Iterative square roots of functions

Abstract

An iterative square root of a self-map f is a self-map g such that $g(g(\cdot ))=f(\cdot )$ . We obtain new characterizations for detecting the non-existence of such square roots for self-maps on arbitrary sets. They are used to prove that continuous self-maps with no square roots are dense in the space of all continuous self-maps for various topological spaces. The spaces studied include those that are homeomorphic to the unit cube in ${\mathbb R}^{m}$ and to the whole of $\mathbb {R}^{m}$ for every positive integer $m.$ However, we also prove that every continuous self-map on a space homeomorphic to the unit cube in $\mathbb {R}^{m}$ with a fixed point on the boundary can be approximated by iterative squares of continuous self-maps.

The non-iterates are dense in the space of continuous self-maps

“…Proof of theorem 1.1 for [0, 1] m and R m . As seen in [4], both C([0, 1] m ) and C(R m ) are metrizable. Further, the technique used to prove theorem 3.8 (resp.…”

“…and R m in their usual topologies (see theorems 3.8 and 4.3 in [4], where it is proven, in particular, that (W(2; X)) c is dense in C(X) for X = [0, 1] m and X = R m , respectively). In this paper we prove stronger versions of these results to show that the same approach can be used to obtain functions with no iterative roots of any order n ⩾ 2 (see theorem 2.1).…”

“…In this paper we prove stronger versions of these results to show that the same approach can be used to obtain functions with no iterative roots of any order n ⩾ 2 (see theorem 2.1). We also generalize the Case (i) of theorem 2.4 in [4] to include all cases of finite cardinality (see (C1) of theorem 2.1). Furthermore, as an application of theorem 2.1, we prove the following result, which is much stronger than theorems 3.8 and 4.3 in [4] and includes an additional case of X = S 1 , the unit circle {z ∈ C : |z| = 1} in C in its usual topology.…”

“…We also generalize the Case (i) of theorem 2.4 in [4] to include all cases of finite cardinality (see (C1) of theorem 2.1). Furthermore, as an application of theorem 2.1, we prove the following result, which is much stronger than theorems 3.8 and 4.3 in [4] and includes an additional case of X = S 1 , the unit circle {z ∈ C : |z| = 1} in C in its usual topology.…”

Iterative roots of two-dimensional mappings

“…Although some regularity conditions for the roots were given there, a general investigation for the iterative roots of two-dimensional mappings is still unknown. For more results on the nonexistence of iterative roots, see references [10, 13].…”