Normal Cones and Thompson Metric

The aim of this paper is to study the basic properties of the Thompson metric $d_T$ in the general case of a real linear space $X$ ordered by a cone $K$. We show that $d_T$ has monotonicity properties which make it compatible with the linear structure. We also prove several convexity properties of $d_T$ and some results concerning the topology of $d_T$, including a brief study of the $d_T$-convergence of monotone sequences. It is shown most of the results are true without any assumption of an Archimedean-type property for $K$. One considers various completeness properties and one studies the relations between them. Since $d_T$ is defined in the context of a generic ordered linear space, with no need of an underlying topological structure, one expects to express its completeness in terms of properties of the ordering, with respect to the linear structure. This is done in this paper and, to the best of our knowledge, this has not been done yet. The Thompson metric $d_T$ and order-unit (semi)norms $|\cdot|_u$ are strongly related and share important properties, as both are defined in terms of the ordered linear structure. Although $d_T$ and $|\cdot|_u$ are only topological (and not metrical) equivalent on $K_u$, we prove that the completeness is a common feature. One proves the completeness of the Thompson metric on a sequentially complete normal cone in a locally convex space. At the end of the paper, it is shown that, in the case of a Banach space, the normality of the cone is also necessary for the completeness of the Thompson metric.Comment: 36 page

Valuations and plurisubharmonic singularities

We extend to higher dimensions some of the valuative analysis of singularities of plurisubharmonic (psh) functions developed by the last two authors. Following Kontsevich and Soibelman we describe the geometry of the space V of all normalized valuations on C[x_1,...,x_n] centered at the origin. It is a union of simplices naturally endowed with an affine structure. Using relative positivity properties of divisors living on modifications of C^n above the origin, we define formal psh functions on V, designed to be analogues of the usual psh functions. For bounded formal psh functions on V, we define a mixed Monge-Ampere operator which reflects the intersection theory of divisors above the origin of C^n. This operator associates to any (n-1)-tuple of formal psh functions a positive measure of finite mass on V. Next, we show that the collection of Lelong numbers of a given germ u of a psh function at all infinitely near points induces a formal psh function u' on V called its valuative transform. When \phi is a psh Holder weight in the sense of Demailly, the generalized Lelong number nu_\phi(u) equals the integral of u' against the Monge-Ampere measure of the valuative transform of \phi. In particular, any generalized Lelong number is an average of valuations. We also show how to compute the multiplier ideal of u and the relative type of u with respect to \phi in the sense of Rashkovskii, in terms of the valuative transforms of u and \phi.Comment: 37 pages, new version. Changed the terminology from convex fonctions to formal psh functions. Corrected statement about the continuity of formal psh functions. Added a proof of the continuity of psh envelopes (theorem 5.13). Clarified the exposition (removed a section on the action of finite maps on formal psh functions). Added new reference

Word length statistics for Teichmuller geodesics and singularity of harmonic measure

Given a measure on the Thurston boundary of Teichmuller space, one can pick a geodesic ray joining some basepoint to a randomly chosen point on the boundary. Different choices of measures may yield typical geodesics with different geometric properties. In particular, we consider two families of measures: the ones which belong to the Lebesgue or visual measure class, and harmonic measures for random walks on the mapping class group generated by a distribution with finite first moment in the word metric. We consider the word length of approximating mapping class group elements along a geodesic ray, and prove that this quantity grows superlinearly in time along almost all geodesics with respect to Lebesgue measure, while along almost all geodesics with respect to harmonic measure the growth is linear. As a corollary, the harmonic and Lebesgue measures are mutually singular. We also prove a similar result for the ratio between the word metric and the relative metric (i.e. the induced metric on the curve complex)

Interval-valued contractive fuzzy negations

In this work we consider the concept of contractive interval-valued fuzzy negation, as a negation such that it does not increase the length or amplitude of an interval. We relate this to the concept of Lipschitz function. In particular, we prove that the only strict (strong) contractive interval-valued fuzzy negation is the one generated from the standard (Zadeh's) negation