Rates of mixing for the measure of maximal entropy of dispersing billiard maps

In a recent work, Baladi and Demers constructed a measure of maximal entropy for finite horizon dispersing billiard maps and proved that it is unique, mixing and moreover Bernoulli. We show that this measure enjoys natural probabilistic properties for H\"older continuous observables, such as at least polynomial decay of correlations and the Central Limit Theorem. The results of Baladi and Demers are subject to a condition of sparse recurrence to singularities. We use a similar and slightly stronger condition, and it has a direct effect on our rate of decay of correlations. For billiard tables with bounded complexity (a property conjectured to be generic), we show that the sparse recurrence condition is always satisfied and the correlations decay at a super-polynomial rate

Behavior of the Escape Rate Function in Hyperbolic Dynamical Systems

For a fixed initial reference measure, we study the dependence of the escape rate on the hole for a smooth or piecewise smooth hyperbolic map. First, we prove the existence and Holder continuity of the escape rate for systems with small holes admitting Young towers. Then we consider general holes for Anosov diffeomorphisms, without size or Markovian restrictions. We prove bounds on the upper and lower escape rates using the notion of pressure on the survivor set and show that a variational principle holds under generic conditions. However, we also show that the escape rate function forms a devil's staircase with jumps along sequences of regular holes and present examples to elucidate some of the difficulties involved in formulating a general theory.Comment: 21 pages. v2 differs from v1 only by additions to the acknowledgment

Slow and fast escape for open intermittent maps

MD was partially supported by NSF grant DMS 1362420. This project was started as part of an RiGs grant through ICMS, Scotland.If a system mixes too slowly, putting a hole in it can completely destroy the richness of the dynamics. Here we study this instability for a class of intermittent maps with a family of slowly mixing measures. We show that there are three regimes: (1) standard hyperbolic-like behavior where the rate of mixing is faster than the rate of escape through the hole, there is a unique limiting absolutely continuous conditionally invariant measure (accim) and there is a complete thermodynamic description of the dynamics on the survivor set; (2) an intermediate regime, where the rate of mixing and escape through the hole coincide, limiting accims exist, but much of the thermodynamic picture breaks down; (3) a subexponentially mixing regime where the slow mixing means that mass simply accumulates on the parabolic fixed point. We give a complete picture of the transitions and stability properties (in the size of the hole and as we move through the family) in this class of open systems. In particular, we are able to recover a form of stability in the third regime above via the dynamics on the survivor set, even when no limiting accim exists.PostprintPeer reviewe

Hitting and escaping statistics : mixing, targets and holes

MD was partially supported by NSF grant DMS 1362420. This project was started as part of an RIG grant through ICMS, Scotland.There is a natural connection between two types of recurrence law: hitting times to shrinking targets, and hitting times to a fixed target (usually seen as escape through a hole). We show that for systems which mix exponentially fast, one can move through a natural parameter space from one to the other. On the other hand, if the mixing is subexponential, there is a phase transition between the hitting times law and the escape law.PostprintPeer reviewe

Complexity on Small Scales II: Metallicities and Ages in the Leo II Dwarf Spheroidal Galaxy

We present metallicities and ages for 52 red giants in the remote Galactic dwarf spheroidal (dSph) galaxy Leo II. These stars cover the entire surface area of Leo II and are radial velocity members. We obtained medium-resolution multi-fiber spectroscopy with ESO/VLT's FLAMES spectrograph. The metallicities were determined based on the near-infrared Ca II triplet. The resulting metallicity distribution (MD) is asymmetric and peaks at [Fe/H]=-1.74 dex on the Carretta & Gratton scale. The full range in metallicities extends from -2.4 to -1.1 dex. As in other dSphs, no extremely metal-poor red giants were found. We compare Leo II's observed MD with model predictions for several other Galactic dSphs from the literature. Leo II clearly exhibits a lack of more metal poor stars, in analogy to the classical G-dwarf problem, which may indicate a comparable `K-giant problem'. Moreover, its evolution appears to have been affected by galactic winds. We use our inferred metallicities as an input parameter for isochrone fits to SDSS photometry and derive approximate ages. The resulting age-metallicity distribution covers the full age range from 2-15 Gyr on our adopted isochrone scale. During the first 7 Gyr relative to the oldest stars [Fe/H] appears to have remained almost constant. The almost constant metallicity at higher ages and a slight drop by about 0.3 dex thereafter may be indicative of rejuvenation by low metallicity gas. Overall, the age-metallicity relation appears to support the formation of Leo II from pre-enriched gas. Evidence for enrichment is seen during the recent 2-4 Gyr. Our findings support earlier photometric findings of Leo II as a galaxy with a prominent old and a dominant intermediate-age population. We do not find a significant radial metallicity gradient nor age gradient in our data.(Abridged)Comment: 23 pages, 12 Figures, accepted for publication in the A

Photocatalytic Nanolithography of Self-Assembled Monolayers and Proteins

Self-assembled monolayers of alkylthiolates on gold and alkylsilanes on silicon dioxide have been patterned photocatalytically on sub-100 nm length-scales using both apertured near-field and apertureless methods. Apertured lithography was carried out by means of an argon ion laser (364 nm) coupled to cantilever-type near-field probes with a thin film of titania deposited over the aperture. Apertureless lithography was carried out with a helium–cadmium laser (325 nm) to excite titanium-coated, contact-mode atomic force microscope (AFM) probes. This latter approach is readily implementable on any commercial AFM system. Photodegradation occurred in both cases through the localized photocatalytic degradation of the monolayer. For alkanethiols, degradation of one thiol exposed the bare substrate, enabling refunctionalization of the bare gold by a second, contrasting thiol. For alkylsilanes, degradation of the adsorbate molecule provided a facile means for protein patterning. Lines were written in a protein-resistant film formed by the adsorption of oligo(ethylene glycol)-functionalized trichlorosilanes on glass, leading to the formation of sub-100 nm adhesive, aldehyde-functionalized regions. These were derivatized with aminobutylnitrilotriacetic acid, and complexed with Ni2+, enabling the binding of histidine-labeled green fluorescent protein, which yielded bright fluorescence from 70-nm-wide lines that could be imaged clearly in a confocal microscope