Entropy estimates for a family of expanding maps of the circle

In this paper we consider the family of circle maps $f_{k,\alpha,\epsilon}:\mathbb{S}^1\rightarrow \mathbb{S}^1$ which when written mod 1 are of the form $f_{k,\alpha,\epsilon}: x \mapsto k x + \alpha + \epsilon \sin(2\pi x)$, where the parameter $\alpha$ ranges in $\mathbb{S}^1$ and $k\geq 2.$ We prove that for small $\epsilon$ the average over $\alpha$ of the entropy of $f_{k,\alpha,\epsilon}$ with respect to the natural absolutely continuous measure is smaller than $\int_0^1 \log|Df_{k,0,\epsilon}(x)|dx,$ while the maximum with respect to $\alpha$ is larger. In the case of the average the difference is of order of $\epsilon^{2k+2}.$ This result is in contrast to families of expanding Blaschke products depending on rotations where the averages are equal and for which the inequality for averages goes in the other direction when the expanding property does not hold, see [4]. A striking fact for both results is that the maximum of the entropies is greater than or equal to $\int_0^1 \log|Df_{k,0,\epsilon}(x)|dx$. These results should also be compared with [3], where similar questions are considered for a family of diffeomorphisms of the two sphere.