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On the problem of classification of polynomial endomorphisms of the plane
M. V. Jakobson
Abstract:
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.
UDC:
519.5
MSC: 37F10,
37B05,
46A45 Received: 21.01.1969
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