large derivatives backward contraction and invariant densities for interval maps in this paper we study the dynamics of a smooth multimodal interval map f with non flat critical points and all periodic points hyperbolic repelling assuming that dfn f c as n holds for all critical points c we show that f satisfies the so called backward contracting property with an arbitrarily large constant and that f has an invariant probability which is absolutely continuous with respect to lebesgue measure and the density of belongs to lp for all p lmax lmax 1 where lmax denotes the maximal critical order of f in the appendix we prove that various growth conditions on the derivatives along the critical orbits imply stronger backward contraction

stability theorem for delay differential equations with impulses abstract in this paper we obtain sufficient conditions for the stability of the zero solution of the delay differential equation with impulses formula

iterative construction of fixed points of asymptotically nonexpansive mappings abstract let t be a completely continuous and asymptotically nonexpansive self mapping of a nonempty closed bounded and convex subset of a hilbert space then under certain conditions the sequence defined by xn 1 ntn xn 1 n x n converges strongly to some fixed point of t a similar result is obtained for asymptotically pseudocontractive mappings whose iterates are lipschitz continuous with respect to some common lipschitz constant

on delay differential equations with impulses abstract sufficient conditions are obtained respectively for the asymptotic stability of the trivial solution of x t ax t j 1 bjx tj t tj t tj and for the existence of a nonoscillatory solution conditions are also obtained for all solutions to be oscillatory the asymptotic behaviour of an impulsively perturbed delay logistic equation is investigated as an extension to a nonlinear equation

maps of intervals with indifferent fixed points thermodynamic formalism and phase transitions we develop the thermodynamic formalism for a large class of maps of the interval with indifferent fixed points for such systems the formalism yields onedimensional systems with many body infinite range interactions for which the thermodynamics is well defined but gibbs states are not piecewise linear systems of this kind yield the soluble in a sense fisher models we prove that such systems exhibit phase transitions the order of which depends on the behavior at the indifferent fixed points we obtain the critical exponent describing the singularity of the pressure and analyze the decay of correlations of the equilibrium states at all temperatures our technique relies on establishing and exploiting a relation between the transfer operators of the original map and its suitable expanding induced version the technique allows one also to obtain a version of the bowen ruelle formula for the hausdorff dimension of repellers for maps with indifferent fixed points and to generalize fisher results to some nonsoluble models

phase transitions for countable markov shifts we study the analyticity of the topological pressure for some one parameter families of potentials on countable markov shifts we relate the non analyticity of the pressure to changes in the recurrence properties of the system we give sufficient conditions for such changes to exist and not to exist we apply these results to the manneville pomeau map and use them to construct examples with different critical behavior

the fibonacci unimodal map the fibonacci recurrence of the critical orbit appeared in the work of branner and hubbard on complex cubic polynomials bh 12 and in yoccoz s work yl y2 on quadratic ones as the worst pattern of recurrence on the other hand a real quadratic fibonacci map was suggested by hofbauer and keller hk as a possible candidate for a map having a wild attractor that is a set a which is the w limit set for lebesgue almost every orbit but is strictly smaller than the w limit set for a generic orbit the w limit set of the critical orbit in hk possesses all known topological properties of wild attractors compare bl2 in fact we will see below that the quadratic fibonacci map does not have a wild attractor however the corresponding question for a map with a degenerate critical point remains open actually the first indication of the fibonacci map appeared in the numerical work of tsuda t related to the belousova zhabotinskii reaction and also in numerical work of shibayama sh more precisely they studied the sequence of fibonacci bifurcations creating the fibonacci map this paper will study topological geometrical and measure theoretical properties of the real fibonacci map our goal was to figure out if this type of recurrence really gives any pathological examples and to compare it with the infinitely renormalizable patterns of recurrence studied by sullivan s it turns out that the situation can be understood completely and is of quite regular nature in particular any fibonacci map with negative schwarzian and nondegenerate critical point has an absolutely continuous invariant measure so we deal with a regular type of chaotic dynamics it turns out also that geometrical properties of the closure of the critical orbit are quite different from those of the feigenbaum map its hausdorff dimension is equal to zero and its geometry is not rigid but depends on one parameter branner and hubbard introduce the concept of a tableau in order to describe recurrence of critical orbits their fibonacci tableau is a basic example which corresponds to one particularly close and regular pattern of recurrence if a complex quadratic map z 1 z2 c realizes this fibonacci tableau then the

nice inducing schemes and the thermodynamics of rational maps we study the thermodynamic formalism of a complex rational map f of degree at least two viewed as a dynamical system acting on the riemann sphere more precisely for a real parameter t we study the existence of equilibrium states of f for the potential t rm ln left f prime right and the analytic dependence on t of the corresponding pressure function we give a fairly complete description of the thermodynamic formalism for a large class of rational maps including well known classes of non uniformly hyperbolic rational maps such as topological collet eckmann maps and much beyond in fact our results apply to all non renormalizable polynomials without indifferent periodic points to infinitely renormalizable quadratic polynomials with a priori bounds and all quadratic polynomials with real coefficients as an application for these maps we describe the dimension spectrum for lyapunov exponents and for pointwise dimensions of the measure of maximal entropy and obtain some level 1 large deviations results for polynomials as above we conclude that the integral means spectrum of the basin of attraction of infinity is real analytic at each parameter in mathbb r with at most two exceptions

equilibrium states of weakly hyperbolic one dimensional maps for holder potentials there is a wealth of results in the literature on the thermodynamic formalism for potentials that are in some sense hyperbolic we show that for a sufficiently regular one dimensional map satisfying a weak hyperbolicity assumption every holder continuous potential is hyperbolic a sample consequence is the absence of phase transitions the pressure function is real analytic on the space of holder continuous functions another consequence is that every holder continuous potential has a unique equilibrium state and that this measure has exponential decay of correlations

strong convergence of asymptotically pseudocontractive semigroup by viscosity iteration in this paper we study the strong convergence of viscosity iteration and modified viscosity iteration process for strongly continuous semigroup of uniformly lipschitzian asymptotically pseudocontractive mappings

equivalence and topological invariance of conditions for non uniform hyperbolicity in the iteration of rational maps we show equivalence of several standard conditions for non uniform hyperbolicity of complex rational functions including the topological collet eckmann condition tce uniform hyperbolicity on periodic orbits exponential shrinking of components of pre images of small discs backward collet eckmann condition at one point positivity of the infimum of lyapunov exponents of finite invariant measures on the julia set the condition tce is stated in purely topological terms so we conclude that all these conditions are invariant under topological conjugacy for rational maps with one critical point in julia set all the conditions above are equivalent to the usual collet eckmann and backward collet eckmann conditions thus the latter ones are invariant by topological conjugacy in the unicritical setting we also prove that neither part of this stronger statement is valid in the multicritical case

large deviations for nonuniformly hyperbolic systems we obtain large deviation estimates for a large class of nonuniformly hyperbolic systems namely those modelled by young towers with summable decay of correlations in the case of exponential decay of correlations we obtain exponential large deviation estimates given by a rate function in the case of polynomial decay of correlations we obtain polynomial large deviation estimates and exhibit examples where these estimates are essentially optimal in contrast with many treatments of large deviations our methods do not rely on thermodynamic formalism hence for holder observables we are able to obtain exponential estimates in situations where the space of equilibrium measures is not known to be a singleton as well as polynomial estimates in situations where there is not a unique equilibrium measure

generalized bounded variation and applications to piecewise monotonic transformations we prove the quasi compactness of the perron frobenius operator of piecewise monotonic transformations when the inverse of the derivative is holder continuous or more generally of bounded p variation

existence and global attractivity of periodic solution for an impulsive delay differential equation with allee effect sufficient conditions are obtained for the existence and global attractivity of positive periodic solution of an impulsive delay differential equation with allee effect the results of this paper improve and generalize noticeably the known theorems in the literature

ergodic properties of invariant measures for piecewise monotonic transformations during the last decade a lot of research has been done on onedimensional dynamics different kinds of invariant measures for certain classes of piecewise monotonic transformations have been considered in most of these cases the perron frobenius operator plays an important role in this paper we try to unify these different examples which are discussed in detail below first we give a discription of the results proved in this paper

zeta functions and transfer operators for piecewise monotone transformations given a piecewise monotone transformationt of the interval and a piecewise continuous complex weight functiong of bounded variation we prove that the ruelle zeta function z of t g extends meromorphically to z 1 where lim g tn 1 g tg 1 n and thatz is a pole of if and only ifz 1 is an eigenvalue of the corresponding transfer operator l we do not assume that l leaves a reference measure invariant

an invariance principle for maps with polynomial decay of correlations we give a general method of deriving statistical limit theorems such as the central limit theorem and its functional version in the setting of ergodic measure preserving transformations this method is applicable in situations where the iterates of discrete time maps display a polynomial decay of correlations

strong convergence theorems for fixed points of asymptotically pseudocontractive semi groups abstract let k be a nonempty closed convex and bounded subset of a real banach space e let t t t t r be a strongly continuous uniformly asymptotically regular and uniformly l lipschitzian semi group of asymptotically pseudocontractive mappings from k into k then for a given u k there exists a sequence y n k satisfying the equation y n 1 n t t n n y n n u for each n n where n 0 1 and t n 0 satisfy appropriate conditions suppose further that e is uniformly convex and has uniformly g teaux differentiable norm under suitable conditions on the mappings t the sequence y n converges strongly to a fixed point of t furthermore an explicit sequence x n generated from x 1 k by x n 1 1 n x n n t t n n x n n n x n x 1 for all integers n 1 where n n are positive real sequences satisfying appropriate conditions converges strongly to a fixed point of t

the persistence of nonoscillatory solutions of delay differential equations under impulsive perturbations abstract we obtain a sufficient condition for the persistence of nonoscillatory solutions of the delay differential equation under impulsive perturbations

stability on nonlinear delay differential equations with impulses abstract in this paper the asymptotic stability for nonlinear delay differential equations with impulses 1 1 x t f t x t t 0 t t k x t k x t k i k x t k k n is studied and 3 2 stability theorems are presented the results here have improved the known results

almost sure invariance principle for nonuniformly hyperbolic systems we prove an almost sure invariance principle that is valid for general classes of nonuniformly expanding and nonuniformly hyperbolic dynamical systems discrete time systems and flows are covered by this result in particular the result applies to the planar periodic lorentz flow with finite horizon statistical limit laws such as the central limit theorem the law of the iterated logarithm and their functional versions are immediate consequences

oscillation criteria for impulsive delay differential equations abstract some new results on oscillation and nonoscillation for delay differential equations with impulses are established a counterexample is given which shows that a known oscillation criterion is incorrect

multifractal analysis of lyapunov exponent for continued fraction and manneville pomeau transformations and applications to diophantine approximation we extend some of the theory of multifractal analysis for conformal expanding systems to two new cases the non uniformly hyperbolic example of the manneville pomeau equation and the continued fraction transformation a common point in the analysis is the use of thermodynamic formalism for transformations with infinitely many branches

exponential stability of linear delay impulsive differential equations for an impulsive delay differential equation with bounded delay and bounded coefficients the following result is established if each solution is bounded on 0 together with its derivative for each bounded right hand side then the equation is exponentially stable an explicit stability theorem is presented

convergence rate of the fuzzy generalized nearest neighbor rule tizzy k nearest neighbor rule k nnr has been applied in a variety of substantive areas yang and chen l described a fuzzy generalized k nn algorithm which is a unified approach to a variety of fuzzy k nnr s they created the strong consistency of posterior risk of the fuzzy generalized nnr in this paper we give their convergence rate that is the convergence rate of posterior risk of the fuzzy generalized nnr is exponentially fast

approximation of fixed points of weakly contractive nonself maps in banach spaces abstract let k be a closed convex subset of a real uniformly smooth banach space e suppose k is a nonexpansive retract of e with p as the nonexpansive retraction let t k e be a d weakly contractive map such that a fixed point x int k of t exists it is proved that a descent like approximation sequence converges strongly to x furthermore if k is a nonempty closed convex subset of an arbitrary real banach space and t k k is a uniformly continuous d weakly contractive map with f t x k tx x it is proved that a descent like approximation sequence converges strongly to x f t

explicit conditions of exponential stability for a linear impulsive delay differential equation for a scalar delay differential equationx t k 1mak t x hk t 0 t 0 x t r hk t t with impulsive conditionsx j bjx j 0 limj j explicit exponential stability results are obtained

stability of nonlinear delay differential equations with impulses abstract by relaxing the restriction used previously we obtain the improved 3 2 stability theorems for the nonlinear delay differential equations with impulses x t f t x t 0 t t k 1 1 x t k x t k i k x t k k n

stability for impulsive delay differential equations abstract in this paper the stability for the scalar impulsive delay differential equation y t a t y t f t y 0 t 0 t k y k y k i k y k k 1 2 lim k k where delay arguments may be bounded or unbounded is investigated some new stability theorems are established which improve and extend several known results in the literature

inequalities in banach spaces with applications well known that many problems in a hilbert space h are solved by applying the following identity ilax 1 a yi12 a412 1 llyl12 a 1 mx yl12 1 1 for all x y in h and 0 5 i 5 1 therefore one natural method to solve problems in a banach space xis to establish equalities and usually inequalities in xanalogous to 1 1 this is in fact shown by some authors recent work see for example reich l poffald and reich 2 lim 3 lim et af 4 prus and smarzewski 5 smarzewski 6 and reich 21 the purpose of this paper is to continue carrying out investigations in this line more precisely in section 3 we establish some inequalities analogous to 1 l in either uniformly convex or uniformly smooth banach spaces as corollaries we obtain new characteristics of p uniform convexity and q uniform smoothness of a banach space x via the functionals i lip and ii llq respectively we give in section 4 applications of the inequalities obtained in section 3 to fixed points of uniformly lipschitzian mappings and to best approximation theory

operator renewal theory and mixing rates for dynamical systems with infinite measure we develop a theory of operator renewal sequences in the context of innite ergodic theory for large classes of dynamical systems preserving an innite measure we determine the asymptotic behaviour of iterates l n of the transfer operator this was previously an intractable problem examples of systems covered by our results include i parabolic rational maps of the complex plane and ii not necessarily markovian nonuniformly expanding interval maps with indierent xed points in addition we give a particularly simple proof of pointwise dual ergodicity asymptotic behaviour of p n 1 l j for the class of systems under consideration in certain situations including pomeau manneville intermittency maps we obtain higher order expansions for l n and rates of mixing also we obtain error estimates in the associated dynkin lamperti arcsine laws

local connectivity and quasi conformal rigidity of non renormalizable polynomials we prove that topologically conjugate non renormalizable polynomials are quasi conformally conjugate from this we derive that each such polynomial can be approximated by a hyperbolic polynomial as a by product we prove that the julia set of a non renormalizable polynomial with only hyperbolic periodic points is locally connected and the branner hubbard conjecture the main tools are the enhanced nest construction developed in a previous joint paper with rigidity for real polynomials ann of math 2 165 2007 749 841 and a lemma of kahn and lyubich for which we give an elementary proof in the real case

asymptotic behavior of solutions of impulsive delay differential equations abstract effective sufficient conditions for the asymptotic stability of the trivial solution of impulsive delay differential equation formula are obtained by investigating respectively the asymptotic behavior of the nonoscillatory solutions and oscillatory solutions of the equation