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On the entropy of holomorphic maps. (English) Zbl 1080.37051
For a space \(X\) let \(X^k\) denote the product \(X\times X\times \cdots\times X\) \((k\) factors). A graph \(\Gamma\) over \(X\) is by definition an arbitrary set \(\Gamma\subset X^2\). For a graph \(\Gamma\) let \(\Gamma_k\subset X^k\) denote the set of strings \((x_1,\dots,x_i, \dots,x_k)\), \(x_i\in X\), where each pair \((x_{i-1},x_i)\in X^2\) is contained in \(\Gamma\). When \(X\) is endowed with a metric, the products in \(X^k\) of balls from \(X\) of radius \(\varepsilon\) are called \(\varepsilon\)-cubes. For a set \(Y\subset X^k\) the symbol \(\text{Cap}_\varepsilon Y\) denotes the minimal number of \(\varepsilon\)-cubes needed to cover \(Y\). When \(f\) is an endomorphism \(X\to X\) its entropy \(h(f)\) is defined as entropy of its graph \(\Gamma\). The author investigates a holomorphic map of the complex projective space \(\mathbb{C} P^m\) onto itself. Every such map is given by \((m+1)\) homogeneous polynomials in \(\mathbb{C}^{m+1}\) each one of the same degree \(p\) and the topological degree \(\deg f\) is equal to \(p^m\). He proves the following theorem: If \(f:\mathbb{C} P^m\to\mathbb{C} P^m\) is holomorphic then \(h(f)=\log(\deg f)\). More precisely, he proves only the inequality \(h(f)\leq \log(\deg f)\), since the inequality \(h(f)\geq\log(\deg f)\) was established by M. Misiurewicz and F. Przytycki [Bull. Acad. Pol. Sci., Sér. Math. Astron. Phys. 25, 573–574 (1977; Zbl 0362.54037)]

MSC:
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
32H50 Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables
37B40 Topological entropy
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