We develop a nonlinear Fredholm alternative theory involving k-ball and k-set perturbations of general homeomorphisms as well as of homeomorphisms that are nonlinear Fredholm maps of index zero. Various generalized first Fredholm theorems are given and finite solvability of general (odd) Fredholm maps of index zero is also studied. We apply these results to the unique and finite solvability of ...

Guba and Sapir asked, in their joint paper [3], if the simultaneous conjugacy problem was solvable in Diagram Groups or, at least, for Thompson’s group F . We give an elementary proof for the solution of the latter question. This relies purely on the description of F as the group of piecewise linear orientationpreserving homeomorphisms of the unit interval with dyadic breakpoints and slopes tha...

We prove that the family of measured dynamical systems which can be realised as uniquely ergodic minimal homeomorphisms on a given manifold (of dimension at least two) is stable under measured extension. As a corollary, any ergodic system with an irrational eigenvalue is isomorphic to a uniquely ergodic minimal homeomorphism on the two-torus. The proof uses the following improvement of Weiss re...

In digital geometry, objects such as digital straight lines can be considered as equivalent classes of Euclidean straight lines with respect to a digitization process. This paper investigates the analysis of the set of straight lines whose digitization is a given digital straight segment with the help of tools from integral geometry. After a definition of a measure on such sets, we illustrate s...

In order to study the geometries of a hyperbolic plane, it is necessary to understand the set of transformations that map from the space to itself. This paper shows that in the Poincaré half-plane model, this group of transformations is the general Möbius group. This is done by showing that the general Möbius group is a group of homeomorphisms that map circles to circles on the extended complex...

This article is concerned with conjugacy problems arising in homeomorphisms group, Hom(F ), of non-compact subsets F of normed vector spaces E. Given two homeomorphisms f and g in Hom(F ), it is shown how the existence of a conjugacy may be related to the existence of a common generalized eigenfunction of the associated Koopman operators. This common eigenfunction serves to build a topology on ...