Abstract
Dynamical Diophantine approximation studies the quantitative properties of the distribution of the orbits in a dynamical system. More precisely, it focuses on the size of dynamically defined limsup sets in the sense of measure and dimension. This quantitative study is motivated by the qualitative nature of the density of the orbits and the connections with the classic Diophantine approximation. In this survey, we collect some recent progress on the dimension theory in dynamical Diophantine approximation. This includes the systems of rational maps on its Julia set, linear map on the torus, beta dynamical system, continued fractions as well as conformal iterated function systems.
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