Factor maps and invariant distributional chaos

The main aim of this article is to show that maps with the specification property have invariant distributionally scrambled sets and that this kind of scrambled set can be transferred from factor to extension under finite-to-one factor maps. This solves some open questions in the literature of the topic.     

Geometry of expanding absolutely continuous invariant measures and the liftability problem

We show that for a large class of maps on manifolds of arbitrary finite dimension, the existence of a Gibbs–Markov–Young structure (with Lebesgue as the reference measure) is a necessary as well as sufficient condition for the existence of an invariant probability measure which is absolutely continuous measure (with respect to Lebesgue) and for which all Lyapunov exponents are positive.     

Gibbs-Markov structures and limit laws for partially hyperbolic attractors with mostly expanding central direction

We consider a partially hyperbolic set K on a Riemannian manifold M whose tangent space splits as TKM=Ecu⊕Es, for which the center-unstable direction Ecu expands non-uniformly on some local unstable disk. We show that under these assumptions f induces a Gibbs-Markov structure. Moreover, the decay of the return time function can be controlled in terms of the time typical points need to achieve some uniform expanding behavior in the center-unstable direction. As an application of the main result we obtain certain rates for decay of correlations, large deviations, an almost sure invariance principle and the validity of the central limit theorem.     

On Reeb components of invariant foliations of projectively Anosov flows

We show that if a C² codimension one foliation on a three-dimensional manifold has a Reeb component and is invariant under a projectively Anosov flow, then it must be a Reeb foliation on S²×S¹.     

Conley index of Poincaré maps in isolating segments

We calculate the discrete-time Conley index of the Poincaré map of a time-periodic ordinary differential equation in an isolated invariant set generated by a periodic isolating segment. As an application, we present results on the existence of bounded solutions of some planar equations.     

A note on the average shadowing property for expansive maps

Let f be a continuous map of a compact metric space. Assuming shadowing for f we relate the average shadowing property of f to transitivity and its variants. Our results extend and complete the work of Sakai [K. Sakai, Various shadowing properties for positively expansive maps, Topology Appl. 131 (2003) 15-31].     

Ergodic properties for some non-expanding non-reversible systems

We study several properties of invariant measures obtained from preimages, for non-invertible maps on fractal sets which model non-reversible dynamical systems. We give two ways to describe the distribution of all preimages for endomorphisms which are not necessarily expanding on a basic set Λ. We give a topological dynamics condition which guarantees that the corresponding measures converge to a unique conformal ergodic borelian measure; this helps in estimating the unstable dimension a.e. with respect to this measure with the help of Lyapunov exponents. When there exist negative Lyapunov exponents of this limit measure, we study the conditional probabilities induced on the non-uniform local stable manifolds by the limit measure, and also its pointwise dimension on stable manifolds.     

Measure-theoretic pressure for subadditive potentials

By using the definition of topological pressure for subadditive potentials, we introduce a new definition of measure-theoretic pressure for ergodic measures of subadditive potentials on a compact metric space. Under some assumption, we show that this definition has a similar formalism to the definition of measure-theoretic pressure for ergodic measures of additive potentials on a compact metric space. As an application, assume that the subadditive potential is Hölder continuous, then for an expansive homeomorphism with the specification property, we study the relationship between the measure-theoretic pressure and the periodic points.     

Transitivities of maps of general topological spaces

In this paper we investigate orbit-transitivity, strong orbit-transitivity, ω-transitivity and open-set-transitivity of maps of general topological spaces. The relation between these transitivities is studied. We discuss various topological spaces, containing pseudo-regular spaces, partially completable spaces, and topological spaces without quasi-isolated points. Several conditions on spaces and on continuity for one transitivity to imply another transitivity are given.     

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).     

Sharkovsky’s program for the classification of triangular maps is almost completed

For a continuous map of the interval, there are more than 50 conditions characterizing zero topological entropy. Some are applicable to the class of triangular maps (x,y)?(f(x),g_x(y)) of the square, but only a few of them are equivalent in this more general setting. In 1989, A.N. Sharkovsky posed the problem of proving or disproving all possible implications between them. During last 20 years, 32 conditions were considered, and most of the work was done. Only 45 relations out of 992 remained not clear. In this paper we give a survey of known results, provide two new examples disproving another 26 possible implications, and spell out the remaining 19 open problems; all but one concern distributional chaos.     

The local variational principle of topological pressure for sub-additive potentials

Let (X,T) be a topological dynamical system and be a sub-additive potential on C(X,R). Let U be an open cover of X. Then for any T-invariant measure μ, let . The topological pressure for open covers U is defined for sub-additive potentials. Then we have a variational principle:

     

An explicit construction of Kleinian groups with small limit sets

This paper provides an explicit construction of Kleinian groups that have small Hausdorff dimension of their limit sets. It is known that such groups exist and they can be constructed by results of Patterson. The purpose here is to work out the methods of calculation.     

Dynamics of systems on infinite lattices

The dynamics of infinite-dimensional lattice systems is studied. A necessary and sufficient condition for asymptotic compactness of lattice dynamical systems is introduced. It is shown that a lattice system has a global attractor if and only if it has a bounded absorbing set and is asymptotically null. As an application, it is proved that the lattice reaction-diffusion equation has a global attractor in a weighted l² space, which is compact as well as contains traveling waves. The upper semicontinuity of global attractors is also obtained when the lattice reaction-diffusion equation is approached by finite-dimensional systems.     

Dynamical systems on lattices with decaying interaction II: Hyperbolic sets and their invariant manifolds

This is the second part of the work devoted to the study of maps with decay in lattices. Here we apply the general theory developed in Fontich et al. (2011) [3] to the study of hyperbolic sets. In particular, we establish that any close enough perturbation with decay of an uncoupled lattice map with a hyperbolic set has also a hyperbolic set, with dynamics on the hyperbolic set conjugated to the corresponding of the uncoupled map. We also describe how the decay properties of the maps are inherited by the corresponding invariant manifolds.     

On topological entropy of transitive triangular maps

In the paper of Alsedà, Kolyada, Llibre and Snoha [L. Alsedà, S.F. Kolyada, J. Llibre, L'. Snoha, Entropy and periodic points for transitive maps, Trans. Amer. Math. Soc. 351 (1999) 1551-1573] there was—among others—proved that a nonminimal continuous transitive map f of a compact metric space (X,ρ) can be extended to a triangular map F on X×I (i.e., f is the base for F) in such a way that F is transitive and has the same entropy as f. The presented paper shows that under certain conditions the extension of minimal maps is guaranteed, too: Let (X,f) be a solenoidal dynamical system. Then there exist a transitive triangular map F such that h(F)=h(f).     

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.     

Trees and hairs for some hyperbolic entire maps of finite order

Let f be an entire transcendental map of finite order, such that all the singularities of f ⁻¹ are contained in a compact subset of the immediate basin B of an attracting fixed point. It is proved that there exist geometric coding trees of preimages of points from B with all branches convergent to points from . This implies that the Riemann map onto B has radial limits everywhere. Moreover, the Julia set of f consists of disjoint curves (hairs) tending to infinity, homeomorphic to a half-line, composed of points with a given symbolic itinerary and attached to the unique point accessible from B (endpoint of the hair). These facts generalize the corresponding results for exponential maps. Research supported by Polish KBN Grant No 2 P03A 034 25.