isometry invariant geodesics and the fundamental group we prove that on closed riemannian manifolds with infinite abelian but not cyclic fundamental group any isometry that is homotopic to the identity possesses infinitely many invariant geodesics we conjecture that the result remains true if the fundamental group is infinite cyclic we also formulate a generalization of the isometry invariant geodesics problem and a generalization of the celebrated weinstein conjecture on a closed contact manifold with a selected contact form any strict contactomorphism that is contact isotopic to the identity possesses an invariant reeb orbit

a conservative discretization of the poisson nernst planck equations on adaptive cartesian grids in this paper we present a novel hybrid finite difference finite volume method for the numerical solution of the nonlinear poisson nernst planck pnp equations on irregular domains the method is described in two spatial dimensions but can be extended to three dimensional problems as well the boundary of the irregular domain is represented implicitly via the zero level set of a signed distance function and quadtree data structures are used to systematically generate adaptive grids needed to accurately capture the electric double layer near the boundary to handle the nonlinearity in the pnp equations efficiently a semi implicit time integration method is utilized an important feature of our method is that total number of ions in the system is conserved by carefully imposing the boundary conditions by utilizing a conservative discretization of the diffusive and more importantly the nonlinear migrative flux term several numerical experiments are conducted which illustrate that the presented method is first order accurate in time and second order accurate in space moreover these tests explicitly indicate that the algorithm is also conservative finally we illustrate the applicability of our method in the study of the charging dynamics of porous supercapacitors

on the average indices of closed geodesics on positively curved finsler spheres in this paper we prove that on every finsler n sphere s n f for n 6 with reversibility and flag curvature k satisfying frac lambda lambda 1 2 k le 1 either there exist infinitely many prime closed geodesics or there exist frac n 2 2 closed geodesics possessing irrational average indices if in addition the metric is bumpy then there exist n 3 closed geodesics possessing irrational average indices provided the number of prime closed geodesics is finite

global surfaces of section in non regular convex energy levels of hamiltonian systems in this paper we prove the existence of global sections of disk type in non regular and strictly convex energy levels of integrable and near integrable hamiltonian systems with two degrees of freedom this extends a result of hofer et al in ann math 2 148 1 197 289 1998 where the same statement is true provided the energy level is regular

loop products and closed geodesics the critical points of the length function on the free loop space m of a compact riemannian manifold m are the closed geodesics on m the length function gives a filtration of the homology of m and we show that the chas sullivan product hi hj hi j n is compatible with this filtration we obtain a very simple expression for the associated graded homology ring grh m when all geodesics are closed or when all geodesics are nondegenerate we also interpret sullivan s coproduct see su1 su2 on c as a product in cohomology h 0 h 0 hi j n 1 0 where 0 m is the constant loop we show that is also compatible with the length filtration and we obtain a similar expression for the ring grh 0 the nonvanishing of products n and n is shown to be determined by the rate at which the morse index grows when a geodesic is iterated we determine the full ring structure h 0 for spheres m s n 3

hamiltonian pseudo rotations of projective spaces the main theme of the paper is the dynamics of hamiltonian diffeomorphisms of mathbb c mathbb p n with the minimal possible number of periodic points equal to n 1 by arnold s conjecture called here hamiltonian pseudo rotations we prove several results on the dynamics of pseudo rotations going beyond periodic orbits using floer theoretical methods one of these results is the existence of invariant sets in arbitrarily small punctured neighborhoods of the fixed points partially extending a theorem of le calvez and yoccoz and of franks and misiurewicz to higher dimensions the other is a strong variant of the lagrangian poincare recurrence conjecture for pseudo rotations we also prove the c 0 rigidity of pseudo rotations with exponentially liouville mean index vector this is a higher dimensional counterpart of a theorem of bramham establishing such rigidity for pseudo rotations of the disk

multiple closed geodesics on 3 spheres abstract this paper is devoted to a study on closed geodesics on finsler and riemannian spheres we call a prime closed geodesic on a finsler manifold rational if the basic normal form decomposition cf y long bott formula of the maslov type index theory pacific j math 187 1999 113 149 of its linearized poincare map contains no 2 2 rotation matrix with rotation angle which is an irrational multiple of or irrational otherwise we prove that if there exists only one prime closed geodesic on a d dimensional irreversible finsler sphere with d 2 it cannot be rational then we further prove that there exist always at least two distinct prime closed geodesics on every irreversible finsler 3 dimensional sphere our method yields also at least two geometrically distinct closed geodesics on every reversible finsler as well as riemannian 3 dimensional sphere we prove also such results hold for all compact simply connected 3 dimensional manifolds with irreversible or reversible finsler as well as riemannian metrics

two elliptic closed geodesics on positively curved finsler spheres abstract in this paper we prove that for every finsler n dimensional sphere s n f with reversibility and flag curvature k satisfying 1 2 k 1 either there exist infinitely many closed geodesics or there exist at least two elliptic closed geodesics and each linearized poincare map has at least one eigenvalue of the form e 1 with being an irrational multiple of

braid floer homology abstract area preserving diffeomorphisms of a 2 disc can be regarded as time 1 maps of non autonomous hamiltonian flows on r z d 2 the periodic flow lines define braid conjugacy classes up to full twists we examine the dynamics relative to such braid classes and define a new invariant for such classes the braid floer homology this refinement of floer homology originally used for the arnol d conjecture yields a morse type forcing theory for periodic points of area preserving diffeomorphisms of the 2 disc based on braiding contributions of this paper include 1 a monotonicity lemma for the behavior of the nonlinear cauchy riemann equations with respect to algebraic lengths of braids 2 establishment of the topological invariance of the resulting braid floer homology 3 a shift theorem describing the effect of twisting braids in terms of shifting the braid floer homology 4 computation of examples and 5 a forcing theorem for the dynamics of hamiltonian disc maps based on braid floer homology

periodic approximations of irrational pseudo rotations using pseudoholomorphic curves we prove that every c 1 smooth area preserving dieomorphism of the closed 2 disk having not more than one periodic point is the uniform limit of periodic c 1 smooth dieomorphisms in particular every smooth irrational pseudo rotation can be c 0 approximated by integrable systems this partially answers a long standing question of a katok regarding zero entropy hamiltonian systems in low dimensions our approach uses pseudoholomorphic curve techniques from symplectic geometry

rotation vectors and fixed points of area preserving surface diffeomorphisms we consider the homological rotation vectors for area preserving diffeomorphisms of compact surfaces which are homotopic to the identity there are two main results the first is that if 0 is in the interior of the convex hull of the rotation vectors for such a diffeomorphism then f has a fixed point of positive index the second result asserts that if f has a vanishing mean rotation vector then f has a fixed point of positive index there are several applications including a new proof of the arnold conjecture for area preserving diffeomorphisms of compact surfaces in this paper we investigate area preserving diffeomorphisms of compact surfaces we are concerned with rotation vectors defined in terms of homology which can be associated to the points of the surface there are two main results the first is that if f is an area preserving diffeomorphism of a compact surface which is homotopic to the identity and 0 is in the interior of the convex hull of the rotation vectors of f then f has a fixed point of positive index the second result concerns the mean rotation vector of f it asserts that if this vector vanishes then f has a fixed point of positive index a special case of this second result when the surface in question is a surface of genus zero was proved in f5 in 6 we give several applications including a new proof of the arnold conjecture for area preserving diffeomorphisms of compact surfaces

finite energy foliations of tight three spheres and hamiltonian dynamics surfaces of sections are a classical tool in the study of 3 dimensional dynamical systems their use goes back to the work of poincare and birkhoff in the present paper we give a natural generalization of this concept by constructing a system of transversal sections in the complement of finitely many distinguished periodic solutions such a system is established for nondegenerate reeb flows on the tight 3 sphere by means of pseudoholomorphic curves the applications cover the nondegenerate geodesic flows on t1s2 rp3 via its double covering s3 and also nondegenerate hamiltonian systems in ir4 restricted to sphere like energy surfaces of contact type

rotation numbers and instability sets translation and rotation numbers have played an interesting and important role in the qualitative description of various dynamical systems in this exposition we are especially interested in applications which lead to proofs of periodic motions in various kinds of dynamics on the annulus the applications include billiards and geodesic flows going beyond this simple qualitative invariant in the study of the dynamics of area preserving annulus maps g d birkhoff was led to the concept of regions of instability for twist maps we discuss the closely related notion of instability sets for a generic area preserving surface diffeomorphism and develop their properties

the conley zehnder index and the saddle center equilibrium abstract we consider hamiltonian systems with two degrees of freedom we suppose the existence of a saddle center equilibrium in a strictly convex component s of its energy level moser s normal form for such equilibriums and a theorem of hofer wysocki and zehnder are used to establish the existence of a periodic orbit in s with several topological properties we also prove the explosion of the conley zehnder index of any periodic orbit that passes close to the saddle center equilibrium

statistical efficiency and optimal design for stepped cluster studies under linear mixed effects models in stepped cluster designs the intervention is introduced into some or all clusters at different times and persists until the end of the study instances include traditional parallel cluster designs and the more recent stepped wedge designs we consider the precision offered by such designs under mixed effects models with fixed time and random subject and cluster effects including interactions with time and explore the optimal choice of uptake times the results apply both to cross sectional studies where new subjects are observed at each time point and longitudinal studies with repeat observations on the same subjects the efficiency of the design is expressed in terms of a cluster mean correlation which carries information about the dependency structure of the data and two design coefficients which reflect the pattern of uptake times in cross sectional studies the cluster mean correlation combines information about the cluster size and the intra cluster correlation coefficient a formula is given for the design effect in both cross sectional and longitudinal studies an algorithm for optimising the choice of uptake times is described and specific results obtained for the best balanced stepped designs in large studies we show that the best design is a hybrid mixture of parallel and stepped wedge components with the proportion of stepped wedge clusters equal to the cluster mean correlation the impact of prior uncertainty in the cluster mean correlation is considered by simulation some specific hybrid designs are proposed for consideration when the cluster mean correlation cannot be reliably estimated using a minimax principle to ensure acceptable performance across the whole range of unknown values 2016 the authors statistics in medicine published by john wiley sons ltd

existence of closed geodesics on finsler n spheres abstract in this paper we prove that on every finsler n sphere s n f with reversibility satisfying f 2 1 2 g 0 and l s n f 1 1 there always exist at least n prime closed geodesics without self intersections where g 0 is the standard riemannian metric on s n with constant curvature 1 and l s n f is the length of a shortest geodesic loop on s n f we also study the stability of these closed geodesics

the fadell rabinowitz index and multiplicity of non contractible closed geodesics on finsler rpn abstract in this paper we prove that for every irreversible finsler n dimensional real projective space r p n f with reversibility and flag curvature k satisfying 16 9 1 2 k 1 with 3 there exist at least n 1 non contractible closed geodesics in addition if the metric f is bumpy with 64 25 1 2 k 1 and 5 3 then there exist at least 2 n 1 2 non contractible closed geodesics which is the optimal lower bound due to katok s example the main ingredients of the proofs are the fadell rabinowitz index theory of non contractible closed geodesics on r p n f and the s 1 equivariant poincare series of the non contractible component of the free loop space on r p n

on a generic property of geodesic flows in this paper we are concerned with a sufficient condition for a riemannian metric on a compact simply connected manifold to have infinitely many geometrically distinct closed geodesics 1969 gromoll meyer proved in gm that for every riemannian metric on a compact simply connected manifold m there are infinitely many geometrically distinct closed geodesics if the sequence bi m f i of betti numbers of the free loop space m is unbounded for some field f using the theory of minimal models vigue poirrier sullivan proved that the rational cohomology algebra h m i q of m is generated by a single element if and only if the sequence bi m i q i of betti numbers of the free loop space m of m is bounded it is a conjecture that the same statement holds for all fields of prime characteristic partial results are due to mccleary ziller mz and halperin vigue poirrier hv now we turn to the manifolds for which the hypothesis of gromoll meyer s theorem does not hold for example spheres and projective spaces then stability properties of the closed geodesics become important a closed geodesic is hyperbolic if the linearized poincare map has no eigenvalue of norm 1 from the bumpy metrics theorem due to abraham ab and anosov an2 and from a pertubation result due to klingenberg takens kt one can conclude a c4 generic metric on a compact manifold has either a non hyperbolic closed geodesic of twist type or all closed geodesics are hyperbolic in the first case there are infinitely many geometrically distinct closed geodesics in every tubular neighborhood of the closed geodesic of twist type due to a theorem by moser mo in the second case there are infinitely many closed geodesics if m is simply connected due to results by hingston hi and the author ra1 hence it follows from these results that a c4 generic metric on a compact simply connected manifold has infinitely many geometrically distinct closed geodesics we remark that it is an open question whether there is a riemannian metric on a simply connected compact manifold all of whose closed geodesics are hyperbolic in ra2 we show that the examples of metrics on the 2

the second closed geodesic on finsler spheres of dimension n 2 we show the existence of at least two geometrically distinct closed geodesics on an n dimensional sphere with a bumpy and non reversible finsler metric for n 2 2000 msc classification 53c22 53c60 58e10

c genericity of positive topological entropy for geodesic flows on s2 we show that there is a c open and dense set of positively curved metrics on s2 whose geodesic flow has positive topological entropy and thus exhibits chaotic behavior the geodesic flow for each of these metrics possesses a horseshoe and it follows that these metrics have an exponential growth rate of hyperbolic closed geodesics the positive curvature hypothesis is required to ensure the existence of a global surface of section for the geodesic flow our proof uses a new and general topological criterion for a surface diffeomorphism to exhibit chaotic behavior very shortly after this manuscript was completed the authors learned about remarkable recent work by hofer wysocki and zehnder 14 15 on three dimensional reeb flows in the special case of geodesic flows on s2 they show that if the geodesic flow has no parabolic closed geodesics this holds for an open and c dense set of riemannian metrics on s2 then it possesses either a global surface of section or a heteroclinic orbit it then immediately follows from the proof of our main theorem that there is a c open and dense set of riemannian metrics on s2 whose geodesic flow has positive topological entropy this concludes a program to show that every orientable compact surface has a c open and dense set of riemannian metrics whose geodesic flow has positive topological entropy

regular and irregular geodesics on spherical harmonic surfaces abstract the behavior of geodesic curves on even seemingly simple surfaces can be surprisingly complex in this paper we use the hamiltonian formulation of the geodesic equations to analyze their integrability properties in particular we examine the behavior of geodesics on surfaces defined by the spherical harmonics using the morales ramis theorem and kovacic algorithm we are able to prove that the geodesic equations on all surfaces defined by the sectoral harmonics are not integrable and we use poincare sections to demonstrate the breakdown of regular motion

geodesics on s2 and periodic points of annulus homeomorphisms we show that an area preserving homeomorphism of the open or closed annulus which has at least one periodic point must in fact have infinitely many interior periodic points a consequence is the theorem that every smooth riemannian metric ons2 with positive gaussian curvature has infinitely many distinct closed geodesics

generalizations of the poincare birkhoff theorem in this article we prove some generalizations of the classical poincar birkhoff theorem on area preserving homeomorphisms of the annulus which satisfy a boundary twist condition the work of g d birkhoff on this theorem and its applications can be found in b1 b2 and chapter v of b3 a more modern treatment can be found in b n we prove a theorem for the open annulus a s1 x 0 1 since as we will see the theorem for the closed annulus can easily be obtained from this result for the open annulus however it is not immediately obvious what should be the analogue of the twist condition it turns out that the most general hypothesis and the most natural from the point of view of our proof involves the notion of positively and negatively returning disks for some lift of f to the covering space a r x 0 1 more precisely if f a a is a lift of f a a we will say that the e is a positively returning disk for f if there is an open disk u c a such that f u n u 0 and fp u n u k 0 forsome n k 0 here u k denotes the set x k t i x t e u thus u is disjoint from its image but under iteration by f returns to a positive translate of itself a negatively returning disk is defined similarly but with k 0 such that fl u n u 0 in section 2 we prove the following

non hyperbolic closed geodesics on positively curved finsler spheres abstract in this paper we prove that for every finsler n dimensional sphere s n f n 3 with reversibility and flag curvature k satisfying 1 2 k 1 there exist at least three distinct closed geodesics and at least two of them are elliptic if the number of prime closed geodesics is finite when n 6 these three distinct closed geodesics are non hyperbolic

simulation of turbulent thermal convection in complicated domains to simulate turbulent flows in complicated enclosed three dimensional domains a fast finite volume high order method is developed in principle the method is based on the chorin temam scheme the poisson solver which is applied to compute the pressure uses the separation of variables together with capacitance matrix technique the developed numerical method generally allows to use hexahedral computational meshes which are non equidistant in all three directions and non regular in any two directions the method was successfully used in three dimensional direct numerical simulations of turbulent high rayleigh number thermal convection in cylindrical and parallelepiped domains with obstacles

existence of infinitely many minimal hypersurfaces in positive ricci curvature in the early 1980s s t yau conjectured that any compact riemannian three manifold admits an infinite number of closed immersed minimal surfaces we use min max theory for the area functional to prove this conjecture in the positive ricci curvature setting more precisely we show that every compact riemannian manifold with positive ricci curvature and dimension at most seven contains infinitely many smooth closed embedded minimal hypersurfaces in the last section we mention some open problems related with the geometry of these minimal hypersurfaces

geometry and dynamics of planar linkages in this article we present an approach to describing the geometry and curvature of the configuration spaces of a class of simple idealized planar linkages this is based on determining the curvature of such configuration spaces canonically embedded into euclidean space and then the behaviour of the dynamics of such linkages can be understood via the associated geodesic flow our objective is to present a method which in principle can be applied to many different examples

a dynamical characterization of universally tight lens spaces we give necessary and sufficient conditions for a closed connected co orientable contact 3 manifold m to be a standard lens space based on assumptions on the reeb flow associated to a defining contact form our methods also provide rational global surfaces of section for non degenerate reeb flows on l p q std with prescribed binding orbits

simple closed geodesics on most alexandrov surfaces abstract we study the existence of simple closed geodesics on most in the sense of baire category alexandrov surfaces with curvature bounded below compact and without boundary we show that it depends on both the curvature bound and the topology of the surfaces

direct solution of the discretized poisson neumann problem on a domain composed of rectangles abstract finite difference approximations for the two dimensional poisson neumann problem 1 div grad p q 0on r n grad p g r on r result in a large system of linear equations lp q the n n matrix l is singular with rank l n 1 a solution exists if q satisfies the consistency condition v t q 0 where l t v 0 v 0 a direct solution scheme is described for the case where r can be decomposed into a set of rectangular domains each having a possibly different but constant material coefficient we assume that this problem has to be solved repeatedly for many vectors q the method is efficient in that the number of operations is of order n log n this efficiency is gained by using fast elliptic solvers for each single rectangle and a proper variant of the socalled influence or capacitance matrix technique here on each rectangle a separate poisson neumann problem has to be solved the essential point is the manner in which the consistency condition is enforced for each rectangle the new method has been successfully applied in a code flux to analyze fluid structure interactions

rotation sets for homeomorphisms and homology in this article we propose a definition of rotation sets for homeomorphisms of arbitrary compact manifolds this approach is based on taking the suspended flow and using ideas of schwartzmann on homology and winding cycles for flows our main application is to give a generalisation of a theorem of libre and mackay for tori to the context of surfaces of higher genus

the existence of two closed geodesics on every finsler 2 sphere in this paper we prove that for every finsler metric on s2 there exist at least two distinct prime closed geodesics

index iteration theories for periodic orbits old and new abstract in this paper we give a survey of the index iteration theories developed for periodic solution orbits of hamiltonian systems in treating various problems from 1950s up to now including some recent developments and some of their applications to periodic solution problems