The main result of this paper is that two large collections of er-godic measure preserving systems, the Odometer Based and the Circular Systems have the same global structure with respect to joinings that preserve underlying timing factors. The classes are canonically isomorphic by a contin-uous map that takes synchronous and anti-synchronous factor maps to synchronous and anti-synchronous factor maps, synchronous and anti-synchro-nous measure-isomorphisms to synchronous and anti-synchronous measure-isomorphisms, weakly mixing extensions to weakly mixing extensions and compact extensions to compact extensions. The first class includes all finite entropy ergodic transformations that have an odometer factor. By results in [6], the second class contains all transformations realizable as dif-feomorphisms using the untwisted Anosov–Katok method. An application of the main result will appear in a forthcoming paper [7] that shows that the diffeomorphisms of the torus are inherently unclassifiable up to measure-isomorphism. Other consequences include the existence of measure distal diffeomorphisms of arbitrary countable distal height.

Construction sequences are a general method of building symbolic shifts that capture cut-and-stack constructions and are general enough to give symbolic representations of Anosov-Katok diffeomorphisms. We show here that any finite entropy system that has an odometer factor can be represented as the limit of a special class of construction sequences, the odometer based construction sequences. These naturally correspond to those cut-and-stack constructions that do not use spacers. The odometer based construction sequences can be constructed to have the small word property and every Choquet simplex can be realized as the simplex of invariant measures of the limit of an odometer based construction sequence.

This paper is the first of a series of papers culminating in the result that measure preserving diffeomorphisms of the disc or 2-torus are unclassifiable. It addresses another classical problem: which abstract measure preserving systems are realizable as smooth diffeomorphisms of a compact manifold? The main result gives symbolic representations of Anosov–Katok diffeomorphisms.

In 1932 von Neumann proposed classifying the statistical behavior of differentiable systems. In modern language this is interpreted as classifying diffeomorphisms of compact manifolds up to measure isomorphism. This paper proves that this is impossible in a rigorous sense.

The crusts of the Moon, Mercury, and many meteorite parent bodies are magnetized. Although the magnetizing field is commonly attributed to that of an ancient core dynamo, a longstanding hypothesized alternative is amplification of the interplanetary magnetic field and induced crustal field by plasmas generated by meteoroid impacts. Here, we use magnetohydrodynamic and impact simulations and analytic relationships to demonstrate that although impact plasmas can transiently enhance the field inside the Moon, the resulting fields are at least three orders of magnitude too weak to explain lunar crustal magnetic anomalies. This leaves a core dynamo as the only plausible source of most magnetization on the Moon.

Methanogenic archaea have been shown to reduce iron from ferric [Fe(III)] to ferrous [Fe(II)] state, but minerals that form during iron reduction by different methanogens remain to be characterized. Here, we show that zerovalent iron (ZVI) minerals, ferrite [α-Fe(0)] and austenite [γ-Fe(0)], appear in the X-ray diffraction spectra minutes after the addition of ferrihydrite to the cultures of a methanogenic archaeon, Methanosarcina barkeri (M. barkeri). M. barkeri cells and redox-active, nonenzymatic soluble organic compounds in organic-rich spent culture supernatants can promote the formation of ZVI; the latter compounds also likely stabilize ZVI. Methanogenic microbes that inhabit organic- and Fe(III)-rich anaerobic environments may similarly reduce Fe(III) to Fe(II) and ZVI, with implications for the preservation of paleomagnetic signals during sediment diagenesis and potential applications in the protection of iron metals against corrosion and in the green synthesis of ZVI.

In this paper we give explicit characterizations, based on the cutting and spacer parameters, of (a) which rank-one transformations factor onto a given finite cyclic permutation, (b) which rank-one transformations factor onto a given odometer, and (c) which rank-one transformations are isomorphic to a given odometer. These naturally yield characterizations of (d) which rank-one transformations factor onto some (unspecified) finite cyclic permutation, (d′) which rank-one transformations are totally ergodic, (e) which rank-one transformations factor onto some (unspecified) odometer, and (f) which rank-one transformations are isomorphic to some (unspecified) odometer.

The existence of numerous iron meteorite groups indicates that some planetesimals underwent melting that led to metal-silicate segregation, sometimes producing metallic cores. Meteorite paleomagnetic records suggest that crystallization of these cores generated dynamo magnetic fields. Here we describe the magnetic history of the partially differentiated IIE iron meteorite parent body. This is the first planetesimal for which we have a time-resolved paleomagnetic record constrained by 40Ar/39Ar chronometry spanning several tens of million years (Ma). We find that the core of the IIE parent body generated a dynamo, likely powered by core crystallization, starting before 78 ± 13 Ma after solar system formation and lasting at least 80 Ma. Such extended core crystallization suggests that the core composed a substantial fraction of the body ((Formula presented.) 13%–19% core-to-body radius ratio depending on the body’s radius), indicating efficient core formation within some partially differentiated planetesimals.

Modern meteorite classification schemes assume that no single planetary body could be source of both unmelted (chondritic) and melted (achondritic) meteorites. This dichotomy is a natural outcome of formation models assuming that planetesimal accretion occurred nearly instantaneously. However, it has recently been proposed that the accretion of many planetesimals lasted over ≳1 million years (Ma). This could have resulted in partially differentiated internal structures, with individual bodies containing iron cores, achondritic silicate mantles, and chondritic crusts. This proposal can be tested by searching for a meteorite group containing evidence for these three layers. We combine synchrotron paleomagnetic analyses with thermal, impact, and collisional evolution models to show that the parent body of the enigmatic IIE iron meteorites was such a partially differentiated planetesimal. This implies that some chondrites and achondrites simultaneously coexisted on the same planetesimal, indicating that accretion was protracted and that apparently undifferentiated asteroids may contain melted interiors.