Scales of Functions and Applications in Dynamical Systems

By a scale we mean a set of real continuous germs at $+\infty$ ($f: [c,\infty)\to\mathbb{R}$) which is linearly ordered by the relation $f>>g$ of eventual dominance (meaning $f(x)>g(x)$, for all $x>c(f,g)$). A classical non-trivial example of a scale is given in G. Hardy's book (1910) "Orders of infinity". The functions (germs) in his class $L$ are defined by fields operations and finite applications of functions $\exp$ and $\log$. The emphasis of the talk is the description of an abstract method to construct large scales of functions. Some classical results in number theory and ergodic theory concerning polynomials extend to functions belonging to functions $f\in L$ (and much larger class $H$ of 'good' functions to be defined). For good $f$ one can often provide precise conditions concerning various distribution properties (e.g., density, uniform distribution) of the sequences $f(n)$ mod 1 (MB, 1994). One can characterize 'good' $f$ which are good for the pointwise ergodic theorem along $[f(n)]$ (joint with M.Wierdl. A. Quas, G. Kolesnik, 2005) which extends Bougains's polynomial point wise ergodic theorem. There is a scale version Szemeredi Theorem (M. Wierdle, N. Frantzikinasis, 2009) which extends V. Bergelson and A. Leibman polynomial Szemeredi Theorem. There is also a scale version for Waring problemm for "good functions" (T. Chan, A. Kumchev and M. Wierdl 2010).

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