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We prove an analog of the Tits alternative for endomorphisms of P1. In particular, we show that if S is a finitely generated semigroup of endomorphisms of P1 over C, then either S has polynomially bounded growth or S contains a nonabelian free semigroup. We also show that if f and g are polarizable maps over any field of any characteristic and Prep.f / 6D Prep.g/, then for all sufficiently large j, the semigroup hf j; gj i is a free semigroup on two generators. [ABSTRACT FROM AUTHOR]

Digital technologies and digitalisation are emerging as new drivers of structural transformation in developing countries. At the firm level, adopting advanced digital technologies offers prospects for improved productivity and competitiveness and, hence, digital industrialisation. However, the determinants of adopting digital technologies in manufacturing firms in developing countries remain anecdotal. This paper uses unique survey data on 516 manufacturing firms in South Africa and a multivariate probit model to examine the determinants of digital technology adoption in South African manufacturing firms. Our results show heterogeneity in the factors explaining digital technology adoption across business functions. Overall, the empirical results reveal that innovation, foreign ownership, exposure to export markets, and higher skilled human capital propel the adoption of digital technologies. In contrast, the lack of capital inhibits the adoption of digital technologies in sampled manufacturing firms in South Africa. We discuss the possible policy implications of these findings and how they fit into the South African Digital Skills policy discourse. [ABSTRACT FROM AUTHOR]

Let G$G$ be a semiabelian variety defined over an algebraically closed field K$K$, endowed with a rational self‐map Φ$\Phi$. Let α∈G(K)$\alpha \in G(K)$ and let Γ⊆G(K)$\Gamma \subseteq G(K)$ be a finitely generated subgroup. We show that the set {n∈N:Φn(α)∈Γ}$\lbrace n\in \mathbb {N} \colon \Phi ^n(\alpha)\in \Gamma \rbrace$ is a union of finitely many arithmetic progressions along with a set S$S$ of Banach density equal to 0$\hskip.001pt 0$. In addition, assuming that Φ$\Phi$ is regular, we prove that the set S$S$ must be finite. [ABSTRACT FROM AUTHOR]

We give examples of birational selfmaps of Pd,d⩾3$\mathbb {P}^d, d \geqslant 3$, whose dynamical degree is a transcendental number. This contradicts a conjecture by Bellon and Viallet. The proof uses a combination of techniques from algebraic dynamics and diophantine approximation. [ABSTRACT FROM AUTHOR]

Let L be a field of characteristic zero, let h:\mathbb {P}^1\to \mathbb {P}^1 be a rational map defined over L, and let c\in \mathbb {P}^1(L). We show that there exists a finitely generated subfield K of L over which both c and h are defined along with an infinite set of inequivalent non-archimedean completions K_{\mathfrak {p}} for which there exists a positive integer a=a(\mathfrak {p}) with the property that for i\in \{0,\ldots,a-1\} there exists a power series g_i(t)\in K_{\mathfrak {p}}[[t]] that converges on the closed unit disc of K_{\mathfrak {p}} such that h^{an+i}(c)=g_i(n) for all sufficiently large n. As a consequence we show that the dynamical Mordell-Lang conjecture holds for split self-maps (h,g) of \mathbb {P}^1 \times X with g an étale self-map of a quasiprojective variety X. [ABSTRACT FROM AUTHOR]