On~the problem of classification of polynomial endomorphisms of the plane

M. V. Jakobson

Sbornik. Mathematics, Tome 9 (1969) no. 3, pp. 345-364

Résumé

The paper is a continuation of the author's paper [1] (Math. Sb. (N.S.) 77(119) (1968), 105–124). § 1 concerns the iterations of a polynomial $P(z)$ of degree $d>1$ on a singular set $\mathscr F$. It is assumed that the critical points of $P^{-1}(z)$ lie either in the domains of attraction of finite attracting cycles or at infinity. The theorems of [1] (Theorem 1 concerning the topological isomorphism of the transformation $P(z)/\mathscr F$ and of a shift on the space of one-sided $d$-ary sequences with a finite number of identifications; Theorem 2: $P/\mathscr F\approx P_\varepsilon/\mathscr F_\varepsilon$) are generalized for the case of a disconnected $\mathscr F$. In § 2 the author investigates the iterations of $P(z)$ on the entire plane $\pi$. He shows (Theorem 3) that the dynamical systems $P/\pi$ and $P_\varepsilon/\pi$ are topologically isomorphic for sufficiently small $|\varepsilon|$ in the case of polynomials satisfying one of the hypotheses of § 1 and a certain “coarse” condition of “nonconjugacy” of the iterations of distinct critical points. Hypothesis: the set of structurally stable mappings $z\to P(z)$ investigated in the paper is everywhere dense in the space of coefficients. Figures : 9. Bibliography: 8 titles.