Stewart Baldwin

Generalizations of a theorem of Sarkovskii on orbits of continuous real-valued functions

Abstract In 1964, Sarkovskii defined a certain linear ordering ⩽ s of the positive integers and proved that m ⩽ s n if every continuous f : R → R having an orbit of size n also has an orbit of size m . This idea is extended to get a partial (but not linear) ordering in which the pattern of the orbit is taken into account. For example if x 1 x 2 x 3 x 4 , then x 1 → x 2 → x 3 → x 4 x → 1 and x 1 → x 3 → x 2 → x 4 → x 1 are both orbits of size 4 but are considered to have distinct patterns in this paper. A combinatorial algorithm which decides the status of any two patterns with respect to the partial ordering is derived, and examples are given for patterns of small size.

Entropy of transitive tree maps

Abstract We obtain lower bounds for the topological entropy of transitive self-maps of trees, depending on the number of endpoints and on the number of edges of the tree.

Countable dense homogeneous spaces under Martin’s axiom

We show that Martin’s axiom for countable partial orders implies the existence of a countable dense homogeneous Bernstein subset of the reals. Using Martin’s axiom we derive a characterization of the countable dense homogeneous spaces among the separable metric spaces of cardinality less thanc. Also, we show that Martin’s axiom implies the existence of a subset of the Cantor set which isλ-dense homogeneous for everyλ

Calculating Topological Entropy

The attempt to find effective algorithms for calculating the topological entropy of piecewise monotone maps of the interval having more than three monotone pieces has proved to be a difficult problem. The algorithm introduced here is motivated by the fact that if f: [0, 1] → [0, 1] is a piecewise monotone map of the unit interval into itself, thenh(f)=limn→∞ (1/n) log Var(fn), where h(f) is the topological entropy off, and Var(fn) is the total variation offn. We show that it is not feasible to use this formula directly to calculate numerically the topological entropy of a piecewise monotone function, because of the slow convergence. However, a close examination of the reasons for this failure leads ultimately to the modified algorithm which is presented in this paper. We prove that this algorithm is equivalent to the standard power method for finding eigenvalues of matrices (with shift of origin) in those cases for which the function is Markov, and present encouraging experimental evidence for the usefulness of the algorithm in general by applying it to several one-parameter families of test functions.

Entropy estimates for transitive maps on trees

Abstract If X is a space, define L(X) to the the infimum of all possible values h( f ) , where h( f ) denotes the topological entropy of f, and f ranges over all transitive functions on X. Various lower and upper bounds are given for L(T) in the case for which T is a tree, which provides an exact value of L(T) for a large class of trees. We also examine the analogous problem when f is required to fix the endpoints of the tree.

Between strong and superstrong

Definition. A cardinal κ is strong iff for every x there is an elementary embedding j : V → M with critical point κ such that x ∈ M . κ is superstrong iff ∃ j : V → M with critical point κ such that V j ( κ ) ∈ M . These definitions are natural weakenings of supercompactness and hugeness respectively and display some of the same relations. For example, if κ is superstrong then V κ ⊨ “∃ proper class of strong cardinals”, but the smallest superstrong cardinal is less than the smallest strong cardinal (if both types exist). (See [SRK] and [Mo] for the arguments involving supercompact and huge, which translate routinely to strong and superstrong.) Given any two types of large cardinals, a typical vague question which is often asked is “How large is the gap in consistency strength?” In one sense the gap might be considered relatively small, since the “higher degree” strong cardinals described below (a standard trick that is nearly always available) and the Shelah and Woodin hierarchies of cardinals (see [St] for a definition of these) seem to be (at least at this point in time) the only “natural” large cardinal properties lying between strong cardinals and superstrong cardinals in consistency strength.

THE <-ORDERING ON NORMAL ULTRAFILTERS

consistent that K has the maximum possible number of normal ultrafilters. Starting with assumptions stronger than measurability, Mitchell [Mi-i] filled in the gap by constructing models of ZFC + GCH satisfying there are exactly i normal ultrafilters over K, where i could be K+ or K++ (measured in the model), or anything < K. Whether or not Mitchells results can be obtained by starting only with a measurable cardinal in the ground model and defining a forcing extension is unknown.

Possible point-open types of subsets of the reals

IfX is a topological space and α is an ordinal, then the point-open game of length α on X, abbreviated Gα(X), is the two person game of length α in which, on the βth move (β<α), the first player (the “point picker”) picks a point of X and the second player (the “open set picker”) picks an open subset of X covering the point just played. The point picker wins iff the open sets thus picked cover X. The point-open type of X, abbreviated pot(X), is defined to be the smallest ordinal α such that the point picker has a winning strategy in Gα(X). This ordinal clearly exists and is no more than the cardinality of X. The main result of this paper is that if we assume the Continuum Hypothesis, then for every limit ordinal α<ω1, there is a subset X of the real numbers such that pot (X) = α. This solves a problem due to Peg Daniels and Gary Gruenhage.

Generalizing the Mahlo Hierarchy, with Applications to the Mitchell Models

On generalise les definitions donnees pour la fonction m, qui peut etre consideree comme une mesure de la grandeur des divers cardinaux

A stable/unstable “manifold” theorem for area preserving homeomorphisms of two manifolds

The stable/unstable manifold theorem for hyperbolic diffeomorphisms has proven to be of extreme importance in differentiable dynamics. We prove a stable/unstable manifold theorem for area preserving homeomorphisms of orientable two manifolds having isolated fixed points of index less than 1. The proof relies uponl the concept of ftee mnodification which was first developed by Morton Brown for homeomorphisms of two manifolds and later extended by Pelikan and Slaminka for area preserving homeomorphisms of two manifolds.