Variations of topological theory and ergodic theory via gap function in non-uniform specification

In [44], we qualitatively studied some classical results implied by the specification property for dynamical systems with non-uniform specification. In this paper, we perform quantitative studies on how properties of topological theory and ergodic theory vary with the gap function in non-uniform specification from different points of view. Firstly, we establish a lower estimation for Bowen topological entropy of irregular sets, determined by the lower asymptotic linear growth rate of the gap function. As a contrast, we prove that under the non-uniform specification property, every non-empty over-saturated set, and thus the completely-irregular set carries full packing topological entropy. We also estimate the Bowen topological entropy of transitive points and the exponential growth rate of periodic orbits. Besides, we provide a constructive mechanism for generating dynamical systems with non-uniform specification, containing an arbitrarily given subshift. This mechanism demonstrates the richness of dynamical systems with non-uniform specification by constructing systems attaining arbitrarily large values of topological entropy. Moreover, we use this mechanism to find examples demonstrating that the lower estimation for Bowen topological entropy is optimal in some cases and that the optimal upper estimation is the full topological entropy. These examples also demonstrate the optimality of the estimation of exponential growth rate and further indicate that, when the lower asymptotic linear growth rate of the gap function is larger than zero, (i) transitive points may not carry full Bowen topological entropy; (ii) the conditional variational principle and the intermediate entropy property may be destroyed. In addition, we study the variations of ergodic optimization.