Continuous Lebesgue measure-preserving maps on one-dimensional manifolds: a survey

We survey the current state-of-the-art about the dynamical behavior of continuous Lebesgue measure-preserving maps on one-dimensional manifolds

S-limit shadowing is generic for continuous Lebesgue measure preserving circle maps

In this paper we show that generic continuous Lebesgue measure preserving circle maps have the s-limit shadowing property. In addition we obtain that s-limit shadowing is a generic property also for continuous circle maps. In particular, this implies that classical shadowing, periodic shadowing and limit shadowing are generic in these two settings as well

Are generic dynamical properties stable under composition with rotations?

In this paper we provide a detailed topological and measure-theoretic study of Lebesgue measure-preserving circle maps that are rotated with inner and outer rotations which are independent of each other. In particular, we analyze the stability of the locally eventually onto and measure-theoretic mixing properties

HALF-HOMOGENEOUS CONTINUA

V magistrskem delu preučujemo polhomogene uverižljive kontinuume in poskušamo najti njihovo klasifikacijo glede na število njihovih krajišč. Najprej se seznanimo z osnovnimi definicijami in zapišemo nekaj enostavnih primerov kontinuumov. Predstavimo nekatere osnovne lastnosti kontinuumov, kot so ireducibilnost, dedna unikoherentnost in homogenost. Definiramo uverižljive kontinuume, konstruiramo nekaj primerov takih kontinuumov in predstavimo nekatere njihove lastnosti. V nadaljevanju predstavimo rezultate iz [5], ki govorijo o polhomogenih uverižljivih kontinuumih s končno mnogo krajišči. Dokažemo, da do homeomorfizma natančno obstajata natanko dva polhomogena uverižljiva kontinuuma z dvema krajiščema. Dokažemo še, da sta ta kontinuuma tudi edina polhomogena uverižljiva kontinuuma z neprazno končno množico krajišč. V zadnjem poglavju predstavimo še nekaj odprtih vprašanj in razpravljamo o možnih rešitvah.In Master\u27s Thesis half-homogeneous chainable continua are studied. More precisely we are discussing the possible classification of such continua with respect to the number of their end points. First we introduce basic definitions and produce some of the basic examples of continua. We also introduce some of the basic properties of continua, such as irreducibility, hereditarily unicoherence and homogenity. We define chainable continua, construct some examples of such continua and show some of their properties. Next we present the results from [5]. More precisely, we present the results about half-homogeneous chainable continua with finitely many end points. We prove that up to homeomorphism there are only two half-homogeneous chainable continua with two end points. We also prove that the two continua are the only half-homogeneous chainable continua with a nonempty finite set of end points. Finally, in the last chapter we introduce some open problems and discuss possible solutions