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A fixed point theorem for branched covering maps of the plane

Alexander Blokh, Lex Oversteegen (2009)

Fundamenta Mathematicae

It is known that every homeomorphism of the plane which admits an invariant non-separating continuum has a fixed point in the continuum. In this paper we show that any branched covering map of the plane of degree d, |d| ≤ 2, which has an invariant, non-separating continuum Y, either has a fixed point in Y, or is such that Y contains a minimal (in the sense of inclusion among invariant continua), fully invariant, non-separating subcontinuum X. In the latter case, f has to be of degree -2 and X has...

A further investigation for Egoroff's theorem with respect to monotone set functions

Jun Li (2003)

Kybernetika

In this paper, we investigate Egoroff’s theorem with respect to monotone set function, and show that a necessary and sufficient condition that Egoroff’s theorem remain valid for monotone set function is that the monotone set function fulfill condition (E). Therefore Egoroff’s theorem for non-additive measure is formulated in full generality.

A Gauss-Kuzmin theorem for the Rosen fractions

Gabriela I. Sebe (2002)

Journal de théorie des nombres de Bordeaux

Using the natural extensions for the Rosen maps, we give an infinite-order-chain representation of the sequence of the incomplete quotients of the Rosen fractions. Together with the ergodic behaviour of a certain homogeneous random system with complete connections, this allows us to solve a variant of Gauss-Kuzmin problem for the above fraction expansion.

A general class of iterative equations on the unit circle

Marek Cezary Zdun, Wei Nian Zhang (2007)

Czechoslovak Mathematical Journal

A class of functional equations with nonlinear iterates is discussed on the unit circle 𝕋 1 . By lifting maps on 𝕋 1 and maps on the torus 𝕋 n to Euclidean spaces and extending their restrictions to a compact interval or cube, we prove existence, uniqueness and stability for their continuous solutions.

A generalisation of Mahler measure and its application in algebraic dynamical systems

Manfred Einsiedler (1999)

Acta Arithmetica

We prove a generalisation of the entropy formula for certain algebraic ℤ d -actions given in [2] and [4]. This formula expresses the entropy as the logarithm of the Mahler measure of a Laurent polynomial in d variables with integral coefficients. We replace the rational integers by the integers in a number field and examine the entropy of the corresponding dynamical system.

A generalization of semiflows on monomials

Hamid Kulosman, Alica Miller (2012)

Mathematica Bohemica

Let K be a field, A = K [ X 1 , ⋯ , X n ] and 𝕄 the set of monomials of A . It is well known that the set of monomial ideals of A is in a bijective correspondence with the set of all subsemiflows of the 𝕄 -semiflow 𝕄 . We generalize this to the case of term ideals of A = R [ X 1 , ⋯ , X n ] , where R is a commutative Noetherian ring. A term ideal of A is an ideal of A generated by a family of terms c X 1 μ 1 ⋯ X n μ n , where c ∈ R and μ 1 , ⋯ , μ n are integers ≥ 0 .

A generalization of the self-dual induction to every interval exchange transformation

Sébastien Ferenczi (2014)

Annales de l’institut Fourier

We generalize to all interval exchanges the induction algorithm defined by Ferenczi and Zamboni for a particular class. Each interval exchange corresponds to an infinite path in a graph whose vertices are certain unions of trees we call castle forests. We use it to describe those words obtained by coding trajectories and give an explicit representation of the system by Rokhlin towers. As an application, we build the first known example of a weakly mixing interval exchange outside the hyperelliptic...

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