Let \(X\) be a compact metric space equipped with a metric \(d\) and \(T\) be a continuous transformation on \(X\). Then \((X,T)\) is called a topological dynamical system (TDS for short). For a positive integer \(n\in \mathbb{N}\), the Bowen metric \(d_n\) is given by \(d_n(x, y) = \max\{d(T^{i}x, T^{i}y) : i = 0, 1, \dots ,n-1\}\) for every \(x, y\in X\). We denote the Bowen ball centred at \(x\) of radius \(\varepsilon\) with length \(n\) by \(B_n(x, \varepsilon) = \{y\in X : d_n(x, y) < \varepsilon\}\). It is well known that there are several formulas for the measure theoretic entropy, such as Shannon-McMillan-Breiman formula, Brin-Katok entropy formula and Katok entropy formula. In this paper, the authors consider the Brin-Katok formula for dynamical systems admitting “mistakes”, thus replacing the Bowen ball with the “mistake Bowen ball”. The authors show that it suffices to study a mistake function which is linear in \(n\) and monotonic with respect to \(\Delta\).
Given a \(T\)-invariant measure \(\mu\), define \[\begin{array}{l} h_{\mu }^+(T,\delta ,x)=\underset{n\rightarrow \infty }{\lim \sup }\frac{-\log \mu (B_n(x,\delta ))}{n}, \\ h_{\mu }^-(T,\delta ,x)=\underset{n\rightarrow \infty }{\lim \inf }\frac{-\log \mu (B_n(x,\delta ))}{n}.\end{array}\] M. Brin and A. Katok [Lect. Notes Math. 1007, 30–38 (1983; Zbl 0533.58020)] proved that for \(\mu\)-almost every \(x\in X\) \[h_{\mu }(T,\delta,x):=\lim_{\delta \rightarrow 0} h_{\mu }^+(T,\delta ,x)=\lim_{\delta \rightarrow 0} h_{\mu }^-(T,\delta ,x).\] In this paper, the authors prove that, for every \(T\)-invariant measure \(\mu\) and mistake function \(g\), the limits \(\lim_{\delta \rightarrow 0} \widetilde{h}_{\mu}^+(T,g,\delta,x)\) and \(\lim_{\delta \rightarrow 0} \widetilde{h}_{\mu}^-(T,g,\delta ,x)\) exist and are equal, for \(\mu\)-almost every \(x\in X\). Denote by \(\widetilde{h}_{\mu}(T,g,.)\) the common value. One has \(h_{\mu }(T)=\int \widetilde{h}_{\mu}(T,g,x)d\mu\), where \(h_{\mu }(T)\) is the measure theoretic entropy.