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 $\frac{1}{2}\leq α\leqβ\leq1$, there is a group $Γ$ with measure $μ$ equidistributed on a finite generating set such that $\liminf\frac{\log H_{Γ,μ}(n)}{\log n}=α,\qquad \limsup \frac{\log H_{Γ,μ}(n)}{\log n}=β.$ 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 $\liminf\frac{{\log}|{\log P}(Y_n=_Γ1)|}{\log n}=\frac{1}{3},\qquad \limsup\frac{{\log}|{\log P}(Y_n=_Γ1)|}{\log n}=1$ and drift satisfying $\liminf\frac{\log {\mathbb{E}}\|Y_n\|}{\log n}=\frac{1}{2},\qquad \limsup\frac{\log {\mathbb{E}}\|Y_n\|}{\log n}=1.$

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