We study the general problem of equidistribution of expanding translates of an analytic curve by an algebraic diagonal flow on the homogeneous space of a semisimple algebraic group G. We define two families of algebraic subvarieties of the associated partial flag variety, which give the obstructions to non-divergence and equidistribution. We apply this to prove that for Lebesgue almost every point on an analytic curve in the space of m × n real matrices whose image is not contained in any subvariety coming from these two families, Dirichlet’s theorem on simultaneous Diophantine approximation cannot be improved. The proof combines geometric invariant theory, Ratner’s theorem on measure rigidity for unipotent flows, and linearization technique.

We show that theMöbius function is disjoint to every analytic skew product dynamical system on T2 over a rotation of the circle.

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For any dynamical system T : X → X of a compact metric space X with g-almost product property and uniform separation property, under the assumptions that the periodic points are dense in X and the periodic measures are dense in the space of invariant measures, we distinguish various periodiclike recurrences and find that they all carry full topological entropy and so do their gap-sets. In particular, this implies that any two kind of periodic-like recurrences are essentially different. Moreover, we coordinate periodic-like recurrences with (ir)regularity and obtain lots of generalized multifractal analyses for all continuous observable functions. These results are suitable for all β-shifts (β > 1), topological mixing subshifts of finite type, topological mixing expanding maps or topological mixing hyperbolic diffeomorphisms, etc. Roughly speaking, we combine many different “eyes” (i.e., observable functions and periodic-like recurrences) to observe the dynamical complexity and obtain a Refined Dynamical Structure for Recurrence Theory and Multi-fractal Analysis.

We prove the dynamical Mordell-Lang conjecture for birational polynomial morphisms on A^2.