Let \(M\) be a smooth compact manifold with some fixed Riemannian metric. Let \(f\colon M\to M\) be a continuous function. Recall that the topological entropy of \(f\) may be defined as follows. Let \(d\) be the metric on \(M\) induced by some Riemannian metric. For \(p\in \mathbb{N}\), define a new metric \(d_{f,p}\) on \(M\) by \(d_{f,p}(x,y)=\max_{0\leq i\leq p} d(f^i(x),f^i(y)).\) Let \(M_f(p,\epsilon)\) be the minimal number of \(\epsilon\)-balls in the \(d_{f,p}\)-metric that cover \(M\). The topological entropy \(h(f)\) is defined by \(h(f)=\lim_{\epsilon\to 0}\limsup_{p\to\infty}\frac{\log_2{M_f(p,\epsilon)}}{p}\).
In this note the authors consider the case when \(M\) is a closed oriented surface \(\Sigma_g\) of genus \(g\). Denote by \(\mathrm{Diff}(\Sigma_g)\) the group of orientation-preserving diffeomorphisms of \(\Sigma_g\). Let \(\mathrm{Ent}(\Sigma_g)\subset \mathrm{Diff}(\Sigma_g)\) be the set of entropy-zero diffeomorphisms. This set is conjugation-invariant and it generates \(\mathrm{Diff}(\Sigma_g)\). In other words, a diffeomorphism of \(\Sigma_g\) is a finite product of entropy-zero diffeomorphisms. One may ask for a minimal decomposition and this question leads to the concept of the entropy norm defined by \(\|f\|_{\mathrm{Ent}}:=\min\{k\in \mathbb{N}\,|\,f=h_1\cdots h_k,\,h_i\in \mathrm{Ent}(\Sigma_g)\}\). The associated bi-invariant metric is denoted by \(d_{\mathrm{Ent}}\). It is known that the entropy norm is bounded in case \(g=0\). In this paper, the authors prove that the entropy norm on the group of diffeomorphisms of a closed orientable surface \(\Sigma_g\) of positive genus \(g>0\) is unbounded.