Titles & Abstracts

Titles and Abstracts - International Workshop on Geometric Methods in Dynamical Systems

Titles and Abstract in PDF format for downloading

Ethan Akin, City College of New York
Title Good measures on Cantor Space
Abstract While there is, up to homeomorphism, only one Cantor space, i.e., one zero-dimensional, perfect, compact, nonempty metric space, there are many measures on Cantor Space which are not topologically equivalent. The "clopen values set" for a full, nonatomic measure µ is the countable dense subset {µ(U) : U is clopen} of the unit interval. It is a topological invariant for the measure. For the class of good measures it is a complete invariant. A full, nonatomic measure µ is "good" if whenever U and V are clopen sets with µ(U) < µ(V), there exists W a clopen subset of V such that µ(W) = µ(U). These measures have interesting dynamical properties. They are exactly the measures which arise from uniquely ergodic minimal systems on Cantor Space. For some of them there is a unique generic measure-preserving homeomorphism. That is, within the Polish group of such homeomorphisms there is a dense, G_δ conjugacy class.
Kathy Alligood, George Mason University
Title Heteroclinic Cycles and Explosions
Abstract An explosion is a discontinuous change in the set of recurrent points as a map parameter is varied. We discuss how certain homoclinic and heteroclinic tangencies in two dimensions and other nontransverse intersections of stable and nonstable manifolds in higher dimensions result in the sudden appearance of recurrent points. Focusing on a parametrized three-dimensional map, we give an example of a global bifurcation in which a chaotic attractor undergoes a crisis, but where there is no tangency at the bifurcation parameter. (This is joint work with Evelyn Sander and Jim Yorke.)
Alex Clark, University of North Texas
Title The linking of minimal sets in the Lorenz Template
Abstract Reporting on joint work with Mike Sullivan, we examine the linking of minimal sets in the Lorenz template.

J. T. Halbert, University of Maryland
Title From Taffy to Pure Chaos
Abstract The Taffy-pulling machine can be considered a "hyperbolic machine" in that each point in the taffy is being stretched in one direction and contracted in another. We will discuss the dynamics of this machine and show that a series of abstractions reduces its action to a pseudo-Anosov map on a sphere. We will show the action of this map and others with animation. (Joint work with Jim Yorke.)
Bill Kalies, Florida Atlantic University
Title Algorithmic Approach to Conley's Decomposition Theorem and Applications to Approximation of Flows
Abstract We discuss recent work with Mischaikow and Vander Vorst on the approximation of the chain recurrent set and Conley's Lyapunov function via spatial discretization which leads to algorithms for generating index filtrations for Morse decompositions and approximations of the corresponding Lyapunov functions. We demonstrate such algorithms in the context of rigorous computation of isolating blocks in flows via multivalued maps on polygonal discretizations of the phase space. The quality of the discretization determines how close the approximation is to the dynamics of the flow, i.e., how closely the recurrent components of the multivalued map approximate the chain recurrent set. We will discuss both theoretical and practical implementation issues and show some examples.
Piotr Kościelniak, Jagiellonian University
Title Covering relations in C⁰ Genericity Theorems
Abstract We outline the proof of the genericity of the periodic shadowing property for homeomorphisms. We use the multidimensional covering relations introduced by Gidea and Zgliczynski. It also turns out that making use of this method we can slightly generalize the theorem (by Akin, Hurley, and Kennedy) stating that some iteration of a generic homeomorphism is semiconjugate (on some subset) to a shift map.
1. P. Kościelniak, The simpler proofs of the genericity of shadowing and periodic shadowing, preprint.
2. P. Kościelniak, M. Mazur, On C⁰ genericity of various shadowing properties, Discrete Contin. Dynam. Syst., to appear.
Marcin Kulczycki, Jagiellonian University
Title Dynamical systems with bounded orbits on a space with measure
Abstract While working on the problem of the existence of a volume preserving fixed point free dynamical system in $\QTR{Bbb}{R}^{3}$ with the diameters of all orbits uniformly bounded, one obtains general theorems about properties of measure preserving systems on metric spaces. This talk outlines several results that stem from this problem.
Krystyna Kuperberg, Auburn University
Title Embedding knots in flows
Abstract We will show several types of modifications of flows on 3-manifolds which introduce new knots and links to the orbit configuration, or embed wild homoclinic orbits. It will be discussed when such a construction can be made to be measure-preserving.
Brian Martensen, University of Texas
Title The topology of tiling spaces and codimension one attractors
Abstract We will state what is known about the topology of one and two dimensional (substitutive) tiling spaces and their relation to hyperbolic expanding attractors. Using this information along with the combinatorics of the tiling space, we give a characterization of codimension one hyperbolic attractors of surfaces. Lastly, we will discuss some partial results and approaches to finding such a characterization for codimension one hyperbolic attractors of three manifolds.
Waclaw Marzantowicz, Adam Mickiewicz University
Title The Entropy conjecture for continuous maps of nilmanifolds
Abstract In 1974 Michael Shub asked the following question: When is the topological entropy of a continuous mapping of a compact manifold into itself estimated from below by the logarithm of the spectral radius of the linear mapping induced in the cohomologies with real coefficients? This estimate has been called the Entropy Conjecture (EC). In 1977 the second author and Michal Misiurewicz proved that EC holds for all continuous mappings on tori. Here we give a proof of EC for all continuous maps of compact nilmanifolds. Also, generalizations for maps of some solvmanifolds and another proof via Lefschetz and Nielsen numbers, under the assumption that the map is not homotopic to a fixed point free map, are provided. (The work is joint with Feliks Przytycki. The research was supported by the Foundation for Polish Science, KBN grant number 2 PO3A 04522.)
Marcin Mazur, Jagiellonian University
Title On the C⁰ Genericity of various shadowing properties
Abstract We review the results of [1,2,3,4] by presenting the methods that were used there in the proofs of C⁰ genericity theorems for various shadowing properties of discrete dynamical systems. In particular, we indicate the usefulness of a handle decomposition of a manifold in proving the genericity of (periodic) shadowing and strong tolerance stability. We also outline the proof of the genericity of weak shadowing that works for systems on general metric spaces satisfying a generalized homogeneity property.
1. P. Kościelniak, M. Mazur, On C⁰ genericity of various shadowing properties, Discrete Contin. Dynam. Syst., to appear.
2. M. Mazur, Tolerance stability conjecture revisited, Topology Appl. 131 (2003) 33-38.
3. M. Mazur, Weak shadowing for discrete dynamical systems on non-smooth manifolds, J. Math. Appl. 281 (2003) 657-662.
4. S. Yu. Pilyugin, O. B. Plamenevskaya, Shadowing is generic, Topology Appl. 97 (1999) 253-266.
Hector Mendez, Universidad Nacional Autónoma de México
Title On the Omega-EP-Property
Abstract A continuum X has the Omega-EP-property provided that for each self-mapping the set of nonwandering points is contained in the closure of the set of eventually periodic points. It is known that the interval has the Omega-EP-property. We show in this talk that the sin(1/x)-curve does not have the Omege-EP-property.
Chris Mouron, Rhodes College
Title Entropy, Crookedness and Wrapping
Abstract Entropy is a measure of the rate of chaotic behavior of a function. I will discuss how positive entropy homeomorphisms require wrapping or bending of continua and how this creates interesting topology. In particular, I will discuss how this creates indecomposable subcontinua.
Marian Mrozek, Jagiellonian University
Title Index Pairs Algorithms
Abstract On the one hand, index pairs constitute the technical machinery needed in the construction of the Conley index. On the other hand, they are the principal tool used in the algorthmic computation of the index. In the literature one can find several definitions of index pairs. Not all of them are good for the algorithmic computation of the Conley index. In the lecture we discuss various definitions of index pairs from the algorithmic point of view. We also introduce the concept of weak index pair and show why it is especially convenient in the computation of the Conley index of numerically unstable dynamical systems.
Helena Nusse, University of Maryland and University of Utrecht
Title Entangled Basin Boundaries, Homoclinic Tangencies, and Prime Ends
Abstract Two-dimensional dynamical systems often have basins with fractal basin boundaries. The purpose of this talk is to describe the structure and properties of basins and their boundaries for two-dimensional diffeomorphisms. Frequently, if not always, there is a periodic saddle on the boundary that is accessible from the basin. Carathéodory, Freudenthal, and many others developed an approach in which an open set (in our case a basin) is compactified using so-called prime end theory. Under the prime end compactification of the basin, boundary points of the basin (prime ends) can be characterized as either type 1, 2, 3, or 4. Most points are of type 1 in all well known examples. If a two-dimensional basin has a basin cell (a trapping region whose boundary consists of pieces of the stable and unstable manifolds of a well chosen periodic orbit), then the basin consists of a central body (the basin cell) and a finite number of channels attached to it, and the basin boundary is fractal. We present a result relating basin cells for a basin to prime ends of type 3, and results relating a finite number of prime ends of either type 2 or 4 to situations when an accessible periodic saddle on the basin boundary has some specified tangencies of its stable and unstable manifolds.
Will Ott, University of Maryland
Title When is a compact subset of Euclidean space a submanifold?
Abstract We address the problem of embedding a compact subset of Euclidean space into a smooth submanifold of minimal dimension. As a consequence of our results, we obtain a proof of the characterization of smooth submanifolds in terms of the notion of tangent direction. These ideas may be applied to the problem of spurious Lyapunov exponents, to embedding theory, and to the Hilbert-Smith conjecture.
Wiktor Radzki, Nicolaus Copernicus University
Title On the structure of the set of bifurcation points of periodic solutions for autonomous Hamiltonian systems
Abstract This talk will concern bifurcation points of 2π-periodic solutions of the Hamilton equation

where is such that for given and every we have and for some (n×n)-matrices A(λ) and B(λ). We shall give a local description of the set of bifurcation points in a neighborhood of given (χ₀,λ₀). The point (χ₀,λ₀) can be degenerate, i.e., the Hessian can be singular. Under some assumptions, the bifurcation points of solutions with periods , can be identified with zeros of the functions defined as . For k = 2 the sets of zeros of F_j consist of analytic curves, the number of which can be computed by using methods of real algebraic geometry.
David Ryden, Baylor University
Title The Sarkowskii ordering for periodic subcontinua of hereditarily decomposable chainable continua
Abstract Piotr Minc and W. R. R. Transue have shown that periodic points of maps of hereditarily decomposable chainable continua follow the Sarkovskii ordering. Examples exist that show the assumption of hereditary decomposability cannot be dropped. Suppose f maps a continuum X onto itself. A (nontrivial) periodic continuum of f is defined to be a (proper) subcontinuum K of X such that f^p(K) = K for some positive integer p. A maximal periodic continuum is a continuum that is maximal with respect to the property of being a nontrivial periodic continuum of f. In this talk, we will show that, apart from a trivial case, the maximal periodic continua of a map from a hereditarily decomposable chainable continuum onto itself follow the Sarkowskii ordering.
Roman Srzednicki, Jagiellonian University
Title Isolating segments and chains
Abstract Isolating segments and chains are the Conley's isolating blocks of special forms suitable for flows generated by non-autonomous equations. They are used in proofs of the existence of orbits satisfying given boundary value data (like the periodic ones), detection of symbolic dynamics, and shadowing of numerically given pseudo-orbits. In the lecture we survey results related to those notions.
Klaudiusz Wojcik, Jagiellonian University and Auburn University
Title Topological horseshoes and delay differential equations
Abstract We show that if an ordinary differential equation x′ = f(x) , where f is a C ¹ vector field on $\QTR{Bbb}{R}^{n}$ has a topological horseshoe for a suitable Poincaré map, then the corresponding delay equation x ′ = f(x(t-h)), for small h, also has a topological horseshoe, i.e., symbolic dynamics and an infinite number of periodic orbits. (This is joint work with Piotr Zgliczynski.)
Wojtek Wojcik, Vrije Universiteit, Amsterdam
Title Chaos in orientation reversing twist maps of the plane
Abstract In this talk we study forcing of periodic points in two dimensional orientation reversing twist maps. We show that if such a map contains a period 4 point of a special type then the system is forced to be chaotic. The main idea behind the proof is to connect the evolution of the twist system with a special flow on braid diagrams and then use symbolic dynamics. We also give some remarks about the sharpness of the theorem, i.e., we study the example of the map having another type of period 4 orbit.
Jim Yorke, University of Maryland
Title From Taffy to Pure Chaos
Abstract The Taffy-pulling machine can be considered a "hyperbolic machine" in that each point in the taffy is being stretched in one direction and contracted in another. We will discuss the dynamics of this machine and show that a series of abstractions reduces its action to a pseudo-Anosov map on a sphere. We will show the action of this map and others with animation. (Joint work with J. T. Halbert.)