| Abstract: |
| Ruelle transfer operators are central to thermodynamic formalism, providing critical insights into phenomena such as the existence and uniqueness of Gibbs measures, calculations of pressures and Hausdorff dimensions, and estimates of correlation decay rates. These operators exhibit distinct behaviors across different function spaces. This discussion will explore the contrast between transfer operators on Holder and Dini function spaces. I will discuss how the spectral gap can be used to analyze the exponential decay of correlations in expanding dynamical systems with Holder potentials. However, the spectral gap vanishes when the dynamical systems are not fully expanding or the potentials are not Holder. In collaboration with Yuan-Ling Ye, we established an optimal quasi-spectral gap condition for studying transfer operators in weakly expanding systems with Dini potentials. This condition enables us to derive precise estimates for the decay rate of correlations. |
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