Behaviors of entropy on finitely generated groups

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 \[\liminf\frac{\log {\mathbb{E}}\|Y_n\|}{\log n}=\frac{1}{2},\qquad \limsup\frac{\log {\mathbb{E}}\|Y_n\|}{\log n}=1.\]