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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
##### Keywords:
entropy; polynomial maps; endomorphisms of complex manifold