The Decoration Conjecture describes the structure of the set of braid types of Smale’s horseshoe map ordered by forcing, providing information about the order in which periodic orbits can appear when a horseshoe is created. A proof of this conjecture is given for the class of so-called lone decorations, and it is explained how to calculate associated braid conjugacy invariants which provide add...

In this chapter we present a survey of the area of set theory in which iterated forcing interacts with elementary embeddings. The original plan was to concentrate on forcing constructions which preserve large cardinal axioms, particularly Reverse Easton iterations. However this plan proved rather restrictive, so we have also treated constructions such as Baumgartner’s consistency proof for the ...

A non-equilibrium form of turbulent Kolmogorov ow is set up by making an instantaneous change in the amplitude of the spatially-periodic forcing. It is found that the response of the ow to this instantaneous change becomes more dramatic as the wavenumber of the forcing is increased, and, at the same time, that the faithfulness with which the large-eddy-simulation results agree with the direct-n...

We investigate the effect of forcing with a coherent Souslin trees on two standard set-theoretic dichotomies and several topological chain conditions.

0 # can be invisibly class generic. 1. Introduction Roughly, the main theorem of this paper is that some instances of any type of non-constructible object are class generic over L, in one sense. Since small large cardinal properties are inherited in L, let us begin by considering the sense in which 0 # can be generic. It is well known that 0 # is not set generic over L, or, indeed, over any inn...

As with all coral reef systems, the ecology of Ningaloo Reef is closely linked to water circulation which transport and disperse key material such as nutrients and larvae. Circulation on coral reefs may be driven by a number of forcing mechanisms including waves, tides, wind, and buoyancy effects. Surface waves interacting with reefs have long been known to dominate the currents on many coral r...

An important problem in the dynamics of surface homeomorphisms is determining the forcing relation between orbits. The forcing relation between periodic orbits can be computed using existing algorithms. Here we consider forcing relations between homoclinic orbits. We outline a general procedure for computing the forcing relation, and apply this to compute the equivalence and forcing relations f...

We prove that if μ is a regular cardinal and P is a μ-centered forcing poset, then P forces that (I[μ++])V generates I[μ++] modulo clubs. Using this result, we construct models in which the approachability property fails at the successor of a singular cardinal. We also construct models in which the properties of being internally club and internally approachable are distinct for sets of size the...

We prove that a wide class of strongly proper forcing posets have quotients with strong properties. Specifically, we prove that quotients of forcing posets which have universal strongly generic conditions on a stationary set of models by certain nice regular suborders satisfy the ω1-approximation property. We prove that the existence of stationarily many ω1-guessing models in Pω2 (H(θ)), for su...

ABSRACT In his book Proper Forcing (1982) Shelah introduced three classes of forcings (complete, proper, and semi-proper) and proved a strong iteration theorem for each of them: The first two are closed under countable support iterations. The latter is closed under revised countable support iterations subject to certain standard restraints. These theorems have been heavily used in modern set th...