Topological Pressure for the Completely Irregular Set of Birkhoff Averages

Abstract: In this paper we mainly study the dynamical complexity of Birkhoff ergodic average under the simultaneous observation of any number of continuous functions. These results can be as generalizations of [6,35] etc. to study Birkhorff ergodic average from one (or finite) observable function to any number of observable functions from the dimensional perspective. For any topological dynamical system with $g-$almost product property and uniform separation property, we show that any {\it jointly-irregular set}(i.e., the intersection of a series of $\phi-$irregular sets over several continuous functions) either is empty or carries full topological pressure. In particular, if further the system is not uniquely ergodic, then the {\it completely-irregular set}(i.e., intersection of all possible {\it nonempty $\phi-$irregular} sets) is nonempty(even forms a dense $G_\delta$ set) and carries full topological pressure. Moreover, {\it irregular-mix-regular sets} (i.e., intersection of some $ \phi-$irregular sets and $ \varphi-$regular sets) are discussed. Similarly, the above results are suitable for the case of BS-dimension. As consequences, these results are suitable for any system such as shifts of finite type or uniformly hyperbolic diffeomorphisms, time-1 map of uniformly hyperbolic flows, repellers, $\beta-$shifts etc..