOI: https://doi.org/10.1007/s11464-018-0720-1

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