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Horseshoes and the Conley Index Spectrum
, 1999
"... Given a continuous map on a locally compact metric space and an isolating neighborhood which is decomposed into two disjoint isolating neighborhoods, it is shown that the spectral information of the associated Conley indices is sufficient to conclude the existence of a semiconjugacy onto the full s ..."
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Given a continuous map on a locally compact metric space and an isolating neighborhood which is decomposed into two disjoint isolating neighborhoods, it is shown that the spectral information of the associated Conley indices is sufficient to conclude the existence of a semiconjugacy onto the full shift dynamics on two symbols.
THE (POULSEN) SIMPLEX OF INVARIANT MEASURES
, 2014
"... Abstract. Two new concepts, closeability with respect to a set of periodic points and linkability of a set of periodic points of a dynamical system are introduced. Examples are provided to show that closeability and linkability are independent properties. Both properties together imply that the se ..."
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Abstract. Two new concepts, closeability with respect to a set of periodic points and linkability of a set of periodic points of a dynamical system are introduced. Examples are provided to show that closeability and linkability are independent properties. Both properties together imply that the set of invariant measures is either a single periodic orbit or the Poulsen simplex — the unique nontrivial Choquet simplex in which extreme points are dense. Moreover, under these conditions every invariant measure has a generic point and an extension of Sigmund’s theorem about generic properties of invariant measures still holds. The periodic specification property implies closeability and linkability for the set of periodic points. The methods apply beyond systems with specification, because all betashifts, all Sgap shifts, and many other dynamical systems are closeable with respect to some linkable sets of periodic points. We study simplices of invariant measures of dynamical systems. A dy