algebraic ℤ actions of entropy rank one we investigate algebraic z d actions of entropy rank one namely those for which each element has finite entropy such actions can be completely described in terms of diagonal actions on products of local fields using standard adelic machinery this leads to numerous alternative characterizations of entropy rank one both geometric and algebraic we then compute the measure entropy of a class of skew products where the fiber maps are elements from an algebraic z d action of entropy rank one this leads via the relative variational principle to a formula for the topological entropy of continuous skew products as the maximum of a finite number of topological pressures we use this to settle a conjecture concerning the relational entropy of commuting toral automorphisms

mahler measure and entropy for commuting automorphisms of compact groups we compute the joint entropy ofd commuting automorphisms of a compact metrizable group letr d u 1 1 d 1 1 be the ring of laurent polynomials ind commuting variables andm be anr d module then the dual groupx m ofm is compact and multiplication onm by each of thed variables corresponds to an action m of d by automorphisms ofx m every action of d by automorphisms of a compact abelian group arises this way iff r d our main formula shows that the topological entropy of alpha r d left langle f right rangle is given by h left alpha r d left langle f right rangle right log m left f right int limits 0 1 ldots int limits 0 1 log left f left e 2 pi it 1 ldots e 2 pi it d right right dt 1 ldots dt d where m f is the mahler measure off this reduces to the classical result for toral automorphisms via jensen s formula while the entropy of a single automorphism of a compact group is always the logarithm of an algebraic integer this no longer seems to hold for joint entropy of commuting automorphisms since values such as 7 3 4 2 occur if p is a non principal prime ideal we show h left alpha r d p right 0 using an analogue of the yuzvinskii thomas addition formula we computeh m for arbitraryr d modulesm and then the joint entropy for an action of d on a not necessarily abelian compact group using a result of boyd we characterize those m which have completely positive entropy in terms of the prime ideals associated tom and show this condition implies that m is mixing of all orders we also establish an analogue of berg s theorem proving that if m has finite entropy then haar measure is the unique measure of maximal entropy if and only if m has completely positive entropy finally we show that for expansive actions the growth rate of the number of periodic points equals the topological entropy

non periodic riemann examples with handles we show the existence of 1 parameter families of non periodic complete embedded minimal surfaces in euclidean space with infinitely many parallel planar ends in particular we are able to produce finite genus examples and quasi periodic examples of infinite genus

entropy geometry and disjointness for zero dimensional algebraic actions we show that many algebraic actions of higher rank abelian groups on zero dimensional compact abelian groups are mutually disjoint the proofs exploit differences in the entropy geometry arising from subdynamics and a form of abramov rokhlin formula for half space entropies

the geometry of minimal surfaces of finite genus ii nonexistence of one limit end examples we demonstrate that a properly embedded minimal surface in 3 with finite genus cannot have one limit end

t t 1 transformation is not loosely bernoulli the t t problem open since 1971 is to demonstrate that the t t transformation a naturally arising k transformation is not bernoulli it is shown here not even to be loosely bernoulli

embedded minimal surfaces of finite topology in this paper we prove that any complete embedded minimal surface m in r 3 with finite topology and compact boundary possibly empty is conformally a compact riemann surface m with boundary punctured in a finite number of interior points and that m can be represented in terms of meromorphic data on its conformal completion m in particular we demonstrate that m is a minimal surface of finite type and describe how this property permits a classification of the asymptotic behavior ofm

topological self joinings of cartan actions by toral automorphisms we show that if r 3 and is a faithful z cartan action on a torus t by automorphisms then any closed subset of t which is invariant and topologically transitive under the diagonal z action by is homogeneous in the sense that it is either the full torus t or a finite set of rational points or a finite disjoint union of parallel translates of some d dimensional invariant subtorus a counterexample is constructed for the rank 2 case

higher genus riemann minimal surfaces riemann surfaces constitute a one parameter family of embedded minimal surfaces which are periodic and have infinitely many horizontal planar ends the surfaces in this family are foliated by circles or straight lines in this paper we prove the existence of a one parameter family of embeded minimal surfaces which have infinitely many horizontal planar ends and have genus k for k 1 37 riemann surfaces as their flux is nearly vertical can be understood as a sequence of parallel planes connected by slightly bent catenoidal neks the surfaces we construct are obtained by replacing one of these catenoidal necks by a member of the family of minimal surfaces discovered by c costa d hoffman and w meek

synchronization points and associated dynamical invariants this paper introduces new invariants for multiparameter dynamical systems this is done by counting the number of points whose orbits intersect at time n under simultaneous iteration of finitely many endomorphisms we call these points synchronization points the resulting sequences of counts together with generating functions and growth rates are subsequently investigated for homeomorphisms of compact metric spaces toral automorphisms and compact abelian group epimorphisms synchronization points are also used to generate invariant measures and the distribution properties of these are analysed for the algebraic systems considered furthermore these systems reveal strong connections between the new invariants and problems of active interest in number theory relating to heights and greatest common divisors

complete embedded minimal surfaces of finite total curvature we will survey what is known about minimal surfaces s r 3 which are complete embedded and have finite total curvature int s k da infty the only classically known examples of such surfaces were the plane and the catenoid the discovery by costa 14 15 early in the last decade of a new example that proved to be embedded sparked a great deal of research in this area many new examples have been found even families of them as will be described below the central question has been transformed from whether or not there are any examples except surfaces of rotation to one of understanding the structure of the space of examples

the geometry of minimal surfaces of finite genus i curvature estimates and quasiperiodicity let mathcal m be the space of properly embedded minimal surfaces in 3 with genus zero and two limit ends normalized so that every surface m mathcal m has horizontal limit tangent plane at infinity and the vertical component of its flux equals one we prove that if a sequence m i i mathcal m has the horizontal part of the flux bounded from above then the gaussian curvature of the sequence is uniformly bounded this curvature estimate yields compactness results and the techniques in its proof lead to a number of consequences concerning the geometry of any properly embedded minimal surface in 3 with finite genus and the possible limits through a blowing up process on the scale of curvature of a sequence of properly embedded minimal surfaces with locally bounded genus in a homogeneously regular riemannian 3 manifold

a note on two types of lyapunov exponents and entropies for zk actions abstract in this paper two types of lyapunov exponents random lyapunov exponents and directional lyapunov exponents and the corresponding entropies random entropy and directional entropy are considered for smooth z k actions the close relation among these quantities are investigated and the formulas of them are given via the lyapunov exponents of the generators

the asymptotic behavior of properly embedded minimal surfaces of finite topology while advances have been made in recent years in the study of properly embedded minimal surfaces of finite topology in r3 3 5 8 9 11 12 13 23 progress has depended in an essential manner on the special structure of such surfaces with finite total curvature a properly immersed minimal surface with finite total curvature is conformally a compact riemann surface punctured in a finite number of points and its gauss map extends to the compact surface as a meromorphic function 22 in particular all the topological ends are conformally equivalent to a punctured disk and there is a well defined limit tangent plane at each end outside of a sufficiently large compact set such an end is a multisheeted graph over the limit tangent plane and if the end is embedded it is asymptotic to either a plane or a half catenoid see e g schoen 24 for a survey of results on properly immersed surfaces of finite total curvature see 9 1 without the assumption of finite total curvature these results are not true in general as an indication of the relative lack of information consider that it is unknown whether or not the helicoid is the only simply connected minimal surface that is properly embedded and nonplanar a question raised by osserman the helicoid has infinite total curvature and a single annular end if a surface has finite topology then all of its ends are annular any general theory of properly embedded minimal surfaces of finite topology must include an understanding of the behavior of these ends in this paper we prove

uniqueness of the riemann minimal examples we prove that a properly embedded minimal surface in r 3 of genus zero with infinite symmetry group is a plane a catenoid a helicoid or a riemann minimal example we introduce the language of hurwitz schemes to understand the underlying moduli space of surfaces in our setting

isomorphism rigidity of irreducible algebraic ℤd actions an irreducible algebraic d action on a compact abelian group x is a d action by automorphisms of x such that every closed invariant subgroup y x is finite we prove the following result if d 2 then every measurable conjugacy between irreducible and mixing algebraic d actions on compact zero dimensional abelian groups is affine for irreducible expansive and mixing algebraic d actions on compact connected abelian groups the analogous statement follows essentially from a result by katok and spatzier on invariant measures of such actions cf 4 and 3 by combining these two theorems one obtains isomorphism rigidity of all irreducible expansive and mixing algebraic d actions with d 2