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Janusz Morawiec

Publications and source records attributed to Janusz Morawiec.

At least 19 recordsLinked to original sources
arXiv2026

Operators arising from invariant measures under some class of multidimensional transformations

We investigate a linear operator associated with a functional equation that arises from studying some class of invariant measures under multidimensional transformations. By examining its iterates, we derive an explicit solution formula for the functional equation in some class of functions and establish a result on the existence of an absolutely continuous invariant measure under a multidimensional transformation that can be viewed as a generalization of classical $p$-adic maps to higher dimensions.

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arXiv2025

Invariant Probability Measures under $p$-adic Transformations

It is well-known that the Lebesgue measure is the unique absolutely continuous invariant probability measure under the $p$-adic transformation. The purpose of this paper is to characterize the family of all invariant probability measures under the $p$-adic transformation and to provide some description of them. In particular, we describe the subfamily of all atomic invariant measures under the $p$-adic transformation as well as the subfamily of all continuous and singular invariant probability measures under the $p$-adic transformation. Iterative functional equations play the base role in our considerations.

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arXiv2024

The betweenness relation distinguishes non-similar pairs of concentric circles

Two subsets $A, B$ of the plane are betweenness isomorphic if there is a bijection $f\colon A\to B$ such that, for every $x,y,z\in A$, the point $f(z)$ lies on the line segment connecting $f(x)$ and $f(y)$ if and only if $z$ lies on the line segment connecting $x$ and $y$. In general, it is quite difficult to tell whether two given subsets of the plane are betweenness isomorphic. We concentrate on the case when the sets $A,B$ belong to the family $ \mathcal A_c$ of unions of pairs of concentric circles in the plane. We prove that $A, B \in \mathcal A_c$ are betweenness isomorphic if and only if they are similar. In particular, there are continuum many betweenness isomorphism classes in $ \mathcal A_c$, and each of these classes consists exactly of all scaled translations of an arbitrary representative of the class. Furthermore, we show that every betweenness isomorphism between sets $A,B\in \mathcal A_c$ is exactly the restriction of a scaled isometry of the plane.

math.MG↗
arXiv2024

Another look at the Matkowski and Wesołowski problem yielding a new class of solutions

The following MW--problem was posed independently by Janusz Matkowski and Jacek Wesołowski in different forms in 1985 and 2009, respectively: Are there increasing and continuous functions $φ\colon [0,1]\to [0,1]$, distinct from the identity on $[0,1]$, such that $φ(0)=0$, $φ(1)=1$ and $φ(x)=φ(\frac{x}{2})+φ(\frac{x+1}{2})-φ(\frac{1}{2})$ for every $x\in[0,1]$? By now, it is known that each of the de Rham functions $R_p$, where $p\in(0,1)$, is a solution of the MW--problem, and for any Borel probability measure $μ$ concentrated on $(0,1)$ the formula $ϕ_μ(x)=\int_{(0,1)}R_p(x) dμ(p)$ defines a solution $ϕ_μ\colon[0,1]\to[0,1]$ of this problem as well. In this paper, we give a new family of solutions of the MW--problem consisting of Cantor-type functions. We also prove that there are strictly increasing solutions of the MW--problem that are not of the above integral form with any Borel probability measure $μ$.

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arXiv2022

Monotone mappings and lines

We study betweenness preserving mappings (we call them \emph{monotone}) defined on subsets of the plane. Once the domain is a convex set, such a mapping is either the restriction of a homography, or its image is contained in the union of a line and a single point, or its image consists of five points, one of them being between two disjoint pairs of the other four points. We also show that an open planar set cannot be mapped in a one-to-one monotone way into the real line. From this we deduce that a one-to-one monotone mapping from a convex planar set with nonempty interior is necessarily a partial homography. Finally, we prove that a set consisting of three pairwise non-parallel lines does not admit a one-to-one monotone mapping into the real line, while on the other hand a set consisting of three closed line segments intersecting at a single point does admit such a mapping.

math.MG↗
arXiv2021

Linear Functional Equations and their Solutions in Lorentz Spaces

Assume that $Ω\subset \mathbb{R}^k$ is an open set, $V$ is a separable Banach space over a field $\mathbb K\in\{\mathbb R,\mathbb C\}$ and $f_1,\ldots,f_N \colonΩ\to Ω$, $g_1,\ldots, g_N\colonΩ\to \mathbb{K}$, $h_0\colon Ω\to V$ are given functions. We are interested in the existence and uniqueness of solutions $φ\colon Ω\to V$ of the linear functional equation $φ=\sum_{k=1}^{N}g_k\cdot(φ\circ f_k)+h_0$ in Lorentz spaces.

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arXiv2019

An application of medial limits to iterative functional equations

Assume that $(Ω,\mathcal A,P)$ is a probability space, $f\colon[0,1] \times Ω\to[0,1]$ is a function such that $f(0,ω)=0$, $f(1,ω)=1$ for every $ω\inΩ$, $g\colon[0,1]\to\mathbb R$ is a bounded function such that $g(0)=g(1)=0$, and $a,b\in\mathbb R$. Applying medial limits we describe bounded solutions $φ\colon[0,1] \to \mathbb R$ of the equation \begin{equation*} φ(x) = \int_Ωφ(f(x,ω)) dP(ω)+g(x) \end{equation*} satisfying the boundary conditions $φ(0)=a$ and $φ(1)=b$.

math.CA↗
arXiv2018

Some Class of Linear Operators Involved in Functional Equations

Fix $N\in\mathbb N$ and assume that for every $n\in\{1,\ldots, N\}$ the functions $f_n\colon[0,1]\to[0,1]$ and $g_n\colon[0,1]\to\mathbb R$ are Lebesgue measurable, $f_n$ is almost everywhere approximately differentiable with $|g_n(x)|<|f'_n(x)|$ for almost all $x\in [0,1]$, there exists $K\in\mathbb N$ such that the set $\{x\in [0,1]:\mathrm{card}{f_n^{-1}(x)}>K\}$ is of Lebesgue measure zero, $f_n$ satisfy Luzin's condition N, and the set $f_n^{-1}(A)$ is of Lebesgue measure zero for every set $A\subset\mathbb R$ of Lebesgue measure zero. We show that the formula $Ph=\sum_{n=1}^{N}g_n\!\cdot\!(h\circ f_n)$ defines a linear and continuous operator $P\colon L^1([0,1])\to L^1([0,1])$, and then we obtain results on the existence and uniqueness of solutions $φ\in L^1([0,1])$ of the equation $φ=Pφ+g$ with a given $g\in L^1([0,1])$.

math.CA↗
arXiv2018

An application of functional equations for generating $\varepsilon$-invariant measures

Let $(X,{\mathcal A},μ)$ be a probability space and let $S\colon X\to X$ be a measurable transformation. Motivated by the paper of K. Nikodem [Czechoslovak Math. J. 41(116) (4) (1991) 565--569], we concentrate on a functional equation generating measures that are absolutely continuous with respect to $μ$ and $\varepsilon$-invariant under $S$. As a consequence of the investigation, we obtain a result on the existence and uniqueness of solutions $φ\in L^1([0,1])$ of the functional equation $$ φ(x)=\sum_{n=1}^{N}|f_n'(x)|φ(f_n(x))+g(x), $$ where $g\in L^1([0,1])$ and $f_1,\ldots,f_N\colon[0,1]\to[0,1]$ are functions satisfying some extra conditions.

math.CA↗
arXiv2018

Means of iterates

We determine continuous bijections $f$, acting on a real interval into itself, whose $k$-fold iterate is the quasi-arithmetic mean of all its subsequent iterates from $f^0$ up to $f^n$ (where $0\le k\le n$). Namely, we prove that if at most one of the numbers $k,n$ is odd, then such functions consist of at most three affine pieces.

math.CA↗
arXiv2018

On a Problem of Janusz Matkowski and Jacek Wesołowski, II

We continue our study started in "On a problem of Janusz Matkowski and Jacek Wesołowski" (see arXiv:1703.08459) of the functional equation \begin{equation*} φ(x)=\sum_{n=0}^{N}φ(f_n(x))-\sum_{n=0}^{N}φ(f_n(0)) \end{equation*} and its increasing and continuous solutions $φ\colon[0,1]\to[0,1]$ such that $φ(0)=0$ and $φ(1)=1$. In this paper we assume that $f_0,\ldots,f_N\colon[0,1]\to[0,1]$ are strictly increasing contractions such that \begin{equation*} 0\leq f_0(0)<f_0(1)\leq f_1(0)<\cdots <f_{N-1}(1)\leq f_N(0)<f_N(1)\leq 1 \end{equation*} and at least one of the weak inequalities is strong.

math.CA↗
arXiv2017

Attractor of Cantor Type with Positive Measure

We construct an iterated function system consisting of strictly increasing contractions $f,g\colon [0,1]\to [0,1]$ with $f([0,1])\cap g([0,1])=\emptyset$ and such that its attractor has positive Lebesgue measure.

math.CA↗
arXiv2017

On a problem of Janusz Matkowski and Jacek Wesołowski

We study the problem of the existence of increasing and continuous solutions $φ\colon[0,1]\to[0,1]$ such that $φ(0)=0$ and $φ(1)=1$ of the functional equation \begin{equation*} φ(x)=\sum_{n=0}^{N}φ(f_n(x))-\sum_{n=1}^{N}φ(f_n(0)), \end{equation*} where $N\in\mathbb N$ and $f_0,\ldots,f_N\colon[0,1]\to[0,1]$ are strictly increasing contractions satisfying the following condition $0=f_0(0)<f_0(1)=f_1(0)<\cdots<f_{N-1}(1)=f_N(0)<f_N(1)=1$. In particular, we give an answer to the problem posed in the article Remark on BV-solutions of a functional equation connected with invariant measures by Janusz Matkowski concerning a very special case of that equation.

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arXiv2015

Reducing the polynomial-like iterative equations order and a generalized Zoltán Boros' problem

We present a technique for reducing the order of polynomial-like iterative equations; in particular, we answer a question asked by Wenmeng Zhang and Weinian Zhang. Our method involves the asymptotic behaviour of the sequence of consecutive iterates of the unknown function at a given point. As an application we solve a problem of Zoltán Boros posed during the 50th ISFE (2012).

math.CA↗
arXiv2015

Integrable solutions of inhomogeneous refinement type equations on intervals

Given a probability measure $P$ on a $σ$-algebra of subsets of a set $Ω$, an interval $I\subset\mathbb R$, $g\in L^1(I)$, and a function $φ\colon I\timesΩ\to I$ fulfilling some conditions we obtain results on the existence of solutions $f\in L^1(I)$ of the inhomogeneous refinement type equation $$ f(x)=\int_Ω\big|φ'_x(x,ω)\big|f(φ(x,ω))dP(ω)+g(x). $$

math.CA↗
arXiv2015

Inhomogeneous refinement equations with random affine maps

Given a probability space $(Ω,{\mathcal A},P)$, random variables $L,M\colonΩ\to\mathbb R$ and $g\in L^1(\mathbb R)$ we obtain two characterizations of these $f\in L^1(\mathbb R)$ which are solutions of the inhomogeneous refinement equation with a random affine map of the form $f(x)=\int_Ω|L(ω)|f(L(ω)x-M(ω))P(dω)+g(x)$.

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