Distribution of periodic points of polynomial diffeomorphisms of C^2

This paper deals with the dynamics of a simple family of holomorphic diffeomorphisms of \C^2: the polynomial automorphisms. This family of maps has been studied by a number of authors. We refer to [BLS] for a general introduction to this class of dynamical systems. An interesting object from the point of view of potential theory is the equilibrium measure $\mu$ of the set $K$ of points with bounded orbits. In [BLS] $\mu$ is also characterized dynamically as the unique measure of maximal entropy. Thus $\mu$ is also an equilibrium measure from the point of view of the thermodynamical formalism. In the present paper we give another dynamical interpretation of $\mu$ as the limit distribution of the periodic points of $f$

Polynomial diffeomorphisms of C^2, IV: The measure of maximal entropy and laminar currents

This paper concerns the dynamics of polynomial automorphisms of ${\bf C}^2$. One can associate to such an automorphism two currents $\mu^\pm$ and the equilibrium measure $\mu=\mu^+\wedge\mu^-$. In this paper we study some geometric and dynamical properties of these objects. First, we characterize $\mu$ as the unique measure of maximal entropy. Then we show that the measure $\mu$ has a local product structure and that the currents $\mu^\pm$ have a laminar structure. This allows us to deduce information about periodic points and heteroclinic intersections. For example, we prove that the support of $\mu$ coincides with the closure of the set of saddle points. The methods used combine the pluripotential theory with the theory of non-uniformly hyperbolic dynamical systems

Observations of the biological communities at Bolsa Chica artificial reef

Bolsa Chica Artificial Reef (BCAR) was constructed in November 1986 with 10,400 tons of concrete rubble and eight concrete and steel barges. Prior to any additional augmentation of BCAR, the u.s. Army Corps of Engineers and the California Coastal Commission required the California Department of Fish and Game (CDFG) to survey the bioloqical communities on and around BCAR. In April 1992, qualitative surveys of the biological communities were conducted on one of the eight modules at BCAR and at a nearby sand-only site. One of the modules, Module D, located in 90 feet of water (MLLW), was surveyed for fish, macroinvertebrates, and turf community organisms (small plants and sessile animals). Twelve species of fish were observed, including kelp bass (Paralabrax clathratus) and barred sand bass (P. nebulifer). Eight macroinvertebrate species were observed, rock scallops (Crassedoma giganteum) being the most abundant. The turf community was comprised of thirteen invertebrate taxa, among which erect ectoprocts (Bugula spp.) were the most numerous. Two species of foliose red algae (Rhodymenia pacifica and Anisocladella pacifica) were also observed. The reef has reached an advanced stage of successional development with fish and invertebrate communities diverse and well established. However, due,.to its depth and the turbidity of surrounding waters, this reef is not likely to ever support a diverse algal community. The diversity and abundance of fish and macroinvertebrates were, as to be expected, much lower in the nearby sand-only site. Only two species of fish and seven macroinvertebrate species were observed. Of these, only the sea pen, Stylatula elongata, was common. Overall, when compared to nearby sand-only habitats, Bolsa Chica Artificial Reef appears to contribute substantially to the local biological productivity. In addition, the concrete rubble used in BCAR' s construction appears to be performing as well as the quarry rock used in all of CDFG's experimental reefs. (Document pdf contains 22 pages

Post-critical set and non existence of preserved meromorphic two-forms

We present a family of birational transformations in $CP_2$ depending on two, or three, parameters which does not, generically, preserve meromorphic two-forms. With the introduction of the orbit of the critical set (vanishing condition of the Jacobian), also called ``post-critical set'', we get some new structures, some "non-analytic" two-form which reduce to meromorphic two-forms for particular subvarieties in the parameter space. On these subvarieties, the iterates of the critical set have a polynomial growth in the \emph{degrees of the parameters}, while one has an exponential growth out of these subspaces. The analysis of our birational transformation in $CP_2$ is first carried out using Diller-Favre criterion in order to find the complexity reduction of the mapping. The integrable cases are found. The identification between the complexity growth and the topological entropy is, one more time, verified. We perform plots of the post-critical set, as well as calculations of Lyapunov exponents for many orbits, confirming that generically no meromorphic two-form can be preserved for this mapping. These birational transformations in $CP_2$, which, generically, do not preserve any meromorphic two-form, are extremely similar to other birational transformations we previously studied, which do preserve meromorphic two-forms. We note that these two sets of birational transformations exhibit totally similar results as far as topological complexity is concerned, but drastically different results as far as a more ``probabilistic'' approach of dynamical systems is concerned (Lyapunov exponents). With these examples we see that the existence of a preserved meromorphic two-form explains most of the (numerical) discrepancy between the topological and probabilistic approach of dynamical systems.Comment: 34 pages, 7 figure

What is the value of a standard?

Standards play a critical role in the procurement of defence, and other, systems. Choosing the most appropriate standard is important but has become more topical given the UK Ministry of Defence policy of “as civilian as possible, as military as necessary”. Whereas historically managers might have selected from classes of defence standards, this choice set is now increased to include civil standards. We develop a model that has been commissioned by the UK Defence Standardisation whose responsibilities include supporting project teams on the selection of standards. Our model is based on an extension of Bayesian Belief Networks, called an Influence Diagram, which allows decisions and consequences to be represented as well as uncertain-ties. We have developed an initial model for a real case to assess the feasibility and use. We outline the con-text of the defence procurement project in our case study and describe the reasoning underpinning the model structure. We have found that it is possible to develop a simple model that captures the views of multiple stakeholders and informs a reasoned choice about the value of alternative standards

Partner orbits and action differences on compact factors of the hyperbolic plane. Part I: Sieber-Richter pairs

Physicists have argued that periodic orbit bunching leads to universal spectral fluctuations for chaotic quantum systems. To establish a more detailed mathematical understanding of this fact, it is first necessary to look more closely at the classical side of the problem and determine orbit pairs consisting of orbits which have similar actions. In this paper we specialize to the geodesic flow on compact factors of the hyperbolic plane as a classical chaotic system. We prove the existence of a periodic partner orbit for a given periodic orbit which has a small-angle self-crossing in configuration space which is a `2-encounter'; such configurations are called `Sieber-Richter pairs' in the physics literature. Furthermore, we derive an estimate for the action difference of the partners. In the second part of this paper [13], an inductive argument is provided to deal with higher-order encounters.Comment: to appear on Nonlinearit

Non-uniqueness of ergodic measures with full Hausdorff dimension on a Gatzouras-Lalley carpet

In this note, we show that on certain Gatzouras-Lalley carpet, there exist more than one ergodic measures with full Hausdorff dimension. This gives a negative answer to a conjecture of Gatzouras and Peres

Discovery of Anion Insertion Electrochemistry in Layered Hydroxide Nanomaterials

Electrode materials which undergo anion insertion are a void in the materials innovation landscape and a missing link to energy efficient electrochemical desalination. In recent years layered hydroxides (LHs) have been studied for a range of electrochemical applications, but to date have not been considered as electrode materials for anion insertion electrochemistry. Here, we show reversible anion insertion in a LH for the first time using Co and Co-V layer hydroxides. By pairing in situ synchrotron and quartz crystal microbalance measurements with a computational unified electrochemical band-diagram description, we reveal a previously undescribed anion-insertion mechanism occurring in Co and Co-V LHs. This proof of concept study demonstrates reversible electrochemical anion insertion in LHs without significant material optimization. These results coupled with our foundational understanding of anion insertion electrochemistry establishes LHs as a materials platform for anion insertion electrochemistry with the potential for future application to electrochemical desalination