Zbigniew Nitecki

Shift automorphisms in the Hénon mapping

We investigate the global behavior of the quadratic diffeomorphism of the plane given byH(x,y)=(1+y−Ax2,Bx). Numerical work by Hénon, Curry, and Feit indicate that, for certain values of the parameters, this mapping admits a “strange attractor”. Here we show that, forA small enough, all points in the plane eventually move to infinity under iteration ofH. On the other hand, whenA is large enough, the nonwandering set ofH is topologically conjugate to the shift automorphism on two symbols.

Topological Dynamics on the Interval

A great deal has been written about maps of the interval, especially in recent years. In addition to many detailed mathematical papers, there have been a number of numerical studies and descriptive works (Co, Fe, GM, HoH, Ma2, MeS, Mr) and studies relating one-dimensional dynamical systems to models in the biological (GOI, HLM, La5–6, Ma1, MaO, WL) and physical (CE, GM1, La3,6, LaR, Lo1–3) sciences. The subject is appealing because it is easy to talk about — very little technical apparatus is needed to pose many problems in the field - and yet one-dimensional systems can exhibit surprizingly complex dynamic behavior.

Non-wandering sets of the powers of maps of the interval

We show that, for maps of the interval, the non-wandering set of the map coincides with the non-wandering set of each of its odd powers, while the nonwandering set of any of its even powers can be strictly smaller.

BRAIDS AND THE NIELSEN-THURSTON CLASSIFICATION

The well-known connection between braids and mapping classes on the n-punctured disc can be exploited to decide, using braid-theoretic techniques, the place of a given isotopy class in the Nielsen-Thurston classification as reducible or irreducible, and in the latter case, as periodic or pseudo-Anosov.

PREIMAGE ENTROPY FOR MAPPINGS

Several entropy-like invariants have been defined for noninvertible maps, based on various ways of measuring the dispersion of preimages and preimage sets in the past. We investigate basic properties of four such invariants, finding that their behavior in some ways differs sharply from the analogous behavior for topological entropy.

Homoclinic and non-wandering points for maps of the circle

For continuous maps ƒ of the circle to itself, we show: (A) the set of nonwandering points of ƒ coincides with that of ƒ n for every odd n ; (B) ƒ has a horseshoe if and only if it has a non-wandering homoclinic point; (C) if the set of periodic points is closed and non-empty, then every non-wandering point is periodic.

Entropy and preimage sets

We study the relation between topological entropy and the dispersion of preimages. Symbolic dynamics plays a crucial role in our investigation. For forward expansive maps, we show that the two pointwise preimage entropy invariants defined by Hurley agree with each other and with topological entropy, and are reflected in the growth rate of the number of preimages of a single point, called a preimage growth point for the map. We extend this notion to that of an entropy point for a system, in which the dispersion of preimages of an

A periodic attractor determined by one function

\varepsilon

Cantorvals and Subsum Sets of Null Sequences

-stable set measures topological entropy. We show that for maps satisfying a weak form of the specification property, every point is an entropy point and that every asymptotically h -expansive homeomorphism (in particular, every smooth diffeomorphism of a compact manifold) has entropy points. Examples are given of maps in which Hurleys invariants differ and of homeomorphisms with no entropy points.

Minimizing topological entropy for maps of the circle

Abstract We construct an example of a C∞ flow φ on R n (n ⩾ 3) and a real-valued function F on R n so that the velocity vectorfield φ φ = X is constant along level sets of F and φ possesses a (nondegenerate) periodic attractor. The example is related to work of R. McGehee and R. Armstrong and others, showing that a proposed formulation of the “competitive exclusion principle” of ecology does not hold in general for non-linear models.