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Recently, Kiriki, Nakano and Soma introduced a concept called pointwiseemergence as a new quantitative perspective into the study of non-existence ofaverages for dynamical systems. In the present paper, we consider the set ofpoints with high pointwise emergence for systems with non-uniform structure andprove that this set carries full topological pressure. For the proof of thisresult, we show that such systems have ergodic measures of arbitraryintermediate pressures.

High pointwise emergence and Katok's conjecture for systems with non-uniform structure