Experiences numeriques sur des billards C1 formes de quatre arcs de cercles

The paper deals with some properties of the dynamical system with two degrees of freedom defined by the motion of a particle in a certain type of billiard. These properties are studied by means of numerical experiments. Most results are represented in the now classical surface of section. One parameter families of billiards with a C¹ boundary constructed with four arcs of circles are defined; we use the property that the four meeting points of such billiards lie on the same circle. These billiards may be convex or non convex. They generalize the ‘oval’ billiard with two axes of symmetry studied by Benettin and Strelcyn. We call them generalized billiards. We find the following results:

1. The periodic orbit along the small diameter of a billiard is stable or unstable in the linear approximation according to the position of the center of each relevant are with respect to the opposite one. This orbit is always stable if the billiard is symmetric with respect to its large diameter.
2. When the center of an arc lies on the opposite arc two different transition patterns from order to chaos are observed for the same billiard. If the billiard is of the Benettin and Strelcyn type three distinct nested chaotic seas are seen two of which are separated by a pseudo-invariant curve generated by a so called cancellation orbit.
3. The total area of non chaotic regions is greater for symmetric billiards.
4. Peanut shaped billiards always look ergodic. It can happen also that strictly convex asymmetric billiards look ergodic. This is important since no strictly convex billiard is known for which ergodicity has been proven. The conjecture is proposed that a generalized billiard with neither 2-periodic nor 4-periodic stable orbit in the linear approximation is ergodic.
5. Transverse invariant curves such as the one found by Hénon and Wisdom seem common for billiards with two axes of symmetry but probably do not exist for asymmetric billiards.

There are therefore several properties which differentiate symmetric billiards from asymmetric ones. We conclude by emphasizing that C¹ generalized billiards are indeed inadequate models for smooth mappings in general.     

Lithology and stratigraphy of the Ordovician facies complexes of Gornyi Altai

In the Ordovician time, the transform margin of the Gornyi Altai region consisted of two bathymetric stages: (1) shelf and upper parts of continental slope; (2) foothills and lower parts of continental slope. The first stage includes the shallow-water facies complexes (terrigenous and terrigenous-carbonate schlieren and variegated flyschoid), while the second stage is composed of deep-water (black shale terrigenous) and subflysch gray (carbonate terrigenous) complexes. Model series of the facies complexes established in our work should be taken into account during the geodynamic analysis of fold zones.     

Archean high-Mg monzodiorite–syenite,epidote skarn,and biotite–sericite gold lodes in the Granny Smith–Wallaby district,Australia: U–Pb and Re–Os chronometry of two intrusion-related hydrothermal systems

The Granny Smith (37 t Au production) and Wallaby deposits (38 t out of a 180 t Au resource) are located northeast of Kalgoorlie, in 2.7 Ga greenstones of the Eastern Goldfields Province, the youngest orogenic belt of the Yilgarn craton, Western Australia. At Granny Smith, a zoned monzodiorite–granodiorite stock, dated by a concordant titanite–zircon U–Pb age of 2,665 ± 3 Ma, cuts across east-dipping thrust faults. The stock is fractured but not displaced and sets a minimum age for large-scale (1 km) thrust faulting (D2), regional folding (D1), and dynamothermal metamorphism in the mining district. The local gold–pyrite mineralization, controlled by fractured fault zones, is younger than 2,665 ± 3 Ma. In augite–hornblende monzodiorite, alteration progressed from a hematite-stained alkali feldspar–quartz–calcite assemblage and quartz–molybdenite–pyrite veins to a late reduced sericite–dolomite–albite assemblage. Gold-related monazite and xenotime define a U–Pb age of 2,660 ± 5 Ma, and molybdenite from veins a Re–Os isochron age of 2,661 ± 6 Ma, indicating that mineralization took place shortly after the emplacement of the main stock, perhaps coincident with the intrusion of late alkali granite dikes. At Wallaby, a NE-trending swarm of porphyry dikes comprising augite monzonite, monzodiorite, and minor kersantite intrudes folded and thrust-faulted molasse. The conglomerate and the dikes are overprinted by barren (<0.01 g/t Au) anhydrite-bearing epidote–actinolite–calcite skarn, forming a 600-m-wide and >1,600-m-long replacement pipe, which is intruded by a younger ring dike of syenite porphyry pervasively altered to muscovite + calcite + pyrite. Skarn and syenite are cut by pink biotite–calcite veins, containing magnetite + pyrite and subeconomic gold–silver mineralization (Au/Ag = 0.2). The veins are associated with red biotite–sericite–calcite–albite alteration in adjacent monzonite dikes. Structural relations and the concordant titanite U–Pb age of the skarn constrain intrusion-related mineralization to 2,662 ± 3 Ma. The main-stage gold–pyrite ore (Au/Ag >10) forms hematite-stained sericite–dolomite–albite lodes in stacked D2 reverse faults, which offset skarn, syenite, and the biotite–calcite veins by up to 25 m. The molybdenite Re–Os age (2,661 ± 10 Ma) of the ore suggests a genetic link to intrusive activity but is in apparent conflict with a monazite–xenotime U–Pb age (2,651 ± 6 Ma), which differs from that of the skarn at the 95% confidence level. The time relationships at both gold deposits are inconsistent with orogenic models invoking a principal role for metamorphic fluids released during the main phase of compression in the fold belt. Instead, mineralization is related in space and time to late-orogenic, magnetite-series, high-Mg monzodiorite–syenite intrusions of mantle origin, characterized by Mg/(Mg + Fe_TOTAL) = 0.31–0.57, high Cr (34–96 ppm), Ni (22–63 ppm), Ba (1,056–2,321 ppm), Sr (1,268–2,457 ppm), Th (15–36 ppm), and rare earth elements (total REE: 343–523 ppm). At Wallaby, shared Ca–K–CO₂ metasomatism and Th-REE enrichment (in allanite) link Au–Ag mineralization in biotite–calcite veins to the formation of the giant epidote skarn, implicating a Th + REE-rich syenite pluton at depth as the source of the oxidized hydrothermal fluid. At Granny Smith, lead isotope data and the Rb–Th–U signature of early hematite-bearing wall-rock alteration point to fluid released by the source pluton of the differentiated alkali granite dikes.     

Petrochemistry of the Paleoproterozoic Udokan copper-bearing sedimentary complex

The maturity of terrigenous material of the Paleoproterozoic Udokan copper-bearing sedimentary complex is considered. The average values of hydrolyzate module (HM), alumina-silica module (AM), and Pettijohn’s coefficient of maturity appreciably vary throughout the Udokan Complex. Among cupriferous sandstones, rocks of the Aleksandrovka Formation are characterized by the highest maturity, whereas rocks of the Chitkanda Formation are distinguished by the lowest maturity. The maturity of cupriferous sandstones of the Sakukan Formation corresponds to that of host rocks. The maturity of cupriferous sandstones from the Aleksandrovka Formation is much higher than that of host rocks. In the Chitkanda Formation, the cupriferous sandstones are much less mature than host rocks. Climatic conditions in provenances estimated from the Nesbitt index of chemical weathering or chemical index of alteration (CIA) may be characterized as temperate ones without prominent climatic features. Most CIA values range from 46 to 66. The formation of copper-bearing sediments was closely related to the periods of volcanic activity.     

Glauconite from Paleogene volcano-terrigenous rocks in Western Kamchatka

It is shown that glauconite-bearing interbeds are widespread in the layer-by-layer studied sections on the Sea of Okhotsk coast (Mainach section) and Kheisliveem River valley (Kavran section), the volcanoterrigenous rocks of the Kovachin, Amanin, and Gakkhin formations of the Paleogene in western Kamchatka (Upper Eocene-Lower Oligocene boundary beds). Detailed mineralogical and structural-crystallochemical characteristics of glauconite from the Amanin Formation are presented. It is suggested that such glauconite should not be used for geochronological purposes.Some specific features of glauconite formation, particularly, the preservation of specific morphological forms at high accumulation rates of volcano-terrigenous rocks, are discussed. Possibility of the formation of glauconite with the active influence of bacterial metabolism is considered.     

Some types of small clinoforms in latest continental deposits

Some rare types of small clinoforms found in the latest continental deposits of lowland platform and mountain regions are described. The clinoforms are represented by prodelta deposits of mountain lakes, oblique-bedded horizons of floodplain alluvium of strongly meandering rivers, thick and short lenses of mountain alluvium, and alluvium horizons of great lowland rivers with oblique bedding grading into horizontal bedding. Such structures bear information on paleogeographic, morphological, and lithodynamic features of continental sedimentation.     

The Mordor Alkaline Igneous Complex,Central Australia: PGE-enriched disseminated sulfide layers in cumulates from a lamprophyric magma

The Mordor Alkaline Igneous Complex (MAIC) is a composite intrusion comprising a body of syenite and a funnel-shaped layered mafic–ultramafic intrusion of lamprophyric parentage, the Mordor Mafic–Ultramafic Intrusion or MMUI. The MMUI is highly unusual among intrusions of lamprophyric or potassic parentage in containing primary magmatic platinum-group element (PGE)-enriched sulfides. The MMUI sequence consists largely of phlogopite-rich pyroxenitic cumulates, with an inward dipping conformable layer of olivine-bearing cumulates divisible into a number of cyclic units. Stratiform-disseminated sulfide accumulations are of two types: disseminated layers at the base of cyclic units, with relatively high PGE tenors; and patchy PGE-poor disseminations within magnetite-bearing upper parts of cyclic units. Sulfide-enriched layers at cycle bases contain anomalous platinum group element contents with grades up to 1.5 g/t Pt+Pd+Au over 1-m intervals, returning to background values of low parts per billion (ppb) on a meter scale. They correspond to reversals in normal fractionation trends and are interpreted as the result of new magma influxes into a continuously replenished magma chamber. Basal layers have decoupled Cu and PGE peaks reflecting increasing PGE tenors up-section, due to increasing R factors during the replenishment episode, or progressive mixing of between resident PGE-poor magma and more PGE-enriched replenishing magma. The presence of PGE enriched sulfides in cumulates from a lamprophyric magma implies that low-degree partial melts do not necessarily leave sulfides and PGEs in the mantle restite during partial melting. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

Fluid and source magma evolution of the Questa porphyry Mo deposit,New Mexico,USA

Combined fluid inclusion microthermometry and microanalysis by laser ablation inductively coupled plasma mass spectrometry (LA-ICPMS) are used to constrain the hydrothermal processes forming a typical Climax-type porphyry Mo deposit. Molybdenum mineralisation at Questa occurred in two superimposed hydrothermal stages, a magmatic-hydrothermal breccia and later stockwork veining. In both stages, texturally earliest fluids were single-phase, of low salinity (~7 wt.% NaCl_equiv.) and intermediate-density. Upon decompression to ~300 bar, they boiled off a vapour phase, leaving behind a residual brine (up to 45 wt.% NaCl_equiv) at temperatures of ~420°C. The highest average Mo concentrations in this hot brine were ~500 μg/g, exceeding the Mo content of the intermediate-density input fluid by about an order of magnitude and reflecting pre-concentration of Mo by fluid phase separation prior to MoS₂ deposition from the brine. Molybdenum concentrations in brine inclusions, then, decrease down to 5 μg/g, recording Mo precipitation in response to cooling of the saline liquid to ~360°C. Molybdenite precipitation from a dense, residual and probably sulphide-depleted brine is proposed to explain the tabular shape of the ore body and the absence of Cu-Fe sulphides in contrast to the more common Cu-Mo deposits related to porphyry stocks. Cesium and Rb concentrations in the single-phase fluids of the breccia range from 2 to 8 and from 40 to 65 μg/g, respectively. In the stockwork veins, Cs and Rb concentrations are significantly higher (45–90 and 110–230 μg/g, respectively). Because Cs and Rb are incompatible and hydrothermally non-reactive elements, the systematic increase in their concentration requires two distinct pulses of fluid exsolution from a progressively more fractionated magma. By contrast, major element and ore metal concentrations of these two fluid pulses remain essentially constant. Mass balance calculations using fluid chemical data from LA-ICPMS suggest that at least 25 km³ of melt and 7 Gt of deep input fluid were necessary to provide the amount of Mo contained in the stockwork vein stage alone. While the absolute amounts of fluid and melt are uncertain, the well-constrained element ratios in the fluids together with empirical fluid/melt partition coefficients derived from the inclusion analyses suggest a high water content of the source melt of ~10%. In line with other circumstantial evidence, these results suggest that initial fluid exsolution may have occurred at a confining pressure exceeding 5 kbar. The source of the molybdenum-mineralising fluids probably was a particularly large magma chamber that crystallised and fractionated in the lower crust or at mid-crustal level, well below the shallow intrusions immediately underlying Questa and other porphyry molybdenum deposits. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

Late Carboniferous porphyry copper mineralization at La Voluntad,Neuquén,Argentina: Constraints from Re–Os molybdenite dating

The La Voluntad porphyry Cu–Mo deposit in Neuquén, Argentina, is one of several poorly known porphyry-type deposits of Paleozoic to Early Jurassic age in the central and southern Andes. Mineralization at La Voluntad is related to a tonalite porphyry from the Chachil Plutonic Complex that intruded metasedimentary units of the Piedra Santa Complex. Five new Re–Os molybdenite ages from four samples representing three different vein types (i.e., quartz–molybdenite, quartz–sericite–molybdenite and quartz–sericite–molybdenite ± chalcopyrite–pyrite) are identical within error and were formed between ~312 to ~316 Ma. Rhenium and Os concentrations range between 34 to 183 ppm and 112 to 599 ppb, respectively. The new Re–Os ages indicate that the main mineralization event at La Voluntad, associated to sericitic alteration, was emplaced during a time span of 1.7 ± 3.2 Ma and that the deposit is Carboniferous in age, not Permian as previously thought. La Voluntad is the oldest porphyry copper deposit so far recognized in the Andes and indicates the presence of an active magmatic arc, with associated porphyry style mineralization, at the proto-Pacific margin of Gondwana during the Early Pennsylvanian.