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There exist one-dimensional transitive cellular automata with non-empty set of strictly temporally periodic points

Matyja J.

Schedae Informaticae

2013

Vol. 22

41--45

In a Cantor metric space BZ; we present a one-sided cellular automaton which positively answers the question Does it exist a transitive cellular automaton (BZ; F) with non-empty set of strictly temporally periodic points? The question can be found in a current and recognized literature of the subject.

On iterative roots of homeomorphisms of the circle

Zdun M.

Bulletin of the Polish Academy of Sciences. Mathematics

2000

Vol. 48, no 2

203-213

In this paper we deal with the problem of existence and uniqueness of continuous iterative roots of homeomorphisms of the circle. Let F : [S^1 --> S^1] be a homeomorphism without periodic points. If the limit set of the orbit [F^k(z), k belongs to Z] equals [S^1], then F has exactly n iterative roots of n-th order. Otherwise F either has no iterative roots of n-th order or F has infinitely many iterative roots depending on an arbitrary function. In this case we determined all iterative roots of n-th order of F.

Rigidity of tame rational functions

Przytycki F., Urbański M.

Bulletin of the Polish Academy of Sciences. Mathematics

1999

Vol. 47, no 2

163-181

We introduce and establish some basic properties of the tame rational functions. The class of these functions contains all the rational functions with no recurrent critical points in their Julia sets. For tame non-exceptional functions we prove that the Lipschitz conjugacy, the same spectra of moduli of derivatives at periodic orbits and conformal conjugavcy are mutually equivalent. We prove also the following rigidity result: If h is a Borel measurable invertible map which conjugates two tame functions f and g a.e. and if h transports conformal measure m[sub f] to a measure equivalent to m[sub g] then h extends from a set of full measure m[sub f] to a conformal homeomorphism of neighbourhoods of respective Julia sets. This extends D. Sullivan's rigidity theorem for holomorphic expanding repellers. We provide also a few lines proof of E. Prado's theorem that two generalized polinomial-like maps at zero Teichmueller's distance are holomorphically conjugate.