Cibles rétrécissantes de rayon $n^{-\frac{1}{d}}$ : propriété du logarithme

Details Cibles rétrécissantes de rayon $n^{-\frac{1}{d}}$ : propriété du logarithme Cibles rétrécissantes de rayon $n^{-\frac{1}{d}}$ : propriété du logarithme

2011-03-23

arxiv.org/abs/1103.4594v1

Mathematics

Dynamical Systems

Scientific paper

A translation on the d-dimensional torus $\mathbb{T}^d$ has the logarithm property if the Shrinking Target Property holds for the sequence of balls with radius $n^{-\frac{1}{d}}$. On $\mathbb{T}^1$ every irrational translations has this property. In higher dimension, we give criterions based upon an unusual Diophantine type. In the two dimensional torus, we construct counter examples to the logarithm with translations vectors of arbitrary small Diophantine type and example of logarithm property with Liouvillian vector

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