on the topological entropy of free semigroup actions abstract we extend the notion the topological entropy of a free semigroup action defined by bufetov 6 to the case of a free semigroup action on a metric space not necessarily compact provide some properties of this extended topological entropy extend the topological analogue of the classical abramov rokhlin formula for the entropy of a skew product transformations with respect to a metric space not necessarily compact and give some bounds for the entropy for some particular systems such as a free semigroup with m generators of affine transformations on p dimensional torus and smooth maps on a riemannian manifold

poincare bendixson theory for leaves of codimension one the level of a local minimal set of a c2 codimension one foliation of a compact manifold is a nonnegative integer defined inductively level zero corresponding to the minimal sets in the usual sense each leaf of a local minimal set at level k is at level k the authors develop a theory of local minimal sets level and how leaves at level k asymptotically approach leaves at lower level this last generalizes the classical poincare bendixson theorem and provides information relating growth topological type and level e g if l is a totally proper leaf at level k then l has exactly polynomial growth of degree k and topological type k

understanding optimising and extending data compression with anisotropic diffusion galic et al journal of mathematical imaging and vision 31 255 269 2008 have shown that compression based on edge enhancing anisotropic diffusion eed can outperform the quality of jpeg for medium to high compression ratios when the interpolation points are chosen as vertices of an adaptive triangulation however the reasons for the good performance of eed remained unclear and they could not outperform the more advanced jpeg 2000 the goals of the present paper are threefold firstly we investigate the compression qualities of various partial differential equations this sheds light on the favourable properties of eed in the context of image compression secondly we demonstrate that it is even possible to beat the quality of jpeg 2000 with eed if one uses specific subdivisions on rectangles and several important optimisations these amendments include improved entropy coding brightness and diffusivity optimisation and interpolation swapping thirdly we demonstrate how to extend our approach to 3 d and shape data experiments on classical test images and 3 d medical data illustrate the high potential of our approach

euler s elastica and curvature based inpainting image inpainting is a special image restoration problem for which image prior models play a crucial role euler s elastica was first introduced to computer vision by mumford algebraic geometry and its applications springer verlag new york 1994 pp 491 506 as a curve prior model by functionalizing the elastica energy masnou and morel proceedings of the 5th ieee international conference image processing 3 1998 pp 259 263 proposed an elastica based variational inpainting model the current paper is intended to contribute to the development of its mathematical foundation and the study of its properties and connections to the earlier works of bertalmio sapiro caselles and ballester siggraph 2000 acm press new york 2000 and chan and shen j visual comm image rep 12 2001 pp 436 449 a computational scheme based on numerical pdes is presented which allows the automatic handling of topologically complex inpainting domains

crossing preserving coherence enhancing diffusion on invertible orientation scores many image processing problems require the enhancement of crossing elongated structures these problems cannot easily be solved by commonly used coherence enhancing diffusion methods therefore we propose a method for coherence enhancing diffusion on the invertible orientation score of a 2d image in an orientation score the local orientation is represented by an additional third dimension ensuring that crossing elongated structures are separated from each other we consider orientation scores as functions on the euclidean motion group and use the group structure to apply left invariant diffusion equations on orientation scores we describe how we can calculate regularized left invariant derivatives and use the hessian to estimate three descriptive local features curvature deviation from horizontality and orientation confidence these local features are used to adapt a nonlinear coherence enhancing crossing preserving diffusion equation on the orientation score we propose two explicit finite difference schemes to apply the nonlinear diffusion in the orientation score and provide a stability analysis experiments on both artificial and medical images show that preservation of crossings is the main advantage compared to standard coherence enhancing diffusion the use of curvature leads to improved enhancement of curves with high curvature furthermore the use of deviation from horizontality makes it feasible to reduce the number of sampled orientations while still preserving crossings

on the ergodic convergence rates of a first order primal dual algorithm we revisit the proofs of convergence for a first order primal dual algorithm for convex optimization which we have studied a few years ago in particular we prove rates of convergence for a more general version with simpler proofs and more complete results the new results can deal with explicit terms and nonlinear proximity operators in spaces with quite general norms

bifurcation analysis of a non cooperative differential game with one weak player we study a bifurcation problem for a system of two differential equations in implicit form for each value of the parameter the solution yields a pair of nash equilibrium strategies in feedback form for a non cooperative differential game when 0 the second player has no power to influence the dynamics of the system and his optimal strategy is myopic the game thus reduces to an optimal control problem for the first player by studying the bifurcation in the solutions to the corresponding system of hamilton jacobi equations one can establish existence and multiplicity of solutions to the differential game as becomes strictly positive

curvature varifolds with boundary we introduce a new class of nonoriented sets in rk endowed with a generalized notion of second fundamental form and boundary proving several compactness and structure properties our work extends the definition and some results of j e hutchinson 13 and can be applied to variational problems involving surfaces with boundary

a variational principle for free semigroup actions abstract in this paper we introduce a notion of measure theoretical entropy for a finitely generated free semigroup action and establish a variational principle when the semigroup is generated by continuous self maps on a compact metric space and has finite topological entropy in the case of semigroups generated by ruelle expanding maps we prove the existence of equilibrium states and describe some of their properties of independent interest are the different ways we will present to compute the metric entropy and a characterization of the stationary measures

the elastic ratio introducing curvature into ratio based image segmentation we present the first ratio based image segmentation method that allows imposing curvature regularity of the region boundary our approach is a generalization of the ratio framework pioneered by jermyn and ishikawa so as to allow penalty functions that take into account the local curvature of the curve the key idea is to cast the segmentation problem as one of finding cyclic paths of minimal ratio in a graph where each graph node represents a line segment among ratios whose discrete counterparts can be globally minimized with our approach we focus in particular on the elastic ratio 0l c i c s c s ds vl c 0l c c s qds that depends given an image i on the oriented boundary c of the segmented region candidate minimizing this ratio amounts to finding a curve neither small nor too curvy through which the brightness flux is maximal we prove the existence of minimizers for this criterion among continuous curves with mild regularity assumptions we also prove that the discrete minimizers provided by our graph based algorithm converge as the resolution increases to continuous minimizers in contrast to most existing segmentation methods with computable and meaningful i e nondegenerate global optima the proposed approach is fully unsupervised in the sense that it does not require any kind of user input such as seed nodes numerical experiments demonstrate that curvature regularity allows substantial improvement of the quality of segmentations furthermore our results allow drawing conclusions about global optima of a parameterization independent version of the snakes functional the proposed algorithm allows determining parameter values where the functional has a meaningful solution and simultaneously provides the corresponding global solution

black hole entropy topological entropy and the baum connes conjecture in k theory we shall try to exhibit a relation between black hole entropy and topological entropy using the famous baum connes conjecture for foliated manifolds which are particular examples of noncommutative spaces

contrast driven elastica for image segmentation minimization of boundary curvature is a classic regularization technique for image segmentation in the presence of noisy image data techniques for minimizing curvature have historically been derived from gradient descent methods which could be trapped by a local minimum and therefore required a good initialization recently combinatorial optimization techniques have overcome this barrier by providing solutions that can achieve a global optimum however curvature regularization methods can fail when the true object has high curvature in these circumstances existing methods depend on a data term to overcome the high curvature of the object unfortunately the data term may be ambiguous in some images which causes these methods also to fail to overcome these problems we propose a contrast driven elastica model including curvature which can accommodate high curvature objects and an ambiguous data model we demonstrate that we can accurately segment extremely challenging synthetic and real images with ambiguous data discrimination poor boundary contrast and sharp corners we provide a quantitative evaluation of our segmentation approach when applied to a standard image segmentation data set

nonexponential leaves at finite level abstracr previous examples of leaves with nonexponential and nonpolynomial growth due to g hector have occurred at infinite level here the same growth types are produced at finite level in open saturated sets of leaves without holonomy such sets consist of leaves with only one or two locally dense ends and it is shown that the exotic growth types only occur in the case of one locally dense end finally c foliations are produced with open saturated sets as above in which the leaves have strictly fractional growth

a linear framework for region based image segmentation and inpainting involving curvature penalization we present the first method to handle curvature regularity in region based image segmentation and inpainting that is independent of initialization to this end we start from a new formulation of length based optimization schemes based on surface continuation constraints and discuss the connections to existing schemes the formulation is based on a cell complex and considers basic regions and boundary elements the corresponding optimization problem is cast as an integer linear program we then show how the method can be extended to include curvature regularity again cast as an integer linear program here we are considering pairs of boundary elements to reflect curvature moreover a constraint set is derived to ensure that the boundary variables indeed reflect the boundary of the regions described by the region variables we show that by solving the linear programming relaxation one gets reasonably close to the global optimum and that curvature regularity is indeed much better suited in the presence of long and thin objects compared to standard length regularity

foliations the ergodic theorem and brownian motion regarding the leaves of a compact foliated riemannian manifold as the orbits of a dynamical system we will prove a birkhoff ergodic theorem asserting that the spatial average of a function equals its time average along random paths in the leaves there are several unexpected developments in the foundations first there always are measures on any compact foliated riemannian manifold m f with respect to which an adequate brownian motion along the leaves ergodic theorem holds second relative to these measures which are called harmonic there is a liouville type theorem for any bounded bore1 function on the manifold which is harmonic on each leaf namely for almost all leaves relative to any one of these harmonic measures this leaf harmonic function is constant on each leaf third the harmonic measures have a very reasonable local characterization in any foliation coordinate system they are transversal sums of h l dx l where h l is a positive harmonic function on the leaf l and dx l is the riemannian measure on l these statements comprise theorem 1 which is stated more precisely later in the introduction the first fact is surprising because up to now the only measures attached to foliations were the holonomy invariant transverse measures of plante 141 and ruelle sullivan 151 and some foliations do not admit these the second fact is unexpected because there are foliations where all the leaves are hyperbolic planes and each hyperbolic plane being conformal to the disk admits 285 0022 1236183 3 00

one dimensional topological dynamical systems of discrete group actions in this survey it is given a brief account of recent research achievements in the study of 1 dimensional topological dynamical systems of discrete group actions the content contains expansiveness ping pong game and geometric entropy invariant sets and invariant measures topological k transitivity sensitivity and devaney s chaos bounded euler classes and classifications some open questions are reported

filling in by joint interpolation of vector fields and gray levels a variational approach for filling in regions of missing data in digital images is introduced the approach is based on joint interpolation of the image gray levels and gradient isophotes directions smoothly extending in an automatic fashion the isophote lines into the holes of missing data this interpolation is computed by solving the variational problem via its gradient descent flow which leads to a set of coupled second order partial differential equations one for the gray levels and one for the gradient orientations the process underlying this approach can be considered as an interpretation of the gestaltist s principle of good continuation no limitations are imposed on the topology of the holes and all regions of missing data can be simultaneously processed even if they are surrounded by completely different structures applications of this technique include the restoration of old photographs and removal of superimposed text like dates subtitles or publicity examples of these applications are given we conclude the paper with a number of theoretical results on the proposed variational approach and its corresponding gradient descent flow

on the topological entropy of a semigroup of continuous maps abstract we introduce the notion of the lower local entropy of a borel probability measure on a compact metric space to estimate the bounds of topological entropy with respect to any subset in the case of finitely generated semigroup of continuous maps moreover we give the notion of the topological entropy of a semigroup generated by finite uniformly continuous maps on a metric space not necessarily compact provide some properties of this topological entropy and estimate the bounds of them for some particular systems such as a semigroup generated by finite affine transformations on the p dimensional torus and a semigroup generated by finite smooth maps on riemannian manifolds

total roto translational variation we consider curvature depending variational models for image regularization such as euler s elastica these models are known to provide strong priors for the continuity of edges and hence have important applications in shape and image processing we consider a lifted convex representation of these models in the roto translation space in this space curvature depending variational energies are represented by means of a convex functional defined on divergence free vector fields the line energies are then easily extended to any scalar function it yields a natural generalization of the total variation to the roto translation space as our main result we show that the proposed convex representation is tight for characteristic functions of smooth shapes we also discuss cases where this representation fails for numerical solution we propose a staggered grid discretization based on an averaged raviart thomas finite elements approximation this discretization is consistent up to minor details with the underlying continuous model the resulting non smooth convex optimization problem is solved using a first order primal dual algorithm we illustrate the results of our numerical algorithm on various problems from shape and image processing

specification and thermodynamical properties of semigroup actions in the present paper we study the thermodynamical properties of finitely generated continuous subgroup actions we propose a notion of topological entropy and pressure functions that do not depend on the growth rate of the semigroup and introduce strong and orbital specification properties under which the semigroup actions have positive topological entropy and all points are entropy points moreover we study the convergence and lipschitz regularity of the pressure function and obtain relations between topological entropy and exponential growth rate of periodic points in the context of semigroups of expanding maps obtaining a partial extension of the results obtained by ruelle for d actions d ruelle trans am math soc 187 237 251 1973 the specification properties for semigroup actions and the corresponding one for its generators and the action of push forward maps are also discussed

entropie geometrique des feuilletages on definit et on etudie l entropie geometrique d un feuilletage generalisant directement l entropie topologique d un flot

the harmonic measures of lucy garnett abstract this paper presents an improved approach to the theory of harmonic measures for foliated spaces introduced by garnett this approach is based on a method for solving elliptic equations on foliated spaces and on the hille yosida theory the diffusion semigroup of a general laplacian and its infinitesimal generator are made explicit applications of the path space to the dynamical study of a foliated space are described in particular the final section studies cocycles on foliated spaces a formula for their asymptotic limit and some analytic and geometric consequences

a note on the complexity function and entropy of pseudogroups abstract we examine the connections between complexity of a pseudogroup its equicontinuity the mixing property and entropy we prove that the entropy of a pseudogroup can be under some additional assumptions computed using a continuous and dynamically generating pseudometric

semigroup actions of expanding maps we consider semigroups of ruelle expanding maps parameterized by random walks on the free semigroup with the aim of examining their complexity and exploring the relation between intrinsic properties of the semigroup action and the thermodynamic formalism of the associated skew product in particular we clarify the connection between the topological entropy of the semigroup action and the growth rate of the periodic points establish the main properties of the dynamical zeta function of the semigroup action and relate these notions to recent research on annealed and quenched thermodynamic formalism meanwhile we examine how the choice of the random walk in the semigroup unsettles the ergodic properties of the action

integral representation of convex functionals on a space of measures abstract in duality pairs such as m b c 0 and w 1 p w 0 1 p a convex integral functional on the space of functions has a polar which admits an integral representation this representation is the sum of a first term involving the absolutely continuous component of the measure and of a second one which is a positively homogeneous function of the singular part the duality is useful in plasticity theory in the sobolev case the study of non parametric integrands is new a description of the sub differential is obtained

sur la dynamique unidimensionnelle en regularite intermediaire using probabilistic methods we prove new rigidity results for groups and pseudo groups of diffeomorphisms of one dimensional manifolds with intermediate regularity class i e between c 1 and c 2 in particular we show some generalizations of denjoy theorem and the classical kopell lemma for abelian groups these techniques are also applied to the study of codimension 1 foliations for instance we obtain several generalized versions of sacksteder theorem in class c 1 we conclude with some remarks about the stationary measure

centennial history of hilbert s 16th problem the second part of hilbert s 16th problem deals with polynomial differential equations in the plane it remains unsolved even for quadratic polynomials there were several attempts to solve it that failed yet the problem inspired significant progress in the geometric theory of planar differential equations as well as bifurcation theory normal forms foliations and some topics in algebraic geometry the dramatic history of the problem as well as related developments are presented below 1 the problem and its counterparts what may be said about the number and location of limit cycles of a planar polynomial vector field of degree n the limit cycle is an isolated closed orbit of a vector field this second part of hilbert s 16th problem appears to be one of the most persistent in the famous hilbert list h second only to the riemann function conjecture traditionally hilbert s question is split into three each one requiring a stronger answer problem 1 is it true that a planar polynomial vector field has but a finite number of limit cycles problem 2 is it true that the number of limit cycles of a planar polynomial vector field is bounded by a constant depending on the degree of the polynomials only the bound on the number of limit cycles in problem 2 is denoted by h n and known as the hilbert number linear vector fields have no limit cycles hence h 1 0 it is still unknown whether or not h 2 exists problem 3 give an upper bound for h n a solution to any of these problems implies a solution for the previous ones only the first problem is solved now the positive answer was established in e92 i91 there are analytic counterparts of problems 1 and 2 received by the editors december 2001 2000 mathematics subject classification primary 34cxx 34mxx 37f75

existence and non existence of length averages for foliations since the pioneering work of ghys et al it has been known that several methods of dynamical systems theory can be adopted to study of foliations our aim in this paper is to investigate complexity of foliations by generalising existence problem of time averages in dynamical systems theory to foliations it has recently been realised that a positive lebesgue measure set of points without time averages only appears for complicated dynamical systems such as dynamical systems with heteroclinic connections or homoclinic tangencies in this paper we introduce the concept of length averages to singular foliations and attempt to collect interesting examples with without length averages in particular we demonstrate that length averages exist everywhere for any codimension one mathscr c 1 orientable singular foliation without degenerate singularities on a compact surface under a mild condition on quasi minimal sets of the foliation which is in strong contrast to time averages of surface flows

anisotropic fast marching on cartesian grids using lattice basis reduction we introduce a modification of the fast marching algorithm which solves the anisotropic eikonal equation associated to an arbitrary continuous riemannian metric cal m on a two or three dimensional domain the algorithm has a complexity cal o n ln n n ln kappa cal m where n is the discrete domain cardinality the logarithmic dependency in the maximum anisotropy ratio kappa cal m of the riemannian metric allows us to handle extreme anisotropies for a limited numerical cost we prove the convergence of the algorithm and illustrate its efficiency by numerical experiments the algorithm relies on the computation at each grid point z of a special system of coordinates a reduced basis of the lattice bbb z m with respect to the symmetric positive definite matrix cal m z encoding the desired anisotropy at this point

mathematical models for local nontexture inpaintings dedicated to stanley osher on the occasion of his 60th birthday abstract inspired by the recent work of bertalmio et al on digital inpaintings siggraph 2000 we develop general mathematical models for local inpaintings of nontexture images on smooth regions inpaintings are connected to the harmonic and biharmonic extensions and inpainting orders are analyzed for inpaintings involving the recovery of edges we study a variational model that is closely connected to the classical total variation tv denoising model of rudin osher and fatemi phys d 60 1992 pp 259 268 other models are also discussed based on the mumford shah regularity comm pure appl math xlii 1989 pp 577 685 and curvature driven diffusions cdd of chan and shen j visual comm image rep 12 2001 the broad applications of the inpainting models are demonstrated through restoring scratched old photos disocclusion in vision analysis text removal digital zooming and edge based image coding