Approximations of continuous functions by squares

Humke, Paul D.; Laczkovich, Miklós

doi:10.1017/s0143385700005599

Cited by 7 publications

(8 citation statements)

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“…We demonstrate the procedure in §3 by showing that the continuous self-maps without square roots are dense in the space of all continuous self-maps of the unit cube in for the supremum metric (see Theorem 3.8). This is a significant generalization of a similar result in [15] (Corollary 5, p. 362) for intervals of the real line. We believe that the most significant contribution of this article is the method itself.…”

Section: Introductionsupporting

confidence: 83%

“…As in §3, we prove that the continuous self-maps with no square roots are dense in the space of all continuous self-maps in a suitable topology, namely the compact-open topology (see Theorem 4.3). Finally, in §5, we prove that the squares of continuous self-maps are dense in the space of all continuous self-maps on , generalizing a result of [15] to higher dimensions.…”

Section: Introductionmentioning

confidence: 72%

“…As seen in §1, much literature is available on iterative square roots of functions on compact real intervals. In this section, we extend some of the results in [15] to the unit cube considered in the metric induced by the norm where . In studying iterative roots of continuous maps on intervals, one observes that it becomes essential to look at piecewise linear maps.…”

Section: Continuous Functions On the Unit Cube Inmentioning

confidence: 81%

“…Humke and Laczkovich considered the above question when with the usual metric This paper is heavily motivated by their results. They proved in [14, 15] that is not dense in , contains no balls of (that is, its complement is dense in ), and is an analytic non-Borel subset of for . Simon proved in [42–44] that is nowhere dense in and is not dense in .…”

Section: Introductionmentioning

confidence: 99%

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Iterative square roots of functions

Bhat¹,

Gopalakrishna²

2022

Ergod. Th. Dynam. Sys.

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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.

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Section: Introductionsupporting

confidence: 83%

Section: Introductionmentioning

confidence: 72%

Section: Continuous Functions On the Unit Cube Inmentioning

confidence: 81%

Section: Introductionmentioning

confidence: 99%

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Iterative square roots of functions

Bhat¹,

Gopalakrishna²

2022

Ergod. Th. Dynam. Sys.

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“…(i) (Humke and Laczkovich [9,10]) The complement of W(2; [0, 1]) is dense in C([0, 1]); W(2; [0, 1]) is not dense in C([0, 1]); and W(n; [0, 1]) is an analytic non-Borel subset of C([0, 1]) for n ≥ 2. (ii) (Simon [15,16,17,18]) W([0, 1]) is of first category and of zero Wiener measure; W(2; [0, 1]) is nowhere dense in C([0, 1]); and W([0, 1]) is not dense in C([0, 1]).…”

mentioning

confidence: 99%

A note on iterated maps of the unit sphere

Gopalakrishna¹

2023

Preprint

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Let C(S m ) denote the set of continuous maps from the unit sphere S m in R m+1 into itself endowed with the supremum norm. We prove that the set {f n : f ∈ C(S m ) and n ≥ 2} of iterated maps is not dense in C(S m ). This, in particular, proves that the periodic points of the iteration operator of order n are not dense in C(S m ) for all n ≥ 2, providing an alternative proof of the result that these operators are not Devaney chaotic on C(S m ) proved in [M. Veerapazham, C. Gopalakrishna, W. Zhang, Dynamics of the iteration operator on the space of continuous self-maps, Proc. Amer. Math. Soc., 149(1) (2021), 217-229].Iteration, which refers to repeating the same action, is not only a crucial operation in contemporary industrial production but also a typical loop program in computer algorithms. Many authors have explored numerous interesting and complex characteristics of this operation through discussions on various aspects such as dynamical systems ([12, 24]), iterative roots ([3, 5, 14, 28]), and solutions to iterative equations ([2, 13, 23, 27]). An iterated map on a non-empty set X is simply a self-map on it of the form f n for some self-map f on X and an integer n ≥ 2, with f k denoting the k-th order iterate of f defined recursively by f 0 = id, the identity map on X, anddenote the set of continuous maps from a locally compact Hausdorff space X into itself in the compact-open topology and W(X) := ∪ ∞ n=2 W(n; X), the set of iterated maps in C(X), where W(n; X) := {f n : f ∈ C(X)} for all n ≥ 2. Then the fact that even complex quadratic polynomials are not iterated maps on C, as shown in [14, Theorem 1], prompts us to ask the natural and interesting question: how large is W(X) in C(X)? Many researchers have investigated this problem from topological and measure theoretic perspectives, and the currently known findings in this regard are summarized in the following.

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On non-monotonicity height of piecewise monotone functions

Zeng

2021

Aequat. Math.

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Approximations of continuous functions by squares

Cited by 7 publications

References 1 publication

Iterative square roots of functions

Iterative square roots of functions

A note on iterated maps of the unit sphere

On non-monotonicity height of piecewise monotone functions

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