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 $f$

Rural trajectories: Diversification and farm-community linkages in Whakatane District, 1999-2003

In New Zealand and elsewhere the interdependence of development in farming and the broader rural community can no longer be taken for granted. Five years ago we conducted a comparative analysis of the interrelated dynamics of change in agriculture and rural communities in the Central North Island. The observed trends from this research suggested that: (i) long and short cycles of change affecting the rural sector are promoting greater diversity in agriculture-community relations; (ii) adjustment processes are ongoing; and (iii) current evidence does not point unambiguously to either the decoupling or re-linking of agriculture and the broader rural community. This paper explores further the ambiguity encountered in the earlier research through a follow-up case study grounded in Whakatane District. The key finding is that as a result of individual effort and the will to diversify, the rural economy of Whakatane District is buoyant and farming remains the major economic activity. However, despite the apparent persistence of strong and pervasive agriculture-community linkages, the district remains vulnerable to forces embedded in short and long cycles of change. In terms of short-cycle change, the pressure on dairy farming from price fluctuation and the increasing attractiveness of conversion to horticulture is affecting the agricultural side of the equation, while the proliferation of lifestyle blocks is notable on the community side. In terms of long-cycle change, the influence of a renaissance of Maori rural living is beginning to be felt on the community side, while the effect of climate change and associated weather extremes is beginning to impact on agriculture

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

Ruppeiner theory of black hole thermodynamics

The Ruppeiner metric as determined by the Hessian of the Gibbs surface provides a geometric description of thermodynamic systems in equilibrium. An interesting example is a black hole in equilibrium with its own Hawking radiation. In this article, we present results from the Ruppeiner study of various black hole families from different gravity theories e.g. 2D dilaton gravity, BTZ, general relativity and higher-dimensional Einstein-Maxwell gravity.Comment: 10 pages, 1 figure. Talk given by N Pidokrajt at ERE2006 in Palma de Mallorca, Spai