Abstract
We determine the form of all semiflows of pairs of weighted quasi-arithmetic means, those over positive dyadic numbers as well as the continuous ones. Then the iterability of such pairs is characterized: necessary and sufficient conditions for a given pair of weighted quasi-arithmetic means to be embeddable into a continuous semiflow are given. In particular, it turns out that surprisingly the existence of a square iterative root in the class of such pairs implies the embeddability.
Highlights
One crucial problem of dynamical system is the embeddability of an individual function into a flow or, more generally, into a semiflow
A function f : X → X is said to be embeddable into a T -flow [T -semiflow ] if there exists a flow [semiflow] F : X × T → X such that f = F (·, 1)
Among others, the questions of how to embed continuous strictly monotonic functions defined on an interval [15], homeomorphisms of the circle [16], two commuting functions [17] and, jointly with Solarz, diffeomorphisms of the plane in a regular iteration semigroup [20]
Summary
One crucial problem of dynamical system is the embeddability of an individual function into a flow or, more generally, into a semiflow. A number of papers devoted to different variants of the embedding problem was written by Zdun He answered, among others, the questions of how to embed continuous strictly monotonic functions defined on an interval [15], homeomorphisms of the circle [16], two commuting functions [17] and, jointly with Solarz, diffeomorphisms of the plane in a regular iteration semigroup [20]. ∼ is an equivalence relation in the set CM(I) Using this notion one can formulate the following result solving the so-called equality problem for weighted quasi-arithmetic means (see [1, Sec. 6.4.3, Theorem 2] [13] by Maksa and Pales; for quasi-arithmetic means the answer was known already in 1934 and can be found in the book [5]).
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