existence of continuous solutions and stability for a general iterative functional equation in this paper c 0 solutions for a general form of iterative functional equation are considered when the coefficient of the first order iterate can be zero which is called the leading coefficient problem for polynomial like iterative equations moreover the stability of those c 0 solutions is also discussed finally as an application we give an algorithm to compute the piecewise linear solutions of polynomial like iterative equations

the set of all iterates is nowhere dense in c 0 1 0 1 we prove that if a mixing map f 0 1 0 1 belongs to the c 0 closure of the set of iterates and f 0 0 f 1 1 then f is an iterate itself together with some one dimensional techniques it implies that the set of all iterates is nowhere dense in c 0 1 0 1 giving the final answer to the question of a bruckner p humke and m laczkovich

forts of quadratic polynomials under iteration abstract since some dynamical behaviors of a one dimensional mapping are influenced by the number of forts attention is paid to the change of the number under iteration for simple computation we work on quadratic polynomials we use the theory of polynomial complete discrimination system to give a symbolic algorithm for the number of forts of iterated polynomials and apply the algorithm to quadratic functions which proves an alternative result that the number either persists to be 1 or tends to infinity under iteration we further compute the number for iterates of order 2 3 7 in the above infinity case and obtain critical values of the parameter at which the number changes those changes with finitely many data display a conjectured fibonacci rule

sharp regularity of linearization for c 1 1 hyperbolic diffeomorphisms c 1 linearization is of special significance because it preserves smooth dynamical behaviors and distinguishes qualitative properties in characteristic directions however c 1 smoothness is not enough to guarantee c 1 linearization for c 1 1 hyperbolic diffeomorphisms on banach spaces c 1 linearization was proved under a gap condition together with a band condition of the spectrum in this paper the result of c 1 linearization in banach spaces is strengthened to c 1 beta linearization with a constant beta 0 under a weaker band condition by a decomposition with invariant foliations the weaker band condition allows the spectrum to be a union of more than two but finitely many bands but restricts those bands to be bounded by a number depending on the supremum of contractive spectrum and the infimum of expansive spectrum furthermore we give an estimate for the exponent beta and prove that the estimate is sharp in the planar case

fractional iterates for n dimensional maps this paper constitutes an extension of results of the paper by narayaninsamy int j bifurcations chaos 6 1 1996 55 67 we study here the solutions of the problem of the fractional iteration for n dimensional maps

local c 1 stability versus global c 1 unstability for iterative roots abstract stability of iterative roots is important in the numerical computation of iterative roots known results show that under some conditions iterative roots of strictly monotonic self mappings are c 0 stable in both the local sense and the global sense in this paper we discuss the c 1 stability for iterative roots of strictly increasing self mappings on a compact interval between two fixed points we prove that those iterative roots are locally c 1 stable but globally c 1 unstable

topological conjugacy of piecewise monotonic functions of nonmonotonicity height 1 abstract the conjugacy problem is an important topic in the theory of dynamical systems and functional equations in this paper we investigate a class of piecewise monotone and continuous maps with nonmonotonicity height 1 we give a sufficient and necessary condition under which any two of these maps are topologically conjugate and construct a topological conjugacy with an extension method if such a conjugacy exists

iterative roots of piecewise monotonic functions of nonmonotonicity height not less than 2 abstract it is known that any piecewise monotonic function of nonmonotonicity height not less than 2 has no continuous iterative roots of order greater than the number of forts of the function so the following problem arises naturally does such a function have an iterative root of order n not greater than the number of forts we consider the case that the number of forts is equal to n in which there appear possibly only two types t 1 and t 2 of iterative roots i e the roots strictly increasing on the interval stretched on all forts of the given function and the roots strictly decreasing on such an interval respectively we characterize all type t 1 roots of order n and give a necessary condition for a root of order n to be of type t 2 a full description of type t 2 is still an open question

l 2 restriction bounds for eigenfunctions along curves in the quantum completely integrable case we show that for a quantum completely integrable system in two dimensions the l2 normalized joint eigenfunctions of the commuting semiclassical pseudodifferential operators satisfy restriction bounds of the form int gamma varphi j hbar 2 ds mathcal o log hbar for generic curves on the surface we also prove that the maximal restriction bounds of burq gerard tzvetkov bgt are generically attained for certain exceptional subsequences of eigenfunctions

a convergent quadratic time lattice algorithm for pricing european style asian options abstract asian options are strongly path dependent derivatives although efficient numerical methods and approximate closed form formulas are available most lack convergence guarantees asian options can also be priced on the lattice all efficient lattice algorithms keep only a polynomial number of states and use interpolation to compensate for the less than full representation of the states let the time to maturity be partitioned into n periods this paper presents the first o n 2 time convergent lattice algorithm for pricing european style asian options it is the most efficient lattice algorithm with convergence guarantees the algorithm relies on the lagrange multipliers to choose optimally the number of states for each node of the lattice the algorithm is also memory efficient extensive numerical experiments and comparison with existing pde analytical and lattice methods confirm the performance claims and the competitiveness of our algorithm this result places the problem of european style asian option pricing in the same complexity class as that of the vanilla option on the lattice

efficient pricing of discrete asian options asian options are popular path dependent financial derivatives this paper uses lattices to price fixed strike european style asian options that are discretely monitored the algorithm proposed can also be applied to floating strike asian options as well because fixed strike and floating strike asian options are related through an equation the discretely monitored version is usually found in practice instead of the continuously monitored version usually encountered in the literature this paper presents the first provably quadratic time convergent lattice algorithm for pricing fixed strike european style discretely monitored asian options it is the most efficient lattice algorithm with convergence guarantees the algorithm relies on the lagrange multipliers to choose the number of states for each node of the lattice extensive numerical experiments and comparisons with many existing numerical methods confirm the performance claims and the competitiveness of our algorithm this result places fixed strike european style discretely monitored asian options in the same complexity class as vanilla options

leading coefficient problem for polynomial like iterative equations abstract because of difficulties in applying fixed point theorems most of known results on the polynomial like iterative equation j 1 n j f j x f x were given by assuming 1 0 and the existence of solutions under the most natural assumption n 0 is an interesting problem called leading coefficient problem for this problem locally expansive invertible c 1 solutions are obtained in the expansive case and the non hyperbolic case in w zhang on existence for polynomial like iterative equations results math 45 2004 185 194 and c 0 increasing solutions are constructed in b xu w zhang construction of continuous solutions and stability for the polynomial like iterative equation j math anal appl 325 2007 1160 1170 in this paper we discuss c 1 solutions for more combinations between expansive mappings and contractive ones and combinations between increasing mappings and decreasing ones

computing iterative roots of polygonal functions based on the iterative root theory for monotone functions an algorithm for computing polygonal iterative roots of increasing polygonal functions was given by j kobza in this paper we not only give an algorithm for roots of decreasing polygonal functions but also generalize kobza s results to the general n furthermore we extend our algorithms for polygonal pm functions a class of non monotonic functions

transversal homoclinic points of a class of conservative diffeomorphisms abstract we develop a global graph transformation to obtain estimates for certain invariant manifolds of a class of area preserving diffeomorphisms with symmetries the existence of homoclinic or heteroclinic points and the analyticity of the angle between the invariant manifolds at them is proved for the henon and the standard maps a lower bound of this angle is given for a large set of values of the parameters

topological conjugacy and transitivity for a class of piecewise monotone maps of the interval on dit qu une application continue f d un intervalle compact dans lui meme est markov lineaire s il est lineaire par morceaux et l ensemble de tous les f k x ou k 0 et x une extremite d un morceau lineaire est fini on donne une classification effective pour des applications de markov lineaires et une procedure effective pour determiner quand une telle application est transitive

iteration and the solution of functional equations for functions analytic in the unit disk this paper considers the classical functional equations of schroeder f o p xf and abel f o p f 1 and related problems of fractional iteration where 9p is an analytic mapping of the open unit disk into itself the main theorem states that under very general conditions there is a linear fractional transformation 4 and a function a analytic in the disk such that a oa p and that with suitable normalization 4 and a are unique in particular the hypotheses are satisfied if 4p is a probability generating function that does not have a double zero at 0 this intertwining relates solutions of functional equations for t to solutions of the corresponding equations for d for example it follows that if 4p has no fixed points in the open disk then the solution space of f o p xf is infinite dimensional for every nonzero x although the discrete semigroup of iterates of t usually cannot be embedded in a continuous semigroup of analytic functions mapping the disk into itself we find that for each z in the disk all sufficiently large fractional iterates of p can be defined at z this enables us to find a function meromorphic in the disk that deserves to be called the infinitesimal generator of the semigroup of iterates of p if the iterates of p can be embedded in a continuous semigroup we show that the semigroup must come from the corresponding semigroup for 4 and thus be real analytic in t the proof of the main theorem is not based on the well known limit technique introduced by koenigs 1884 but rather on the construction of a riemann surface on which an extension of 9p is a bijection much work is devoted to relating characteristics of p to the particular linear fractional transformation constructed in the theorem

how violently do smooth iterative roots oscillate abstract a crisis in the smoothness of iterative roots appears at fixed points the local c r smoothness of iterative roots near a hyperbolic fixed point is well known but its global c r extension to both fixed points is generically impossible because of their violent oscillation near the next hyperbolic fixed point in this study we consider how violently these smooth iterative roots oscillate we prove that their first order derivatives are bounded but their r th r 2 order derivatives are generically unbounded in addition we prove that the global c 1 smoothness is of vital significance for the global higher order derivatives as a corollary we show that iterative roots on the maximal interval oscillate with c 1 boundedness but c r r 2 unboundedness

on fractional iterates of a brouwer homeomorphism embeddable in a flow we present a method for finding continuous and consequently homeomorphic orientation preserving iterative roots of a brouwer homeomorphism which is embeddable in a flow to obtain the roots we use a countable family of maximal parallelizable regions of the flow which is a cover of the plane the maximal parallelizable regions are unions of equivalence classes of an appropriate equivalence relation we show that if an equivalence class is invariant under the nth iterate of a brouwer homeomorphism g then it is invariant under g we use this fact to prove that each maximal parallelizable region of the flow must be invariant under all homeomorphic orientation preserving iterative roots of the given brouwer homeomorphism

boundedness of projection operators and cesaro means in weighted l p space on the unit sphere for the weight function d 1 i 1 x i 2 i on the unit sphere sharp local estimates of the orthogonal projection operators are obtained and used to prove the convergence of the cesaro c means in the weighted l p space for above the critical index similar results are also proved for corresponding weight functions on the unit ball and on the simplex

near sharp strichartz estimates with loss in the presence of degenerate hyperbolic trapping we consider an n dimensional spherically symmetric asymptotically euclidean manifold with two ends and a codimension 1 trapped set which is degenerately hyperbolic by separating variables and constructing a semiclassical parametrix for a time scale polynomially beyond ehrenfest time we show that solutions to the linear schrodinger equation with initial conditions localized on a spherical harmonic satisfy strichartz estimates with a loss depending only on the dimension n and independent of the degeneracy the strichartz estimates are sharp up to an arbitrary 0 loss this is in contrast to christianson and wunsch amer j math 2013 where it is shown that solutions satisfy a sharp local smoothing estimate with loss depending only on the degeneracy of the trapped set independent of the dimension

strichartz estimates for the fractional schrodinger and wave equations on compact manifolds without boundary we firstly prove strichartz estimates for the fractional schrodinger equations on mathbb r d d geq 1 endowed with a smooth bounded metric g we then prove strichartz estimates for the fractional schrodinger and wave equations on compact riemannian manifolds without boundary m g this result extends the well known strichartz estimate for the schrodinger equation given in 7 we finally give applications of strichartz estimates for the local well posedness of the pure power type nonlinear fractional schrodinger and wave equations posed on m g

the question on characteristic endpoints for iterative roots of pm functions abstract iterative roots of mappings are of special interests because it defines fractional iteration displays middle procedure of evolution and proposes a weak version of the embedding flow problem for pm functions of height 1 the class of 1 dimensional mappings having the simplest nonmonotonicity the existence of continuous iterative roots of any order was obtained under the characteristic endpoints condition and the condition was proved to be necessary for those orders greater than the number of forts plus 1 this suggests an open question about iterative roots without that condition called the question on characteristic endpoints in this paper the question is answered completely in the case that the number of forts is equal to the order although results of nonexistence are also obtained for the case that the number of forts is greater than the order a full description is still open

pricing asian option by the fft with higher order error convergence rate under levy processes pricing asian options is a long standing hard problem there is no analytical formula for the probability density of its payoff even when the process of the underlying asset follows the simple lognormal diffusion process it is known that the density function of a discretely sampled asian option s payoff can be efficiently approximated by the fast fourier transform fft as a result we can accurately price the option under more general levy processes this paper shows that the pricing error of this approach called the fft approach can be decomposed into the truncation error the integration error and the interpolation error we prove that previous algorithms that follow the fft approach converge quadratically to improve the error convergence rate our proposed algorithms reduce the integration error by the higher order newton cotes formulas and new integration rules derived from the lagrange interpolating polynomial the interpolation error is reduced by the higher order newton divided difference interpolation formula consequently our algorithms can be sped up by the fft to achieve the same time complexity as previous algorithms but with a faster error convergence rate numerical results are given to verify the efficiency and the fast convergence of our algorithms

non monotonic iterative roots extended from characteristic intervals it has been treated as a difficult problem to find iterative roots of non monotonic functions for some pm functions which do not increase the number of forts under iteration a method was given to obtain a non monotonic iterative root by extending a monotone iterative root from the characteristic interval in this paper we prove that every continuous iterative root is an extension from the characteristic interval and give various modes of extension for those iterative roots of pm functions

the general exact bijective continuous solution of feigenbaum s functional equation solutionsf of feigenbaum s functional equationf f x 1 f x where 0 is a fixed real number account for many of the fascinating properties of the behaviour of successive iterates of one parameter families of nonlinear maps in connection with the phenomenon of intermittency interesting families of exact solutions have recently been found for 0 these solutions can all be derived from continuous bijective solutions which are topologically equivalent to translations in this paper thegeneral exact continuous bijective solution is found for any 0 positive or negative in particular it is shown that forany 0 there are solutions which areinequivalent to translations and it is shown that bijective solutions equivalent to translations exist only when 0 1 these results considerably enlarge the stock of available exact solutions of feigenbaum s equation

construction of continuous solutions and stability for the polynomial like iterative equation abstract most of known results such as existence uniqueness and stability for polynomial like iterative equations were given under the assumption that the coefficient of the first order iteration term does not vanish the existence with a non zero leading coefficient was therefore raised as an open problem it was positively answered for local c 1 solutions later in this paper this problem is answered further by constructing c 0 solutions moreover we discuss the stability of those c 0 solutions which consequently implies a result of the stability for iterative roots

pricing exotic derivatives exploiting structure in this paper we introduce a new fast and accurate numerical method for pricing exotic derivatives when discrete monitoring occurs and the underlying evolves according to a markov one dimensional stochastic processes the approach exploits the structure of the matrix arising from the numerical quadrature of the pricing backward formulas to devise a convenient factorization that helps greatly in the speed up of the recursion the algorithm is general and is examined in detail with reference to the cev constant elasticity of variance process for pricing different exotic derivatives such as asian barrier bermudan lookback and step options for which up to date no efficient procedures are available extensive numerical experiments confirm the theoretical results the matlab code used to perform the computation is available online at http www1 mate polimi it marazzina bp htm 2013 elsevier b v all rights reserved

an efficient convergent lattice algorithm for european asian options financial options whose payoff depends critically on historical prices are called path dependent options their prices are usually harder to calculate than options whose prices do not depend on past histories asian options are popular path dependent derivatives and it has been a long standing problem to price them efficiently and accurately no known exact pricing formulas are available to price them under the continuous time black scholes model although approximate pricing formulas exist they lack accuracy guarantees asian options can be priced numerically on the lattice a lattice divides the time to maturity into n equal length time steps the option price computed by the lattice converges to the option value under the black scholes model as n unfortunately only subexponential time algorithms are available if asian options are to be priced on the lattice without approximations efficient approximation algorithms are available for the lattice the fastest lattice algorithm published in the literature runs in o n 3 5 time whereas for the related pde method the fastest one runs in o n 3 time this paper presents a new lattice algorithm that runs in o n 2 5 time the best in the literature for such methods our algorithm exploits the method of lagrange multipliers to minimize the approximation error numerical results verify its accuracy and the excellent performance

continuity of iteration and approximation of iterative roots motivated by computing iterative roots for general continuous functions in this paper we prove the continuity of the iteration operators t n defined by t nf f n we apply the continuity and introduce the concept of continuity degree to answer positively the approximation question if lim m f m f can we find an iterative root f m of f m of order n for each email protected n such that the sequence f m tends to the iterative root of f of order n associated with a given initial function we not only give the construction of such an approximating sequence f m but also illustrate the approximation of iterative roots with an example some remarks are presented in order to compare our approximation with the hyers ulam stability

sharpness for c 1 linearization of planar hyperbolic diffeomorphisms abstract c 1 linearization preserves smooth dynamical behaviors and distinguishes qualitative properties in characteristic directions planar hyperbolic diffeomorphisms are the most elementary ones of representatively technical difficulties in the study of c 1 linearization in the poincare domain both eigenvalues inside the unit circle s 1 a lower bound 0 was given such that c 1 smoothness with 0 1 admits c 1 linearization our first purpose of this paper is to prove the sharpness of 0 and give a weaker linearization for 0 in the siegel domain one eigenvalue inside s 1 but the other outside s 1 it is known that c 1 smoothness admits c 1 linearization for all 0 1 the second purpose is to prove that the c 1 linearization is actually a c 1 linearization and give sharp estimates for

accurate and efficient lattice algorithms for american style asian options with range bounds abstract asian options are popular path dependent options and it has been a long standing problem to price them efficiently and accurately since there is no known exact pricing formula for asian options numerical pricing formulas like lattice models must be employed a lattice divides a certain time interval into n time steps and the pricing results generated by the lattice called desired option values for convenience converge to the theoretical option value as n since a brute force lattice pricing algorithm runs in subexponential time in n some heuristics like interpolation method are used to strike the balance between the efficiency and the accuracy but the pricing results might not converge due to the accumulation of interpolation errors for pricing european style asian options the evaluation on the major part of the lattice can be done by a simple formula and the interpolation method is only required on the minor part of the lattice thus polynomial time algorithms with convergence guarantee for european style asian options can be derived however such a simple formula does not exist for american style asian options this paper suggests an efficient range bound algorithm that bracket the desired option value by taking advantages of the early exercise property of american style options we show that part of the lattice can be evaluated by a simple formula the interpolation method is required on the remaining part of the lattice and the upper and the lower bounds option values produced by the proposed algorithm are essentially numerically identical thus the theoretical option value is said to be obtained practically when the range bound algorithm runs on a lattice with large number of time steps

iterative roots of piecewise monotonic functions with finite nonmonotonicity height abstract it is known that any continuous piecewise monotonic function with nonmonotonicity height not less than 2 has no continuous iterative roots of order n greater than the number of forts of the function in this paper we consider the problem of iterative roots in the case that the order n is less than or equal to the number of forts by investigating the trajectory of possible continuous roots we give a general method to find all iterative roots of those functions with finite nonmonotonicity height

logarithmic improvements in l p bounds for eigenfunctions at the critical exponent in the presence of nonpositive curvature we consider the problem of proving l p bounds for eigenfunctions of the laplacian in the high frequency limit in the presence of nonpositive curvature and more generally manifolds without conjugate points in particular we prove estimates at the critical exponent p c frac 2 d 1 d 1 where a spectrum of scenarios for phase space concentration must be ruled out our work establishes a gain of an inverse power of the logarithm of the frequency in the bounds relative to the classical l p bounds of the second author

a note on iterative roots of pm functions abstract in this paper we study iterative roots of pm functions a special class of non monotone functions problem 2 in w zhang pm functions their characteristic intervals and iterative roots ann polon math lxv 1997 119 128 is solved partly and theorem 4 in that paper is generalized

on the node fractional iterates let m be a bounded convex set of r n let f m r n be a continuous non bijective map we study solutions g of the functional equation g m x g g m 1 x f x m 2 this functional equation represents the problem of the fractional iteration for f

a connection between fractional iteration and graph theory we consider the problem of the fractional iteration for n dimensional maps we have two objectives in this paper the first objective is to establish a connection between fractional iteration and graph theory the second objective is to make a study concerning the problem of fractional iteration for n dimensional maps

non trivial l q solutions to the ginzburg landau equation it is shown that the contour problem for the stationary ginzburg landau equation where x r with r x is well posed in l 4 n for a class of small data f l 2 s n 1

iterative functional equation x x t f t with f t piecewise linear the equation x x t f t is discussed in many papers purely from the theoretical point of view existence properties we consider the properties of the solution and corresponding algorithms for computing the exact solution of such equations in the case of f t being a line or polygon

sharp resolvent estimates outside of the uniform boundedness range in this paper we are concerned with resolvent estimates for the laplacian delta in euclidean spaces uniform resolvent estimates for delta were shown by kenig et al duke math j 55 2 329 347 1987 who established rather a complete description of the lebesgue spaces allowing such estimates however the problem of obtaining sharp l p l q bounds depending on z has not been considered in a general framework which admits all possible p q in this paper we present a complete picture of sharp l p l q resolvent estimates which may depend on z we also obtain the sharp resolvent estimates for the fractional laplacians and a new result for the bochner riesz operators of negative index