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Topological pressure for the completely irregular set of birkhoff averages

The author is supported by National Natural Science Foundation of China (grant no. 11671093, 11301088) and Specialized Research Fund for the Doctoral Program of Higher Education (No. 20130071120026).
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  • It is well-known that for certain dynamical systems (satisfying specification or its variants), the set of irregular points w.r.t. a continuous function $\phi$ (i.e. points with divergent Birkhoff ergodic averages observed by $\phi$ ) either is empty or carries full topological entropy (or pressure, see [6,17,36,37] etc. for example). In this paper we study the set of irregular points w.r.t. a collection $D$ of finite or infinite continuous functions (that is, points with divergent Birkhoff ergodic averages simultaneously observed by all $\phi∈D$ ) and obtain some generalized results. As consequences, these results are suitable for systems such as mixing shifts of finite type, uniformly hyperbolic diffeomorphisms, repellers and $β-$ shifts.

    Mathematics Subject Classification: 37C50, 37C45, 37B10, 37D20.


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