A variety of behaviors of entropy functions of random walks on finitely generated groups is presented, showing that for any $\frac{ 1}{ 2}\leq \alpha\leq\beta\leq1$, there is a group $\Gamma$ with measure $\mu$ equidistributed on a finite generating set such that \[\liminf\frac{ \log H_{ \Gamma ,\mu}(n)}{ \log n}=\alpha ,\qquad \limsup \frac{ \log H_{ \Gamma ,\mu}(n)}{ \log n}=\beta .\] The groups involved are finitely generated subgroups of the group of automorphisms of an extended rooted tree. The return probability and the drift of a simple random walk $Y_n$ on such groups are also evaluated, providing an example of group with return probability satisfying \[\liminf\frac{ { \log}|{ \log P}(Y_n=_{ \Gamma}1)|}{ \log n}=\frac{ 1}{ 3},\qquad \limsup\frac{ { \log}|{ \log P}(Y_n=_{ \Gamma}1)|}{ \log n}=1\] and drift satisfying \[\liminf\frac{ \log { \mathbb{ E}}\|Y_n\|}{ \log n}=\frac{ 1}{ 2},\qquad \limsup\frac{ \log { \mathbb{ E}}\|Y_n\|}{ \log n}=1.\]
Categories: math.GR math.PR