Marcy Barge

Self-similarity in inverse limit spaces of the tent family

Taking inverse limits of the one-parameter family of tent maps of the interval generates a one-parameter family of inverse limit spaces. We prove that, for a dense set of parameters, these spaces are locally, at most points, the product of a Cantor set and an arc. On the other hand, we show that there is a dense Gδ set of parameters for which the corresponding space has the property that each neighborhood in the space contains homeomorphic copies of every inverse limit of a tent map. In 1967, R. F. Williams ([10]) proved that hyperbolic one-dimensional attractors are inverse limits of maps on branched one-manifolds. These attractors have the solenoid-like property of being everywhere locally homeomorphic with the product of a Cantor set and an arc. Also, for dissipation parameter near zero, most of the full attracting sets for maps in the Hénon family are homeomorphic with inverse limits of unimodal maps of the interval ([1]). Except at finitely many points (the points of a stable periodic orbit), these sets are locally homeomorphic with the product of a Cantor set and an arc (see the comment following Theorem 1). Computer-generated pictures, at first glance, suggest that other one-dimensional (but non-hyperbolic) attractors might have a similar local structure. In particular, the transitive Hénon attractors appear to be, at most points, locally the product of a Cantor set and an arc. However, ‘blowing up’ computer pictures of these attractors usually indicates the presence of ‘hooks’ in the midst of regions that, under less scrutiny, look like a Cantor set of nearly parallel arcs. In this paper we consider the local topological properties of a one-parameter family of conceptual models for the Hénon attractors, inverse limits of tent maps. We find the following: for a dense set of parameters, the inverse limit space is, except at finitely many points, the product of a Cantor set and an arc (Theorem 1). However, for a dense Gδ set of parameters, the inverse limit space is nowhere locally homeomorphic with the product of a Cantor set and an arc. In this second case, the inverse limit spaces display a remarkable form of self-similarity and local recapitulation of the entire family: not only does every open set in each space contain a homeomorphic copy of the entire space, each open set also contains a homeomorphic copy of every other inverse limit space appearing in the tent family (Corollary 6). In a forthcoming paper, we prove that the set of parameters for which this holds has full measure. Received by the editors May 16, 1995. 1991 Mathematics Subject Classification. Primary 54F15, 58F03, 58F12. The first author was supported in part by NSF-DMS-9404145. c ©1996 American Mathematical Society

The dynamics of continuous maps of finite graphs through inverse limits

Suppose that f: G -* G is a continuous piecewise monotone function on a finite graph G. Then the following are equivalent: (i) f has positive topological entropy; (ii) there are disjoint intervals I1 and I2 and a positive integer n with I1 U I2 C fn(JI) n fn(I2); (iii) the inverse limit space constructed by using f on G as a single bonding map contains an indecomposable subcontinuum. This result generalizes known results for the interval and circle.

Inverse limits on [0,1] using logistic bonding maps

Abstract In this paper we investigate inverse limits on [0, 1] using a single bonding map chosen from the logistic family, fλ (x) = 4λx(1 − x) for 0 ⩽ λ ⩽ 1. Many interesting continua occur as such inverse limits from arcs to indecomposable continua. Among other things we observe that up through the Feigenbaum limit the inverse limit is a point or is hereditarily decomposable and otherwise the inverse limit contains an indecomposable continuum.

Asymptotic orbits of primitive substitutions

A primitive, aperiodic substitution on d letters has at most d2 asymptotic orbits; this bound is sharp. Since asymptotic arc components in tiling spaces associated with substitutions are in 1-1 correspondence with asymptotic words, this provides a bound for those as well.

Rotation and periodicity in plane separating continua

We prove that if F is an orientation-preserving homeomorphism of the plane that leaves invariant a continuum Λ which irreducibly separates the plane into exactly two domains, then the convex hull of the rotation set of F restricted to Λ is a closed interval and each reduced rational in this interval is the rotation number of a periodic orbit in Λ. We also show that the interior and exterior rotation numbers of F associated with Λ are contained in the convex hull of the rotation set of F restricted to Λ and that if this set is nondegenerate then Λ is an indecomposable continuum.

Inverse limit spaces of infinitely renormalizable maps

Abstract There are uncountably many distinct inverse limit spaces that can be formed with unimodal maps as bonding maps.

Circle maps and inverse limits

Abstract The results of this paper relate the dynamics of a continuous map ƒ of the circle and the topology of the inverse limit space with bonding map ƒ. When the dynamics of ƒ are “chaotic” the inverse limit space contains indecomposable subcontinua. The converse is also true if ƒ has finitely many turning points.

Equicontinuous Factors, Proximality and Ellis Semigroup for Delone Sets

We discuss the application of various concepts from the theory of topological dynamical systems to Delone sets and tilings. In particular, we consider the maximal equicontinuous factor of a Delone dynamical system, the proximality relation and the enveloping semigroup of such systems.

Asymptotic structure in substitution tiling spaces

Every sufficiently regular non-periodic space of tilings of

A fixed point theorem for plane separating continua

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