On extensions of transitive maps

On a compact real interval, the spaces of all transitive maps, all piecewise monotone transitive maps and all piecewise linear transitive maps are considered with the uniform metric. It is proved that they are contractible and uniformly locally arcwise connected. Then the spaces of all piecewise monotone transitive maps with given number of pieces as well as various unions of such spaces are considered and their connectedness properties are studied.

Consider a one-parameter family of circle diffeomorphisms which unfolds a saddle-node periodic orbit at the edge of an ‘Arnold tongue’. Recently it has been shown that homoclinic orbits of the saddle-node periodic points induce a ‘transition map’ which completely describes the smooth conjugacy classes of such maps and determines the universalities of the bifurcations resulting from the disappearance of the saddle-node periodic points. We show that after the bifurcation the relative density (measure) of parameter values corresponding to irrational rotation numbers is completely determined by the transition map and give a formula for this density. It turns out that this density is always less than 1 and generically greater than 0, with the exceptional cases having infinite co-dimension.

Transitive and series transitive maps on [formula omitted

We give a new formula for the topological entropy of a continuous map f of the interval to itself with f2 topologically transitive. This leads to stronger forms of well-known results about horseshoes and periodic orbit patterns for continuous maps of the interval in the special case of topologically transitive maps.

We study the totally transitive graph maps which have periodic points. In particular we show that the cofiniteness of the set of periods characterizes these maps among transitive graph maps.

For continuous self-maps of compact metric spaces, we explore the relationship among the shadowable points, sensitive points, and entropy points. Specifically, we show that (1) if the set of shadowable points is dense in the phase space, then any interior point of the set of sensitive points is an entropy point; and (2) if the topological entropy is zero, then the denseness of the set of shadowable points is equivalent to almost chain continuity. In addition, we present a counter-example to a question raised by Ye and Zhang regarding entropy points.

b-7 - Cleavable (Splittable) Spaces

Intra-orbit separation of dense orbits of dendrite maps

We prove that the Möbius disjointness conjecture holds for graph maps with zero topological entropy and for all monotone local dendrite maps. We further show that this also holds for continuous maps on certain class of dendrites. Moreover, we see that there is an example of transitive dendrite map with zero entropy for which Möbius disjointness conjecture holds.

<p>We prove that a continuous triangular map G of the n-dimensional cube I<sup>n</sup> has only fixed points and no other periodic points if and only if G has a common fixed point with every continuous triangular map F that is nontrivially compatible with G. This is an analog of Jungck theorem for maps of a real compact interval. We also discuss possible extensions of Jungck theorem, Jachymski theorem and some related results to more general spaces. In particular, the spaces with the fixed point property and the complete invariance property are considered.</p>

In this paper, we work on the existence of approximate solution for the fixed point equation Rx + Qx = x, when R is a weak contractive mapping and Q is a continuous self-map which is a new topic related to the background literature. Till now, no results were found to find the existence of approximate solution for such a kind of mappings in literature to the best of our knowledge. Here we prove the existence of best proximity point for the sum of a weak contractive non self map and a continuous self map. Also, we provide an example to support our main theorem. And also, we prove the existence of best proximity point for the sum of Geraghty-contraction and continuous map.

The aim of this paper is to study some fixed point results in compact metric spaces. It discusses the existence and uniqueness of fixed point of a self-map on a metric space. In this paper we defined generalized contraction condition on the continuous self-mappings on compact metric spaces. Our result generalizes the results of Edelstein and Fisher. Keywords: Compact Space, Metric Space Contraction, Fixed Point, Convergence sequence

We call a \emph{comb} a map $f:I\to [0,\infty)$, where $I$ is a compact interval, such that $\{f\ge \varepsilon\}$ is finite for any $\varepsilon$. A comb induces a (pseudo)-distance $\dtf$ on $\{f=0\}$ defined by $\dtf(s,t) = \max_{(s\wedge t, s\vee t)} f$. We describe the completion $\bar I$ of $\{f=0\}$ for this metric, which is a compact ultrametric space called \emph{comb metric space}. Conversely, we prove that any compact, ultrametric space $(U,d)$ without isolated points is isometric to a comb metric space. We show various examples of the comb representation of well-known ultrametric spaces: the Kingman coalescent, infinite sequences of a finite alphabet, the $p$-adic field and spheres of locally compact real trees. In particular, for a rooted, locally compact real tree defined from its contour process $h$, the comb isometric to the sphere of radius $T$ centered at the root can be extracted from $h$ as the depths of its excursions away from $T$.

Optimal fixed size confidence procedures are derived for the mean of a normal random variable with known variance, when the mean is restricted to a compact interval. These confidence procedures are, in turn, based on the solution of a related minimax decision problem which is characterized by a zero-one loss function and a compact interval parameter space. The minimax rules obtained are nonrandomized, admissible, Bayes procedures. The decision-theoretic results are extended in two ways: (i) structurally similar (admissible) Bayes minimax rules are also obtained when the sampling distribution has a density function which is unimodal, symmetric about the location parameter and possesses a (strictly) monotone likelihood ratio; (ii) structurally similar minimax rules (minimax within the class of nonrandomized, odd, monotone procedures) are again obtained when the assumption of a monotone likelihood ratio is relaxed.

A pair of spaces of upper semi-continuous maps and continuous maps