Come with us to read a new book that is coming recently. Yeah, this is a new coming book that many people really want to read will you be one of them? Of course, you should be. It will not make you feel so hard to enjoy your life. Even some people think that reading is a hard to do, you must be sure that you can do it. Hard will be felt when you have no ideas about what kind of book to read. Or sometimes, your reading material is not interesting enough.

No abstract

We investigate homeomorphisms of a compact interval, applied randomly. We consider this system as a skew product with the two-sided Bernoulli shift in the base. If on the open interval there is a metric in which almost all maps are contractions, then (with mild additional assumptions) there exists a global pullback attractor, which is a graph of a function from the base to the fiber. It is also a forward attractor. However, the value of this function depends only on the past, so when we take the one-sided shift in the base, it disappears. We illustrate those phenomena on an example, where there are two piecewise linear homeomorphisms, one moving points to the right and the other one to the left.

This survey article is concerned with the study of bifurcations of piecewise-smooth maps. We review the literature in circle maps and quasicontractions and provide paths through this literature to prove sufficient conditions for the occurrence of two types of bifurcation scenarios involving rich dynamics. The first scenario consists of the appearance of periodic orbits whose symbolic sequences and "rotation" numbers follow a Farey tree structure; the periods of the periodic orbits are given by consecutive addition. This is called the period adding bifurcation, and its proof relies on results for maps on the circle. In the second scenario, symbolic sequences are obtained by consecutive attachment of a given symbolic block and the periods of periodic orbits are incremented by a constant term. It is called the period incrementing bifurcation, in its proof relies on results for maps on the interval. We also discuss the expanding cases, as some of the partial results found in the literature also hold when these maps lose contractiveness. The higher dimensional case is also discussed by means of quasi-contractions. We also provide applied examples in control theory, power electronics and neuroscience where these results can be applied to obtain precise descriptions of their dynamics.

The main results in this paper concern the minimal sets of periods possible in a given homotopy class of torus maps. For maps on the 2-torus, we provide a complete description of these minimal sets. A number of results on higher dimensional tori are also proved; including criteria for every map in a given homotopy class to have all periods, or all but finitely many periods.

It is well known that a continuous piecewise monotone interval map with positive topological entropy is semiconjugate to a map of a constant slope and the same entropy, and if it is additionally transitive then this semiconjugacy is actually a conjugacy. We generalize this result to piecewise continuous piecewise monotone interval maps, and as a consequence, get it also for piecewise monotone graph maps. We show that assigning to a continuous transitive piecewise monotone map of positive entropy a map of constant slope conjugate to it defines an operator, and show that this operator is not continuous.Date: August 6, 2014. 2010 Mathematics Subject Classification. Primary: 37E05, 37B40. Key words and phrases. Piecewise monotonotone maps, semiconjugacy to a map of constant slope, topological entropy, interval Markov maps, measure of maximal entropy.

No abstract

No abstract