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Thomas Zürcher

Publications and source records attributed to Thomas Zürcher.

18 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.

math.FA↗
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.

math.CA↗
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 $μ$.

math.CA↗
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.

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

On sets where $\operatorname{lip} f$ is finite

Given a function $f\colon \mathbb{R}\to \mathbb{R}$, the so-called "little lip" function $\operatorname{lip} f$ is defined as follows: \begin{equation*} \operatorname{lip} f(x)=\liminf_{r{\scriptscriptstyle \searrow} 0}\sup_{|x-y|\le r} \frac{|f(y)-f(x)|}{r}. \end{equation*} We show that if $f$ is continuous on $\mathbb{R}$, then the set where $\operatorname{lip} f$ is infinite is a countable union of a countable intersection of closed sets (that is an $F_{σδ}$ set). On the other hand, given a countable union of closed sets $E$, we construct a continuous function $f$ such that $\operatorname{lip} f$ is infinite exactly on $E$. A further result is that for the typical continuous function $f$ on the real line $\operatorname{lip} f$ vanishes almost everywhere.

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.

math.CA↗
arXiv2013

Space-Filling vs. Luzin's Condition (N)

Let us assume that we are given two metric spaces, where the Hausdorff dimension of the first space is strictly smaller than the one of the second space. Suppose further that the first space has sigma-finite measure with respect to the Hausdorff measure of the corresponding dimension. We show for quite general metric spaces that for any measurable surjection from the first onto the second space, there is a set of measure zero that is mapped to a set of positive measure (both measures are the Hausdorff measures corresponding to the Hausdorff dimension of the first space). We also study more general situations where the measures on the two metric spaces are not necessarily the same and not necessarily Hausdorff measures.

math.CA↗
arXiv2013

Luzin's Condition (N) and Modulus of Continuity

In this paper, we establish Luzin's condition (N) for mappings in certain Sobolev-Orlicz spaces with certain moduli of continuity. Further, given a mapping in these Sobolev-Orlicz spaces, we give bounds on the size of the exceptional set where Luzin's condition (N) may fail. If a mapping violates Luzin's condition (N), we show that there is a Cantor set of measure zero that is mapped to a set of positive measure.

math.CA↗
arXiv2013

Sharp differentiability results for lip

We give a sharp condition on the lower local Lipschitz constant of a mapping from a metric space supporting a Poincaré inequality to a Banach space with the Radon-Nikodym property that guarantees differentiability at almost every point. We apply these results to obtain a non-embedding theorem for a corresponding class of mappings.

math.MG↗
arXiv2010

Static replications with traffic light options

It is well known that any sufficiently regular one-dimensional payoff function has an explicit static hedge by bonds, forward contracts and lots of vanilla options. We show that the natural extension of the corresponding representation leads to a static hedge based on the same instruments along with traffic light options, which have recently been introduced in the market. One big advantage of these replication strategies is the easy structure of the hedge. Hence, traffic light options are particularly powerful building blocks for more complicated bivariate options. While it is well known that the second strike derivative of non-discounted prices of vanilla options are related to the risk-neutral density of the underlying asset price in the corresponding absolutely continuous settings, similar statements hold for traffic light options in sufficiently regular bivariate settings.

q-fin.RM↗