The complex of pant decompositions of a surface

We exhibit a set of edges (moves) and 2-cells (relations) making the complex of pant decompositions on a surface a simply connected complex. Our construction, unlike the previous ones, keeps the arguments concerning the structural transformations independent from those deriving from the action of the mapping class group. The moves and the relations turn out to be supported in subsurfaces with 3g−3+n=1,2 (where g is the genus and n is the number of boundary components), illustrating in this way the so-called Grothendieck principle.     

Expansive homeomorphisms and plane separating continua

A homeomorphism is expansive provided that there exists a constant c>0 and for every x,y∈X there exists an integer n, dependent only on x and y, such that d(hn(x),hn(y))>c. It is shown that if X is a 1-dimensional continuum that separates the plane into 2 pieces, then h cannot be expansive.     

Signatures of foliated surface bundles and the symplectomorphism groups of surfaces

For any closed oriented surface Σ_g of genus g?3, we prove the existence of foliatedΣ_g-bundles over surfaces such that the signatures of the total spaces are non-zero. We can arrange that the total holonomy of the horizontal foliations preserve a prescribed symplectic form ω on the fiber. We relate the cohomology class represented by the transverse symplectic form to a crossed homomorphism which is an extension of the flux homomorphism from the identity component to the whole group of symplectomorphisms of Σ_g with respect to the symplectic form ω.     

Nielsen classes of orientation reversing homeomorphisms and complementary domains of plane separating invariant continua

Let h be an orientation reversing planar homeomorphism and X be an invariant plane separating continuum. We show that there is a natural linear order on the invariant components of R²?X that resemble the one found in connected unions of circles invariant under the reflection r(x,y)=(−x,y). The main result relates to the Nielsen fixed point theory and work of Krystyna Kuperberg on fixed points of planar homeomorphisms in invariant continua.     

Extending homeomorphisms from punctured surfaces to handlebodies

Let Hg be a genus g handlebody and MCG2n(Tg) be the group of the isotopy classes of orientation preserving homeomorphisms of Tg=∂Hg, fixing a given set of 2n points. In this paper we find a finite set of generators for , the subgroup of MCG2n(Tg) consisting of the isotopy classes of homeomorphisms of Tg admitting an extension to the handlebody and keeping fixed the union of n disjoint properly embedded trivial arcs. This result generalizes a previous one obtained by the authors for n=1. The subgroup turns out to be important for the study of knots and links in closed 3-manifolds via (g,n)-decompositions. In fact, the links represented by the isotopy classes belonging to the same left cosets of in MCG2n(Tg) are equivalent.     

Hyperbolic sections in surface bundles

We give conditions assuring that the given section in a surface bundle over the circle is hyperbolic in terms of the ‘‘projection" in the fiber surface according to the Nielsen-Thurston types of the monodromies.     

The Euler-Poincaré characteristic of index maps

We apply the concept of the Euler-Poincaré characteristic and the periodicity number to the index map of an isolated invariant set in order to obtain a new criterion for the existence of periodic points of a continuous map in a given set.     

Dynamical systems which realize given random bi-sequences of points on their orbits

A dynamical system consists of a phase space of possible states, together with an evolution rule that determines all future states and all past states given a state at any particular moment. In this paper, we show that for any countable random infinite bi-sequences of states of some phase space, there exists an evolution rule in C⁰-topology which realizes precisely the given sequences of states on their orbits and satisfies some regular conditions on the times to realize the states.     

Homotopy Conley index for discrete multivalued dynamical systems

We define an index of Conley type for a certain class of upper semicontinuous multivalued dynamical systems. We use the Szymczak functor and apply techniques introduced by Reineck, Mrozek and Srzednicki for the index over the base. Moreover we introduce the notion of the homotopy partial functor for the usc maps. We show that the index possesses Wa?ewski and homotopy properties. We also give four examples that exhibit the benefits of our index over the cohomological index defined by Mrozek and Kaczyński.     

A trace formula for the forcing relation of braids

The forcing relation of braids has been introduced for a 2-dimensional analogue of the Sharkovskii order on periods for maps of the interval. In this paper, by making use of the Nielsen fixed point theory and a representation of braid groups, we deduce a trace formula for the computation of the forcing order.     

The Hausdorff dimension of chaotic sets for interval self-maps

We obtain two sufficient conditions for an interval self-map to have a chaotic set with positive Hausdorff dimension. Furthermore, we point out that for any interval Lipschitz maps with positive topological entropy there is a chaotic set with positive Hausdorff dimension.     

Zig-zag chains and metric equivalences between ultrametric spaces

We study the classification of ultrametric spaces based on their small scale geometry (uniform homeomorphism), large scale geometry (coarse equivalence) and both (bi-uniform equivalences). Using a combinatoric approach we consider every ultrametric space as the end space of a chain and prove that all these equivalences can be characterized by the existence of a common zig-zag chain.     

Topological entropy of maps on regular curves

In [G.T. Seidler, The topological entropy of homeomorphisms on one-dimensional continua, Proc. Amer. Math. Soc. 108 (1990) 1025-1030], G.T. Seidler proved that the topological entropy of every homeomorphism on a regular curve is zero. Also, in [H. Kato, Topological entropy of monotone maps and confluent maps on regular curves, Topology Proc. 28 (2) (2004) 587-593] the topological entropy of confluent maps on regular curves was investigated. In particular, it was proved that the topological entropy of every monotone map on any regular curve is zero. In this paper, furthermore we investigate the topological entropy of more general maps on regular curves. We evaluate the topological entropy of maps f on regular curves X in terms of the growth of the number of components of f−n(y) (y∈X).     

The configuration space of planar spidery linkages

The configuration space of the mechanism of a planar robot is studied. We consider a robot which has n arms such that each arm is of length 1+1 and has a rotational joint in the middle, and that the endpoint of the kth arm is fixed to . Generically, the configuration space is diffeomorphic to an orientable closed surface. Its genus is given by a topological way and a Morse theoretical way. The homeomorphism types of it when it is singular is also given.     

Coincidence Nielsen numbers for covering maps for smooth manifolds

Let be maps between closed smooth manifolds of the same dimension, and let and be finite regular covering maps. If the manifolds are nonorientable, using semi-index, we introduce two new Nielsen numbers. The first one is the Linear Nielsen number NL(f,g), which is a linear combination of the Nielsen numbers of the lifts of f and g. The second one is the Nonlinear Nielsen number NED(f,g). It is the number of certain essential classes whose inverse images by p are inessential Nielsen classes. In fact, N(f,g)=NL(f,g)+NED(f,g), where by abuse of notation, N(f,g) denotes the coincidence Nielsen number defined using semi-index.     

Strongly countable dimensional compacta form the Hurewicz set

The hyperspaces of strongly countable dimensional compacta of positive dimension and of strongly countable dimensional continua of dimension greater than 1 in the Hilbert cube are homeomorphic to the Hurewicz set of all nonempty countable closed subsets of the unit interval [0,1]. These facts hold true, in particular, for covering dimension dim and cohomological dimension dimG, where G is any Abelian group.     

The hyperspace of hereditarily decomposable subcontinua of a cube is the Hurewicz set

The hyperspaces of hereditarily decomposable continua and of decomposable subcontinua without pseudoarcs in the cube of dimension greater than 2 are homeomorphic to the Hurewicz set of all nonempty countable closed subsets of the unit interval [0,1]. Moreover, in such a cube, all indecomposable subcontinua form a homotopy dense subset of the hyperspace of (nonempty) subcontinua.     

A note on monotonically metacompact spaces

We show that any metacompact Moore space is monotonically metacompact and use that result to characterize monotone metacompactness in certain generalized ordered (GO) spaces. We show, for example, that a generalized ordered space with a σ-closed-discrete dense subset is metrizable if and only if it is monotonically (countably) metacompact, that a monotonically (countably) metacompact GO-space is hereditarily paracompact, and that a locally countably compact GO-space is metrizable if and only if it is monotonically (countably) metacompact. We give an example of a non-metrizable LOTS that is monotonically metacompact, thereby answering a question posed by S.G. Popvassilev. We also give consistent examples showing that if there is a Souslin line, then there is one Souslin line that is monotonically countable metacompact, and another Souslin line that is not monotonically countably metacompact.     

A note on monotone covering properties

In this note, we show that a monotonically normal space that is monotonically countably metacompact (monotonically meta-Lindelöf) must be hereditarily paracompact. This answers a question of H.R. Bennett, K.P. Hart and D.J. Lutzer. We also show that any compact monotonically meta-Lindelöf T₂-space is first countable. In the last part of the note, we point out that there is a gap in Proposition 3.8 which appears in [H.R. Bennett, K.P. Hart, D.J. Lutzer, A note on monotonically metacompact spaces, Topology Appl. 157 (2) (2010) 456-465]. We finally give a detailed proof of how to overcome the gap.     

Positive entropy on nonautonomous interval maps and the topology of the inverse limit space

Entropy on nonautonomous maps of the interval is defined 2 ways. Under one definition, called forward entropy, it is shown that positive entropy implies that the inverse limit space of contains an indecomposable subcontinuum. Under the second definition, called backwards entropy, it is shown that the inverse limit space of is not locally connected.