functional equations for circle homeomorphisms with golden ratio rotation number the investigation of a scaling limit for mappings of the circle to itself with golden ratio rotation number leads to a pair of functional equations with at least a formal resemblance to the functional equation using the accumulation of period doubling bifurcations we discuss the general theory of these functional equations assuming that solutions exist

non uniform specification and large deviations for weak gibbs measures we establish bounds for the measure of deviation sets associated to continuous observables with respect to not necessarily invariant weak gibbs measures under some mild assumptions we obtain upper and lower bounds for the measure of deviation sets of some non uniformly expanding maps including quadratic maps and robust multidimensional non uniformly expanding local diffeomorphisms for that purpose a measure theoretical weak form of specification is introduced and proved to hold for the robust classes of multidimensional non uniformly expanding local diffeomorphisms and viana maps

non lacunary gibbs measures for certain fractal repellers in this paper we study non uniformly expanding repellers constructed as the limit sets for a non uniformly expanding dynamical systems we prove that given a holder continuous potential satisfying a summability condition there exists non lacunary gibbs measure for with positive lyapunov exponents and infinitely many hyperbolic times almost everywhere moreover this non lacunary gibbs measure is an equilibrium measure for

ergodic theory for markov fibred systems and parabolic rational maps a paraboiic rational map of the riemann sphere admits a non atomic h conformal measure on its julia set where h the hausdorff dimension of the julia set and satisfies 1 2 h 2 with respect to this measure the rational map is conservative exact and there is an equivalent a finite invariant measure finiteness of the measure is characterised central limit theorems are proved in the case of a finite invariant measure and return sequences are identified in the case of an infinite one a theory of markov fibred systems is developed and parabolic rational maps are considered within this framework

upper bounds for ergodic sums of infinite measure preserving transformations for certain conservative infinite measure preserving transformations t we identify increasing functions a for which lim sup n 1 a n k 1 n f o t k x fd a e holds for any nonnegative integrable function f in particular the results apply to some markov shifts and number theoretic transformations and include the other law of the iterated logarithm

herman s theory revisited we prove that a c 2 smooth orientation preserving circle diffeomorphism with rotation number in diophantine class d 0 1 is c 1 smoothly conjugate to a rigid rotation we also derive the most precise version of denjoy s inequality for such diffeomorphisms

parabolic orbifolds and the dimension of the maximal measure for rational maps let f g g be a rational map of the riemann sphere d e g f 2 a natural invariant measure m the measure of maximal entropy was constructed by ljubich lju and independently by freire lopes and mafi6 flm the aim of this paper is to compare this measure with some hausdorff measures first recall the following definition for a probabili ty measure v on or more generally on a smooth manifold the hausdorff dimension of v is defined by a formula hd v inf hd y y v y l

dynamic analysis of space frames the method of reverberation ray matrix and the orthogonality of normal modes the formulation of reverberation ray matrix analysis has been proposed to study wave propagation in planar frames it is applied here to modal analysis of complex three dimensional framed structures optionally with lumped masses and or elastic supports furthermore by means of betti s reciprocity theorem orthogonal conditions are established for different natural modes and hence transient response analysis based on mode superposition is developed both the reverberation ray matrix analysis for free vibration and the mode superposition method for transient response are illustrated by numerical examples

rational rotation numbers for maps of the circle we consider families of maps of the circle of degree 1 which are homeomorphisms but not diffeomorphisms that is maps like x to x t frac c 2 pi sin 2 pi x bmod 1 withc 1 we prove that the set of parameter values corresponding to irrational rotation numbers has lebesgue measure 0 in other words the intervals on which frequency locking occurs fill up the set of full measure

invariant representation of propagation properties for bi coupled periodic structures general bi coupled periodic systems are dealt with by means of transfer matrices of single units the solutions of the associated characteristic equation are discussed in terms of invariant quantities by exploiting the well known reversibility of its coefficients an exhaustive description of the free wave propagation patterns is given on the invariant plane where propagation domains with qualitatively different character are identified the asymptotic behavior of the roots of the characteristic equation when the invariants tend to infinity is analyzed the contour plot of the real part of the propagation constants responsible for the amount of attenuation of the characteristic waves is illustrated on the invariants plane next several models of bi coupled periodic structures made up of beams resting on elastic supports are considered a non linear mapping from the invariants plane to the physical parameters plane provides a concise representation of the pattern of the propagation domains a mechanical interpretation associated with the boundaries of these regions is given finally the proper selection of the physical parameters governing the propagation modes is discussed

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

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

critical circle maps near bifurcation we estimate harmonic scalings in the parameter space of a one parameter family of critical circle maps these estimates lead to the conclusion that the hausdorff dimension of the complement of the frequency locking set is less than 1 but not less than 1 3 moreover the rotation number is a holder continuous function of the parameter

on the numerical computation of diophantine rotation numbers of analytic circle maps in this paper we present a numerical method to compute diophantine rotation numbers of circle maps with high accuracy we mainly focus on analytic circle diffeomorphisms but the method also works in the case of enough finite differentiability the keystone of the method is that under these conditions the map is conjugate to a rigid rotation of the circle moreover although it is not fully justified by our construction the method turns out to be quite efficient for computing rational rotation numbers we discuss the method through several numerical examples

large deviations for countable to one markov systems in this paper we study large deviation properties for countable to one markov systems associated to weak gibbs measures for non holder potentials furthermore we establish multifractal large deviation laws for countable to one piecewise conformal markov systems which are derived systems constructed over hyperbolic regions for certain nonhyperbolic systems exhibiting intermittency we apply our results to higher dimensional number theoretical transformations

thermodynamics of towers of hyperbolic type we introduce a class of continuous maps f of a compact topological space x admitting inducing schemes of hyperbolic type and describe the associated tower constructions we then establish a thermodynamic formalism i e we describe a class of real valued potential functions on x such that f possess a unique equilibrium measure associated to each which minimizes the free energy among the measures that are liftable to the tower we also describe some ergodic properties of equilibrium measures including decay of correlations and the central limit theorem we then study the liftability problem and show that under some additional assumptions on the inducing scheme every measure that charges the base of the tower and has sufficiently large entropy is liftable our results extend those obtained in ps05 ps08 for inducing schemes of expanding types and apply to certain multidimensional maps applications include obtaining the thermodynamic formalism for young s diffeomorphisms the henon family at the first birfucation and the katok map in particular for the katok map we obtain the exponential decay of correlations for equilibrium measures associated to the geometric potentials with 0 t 1

weak expansion properties and large deviation principles for expanding thurston maps abstract in this paper we prove that an expanding thurston map f s 2 s 2 is asymptotically h expansive if and only if it has no periodic critical points and that no expanding thurston map is h expansive as a consequence for each expanding thurston map without periodic critical points and each real valued continuous potential on s 2 there exists at least one equilibrium state for such maps we also establish large deviation principles for iterated preimages and periodic points it follows that iterated preimages and periodic points are equidistributed with respect to the unique equilibrium state for an expanding thurston map without periodic critical points and a potential that is holder continuous with respect to a visual metric on s 2

vibration reduction in piecewise bi coupled periodic structures the problem of minimizing transmitted vibrations through finitely long periodic structures is addressed bi coupled periodic element properties and arrangement are tailored to localize the response around the excitation source within any assigned frequency range bi dimensional analytical maps of the single unit free wave propagation domains stop pass and complex domains provide the optimal choice of the cell properties and ordering moreover the amount of vibration suppression along the periodic structure is also controlled as it can be described through iso attenuation curves representing the contour plot of the real part of the propagation constants applications to both undamped and damped beams resting on elastic supports are illustrated the response of the periodic structures to harmonic excitations is expressed through the wave vector method taking into account the effects of wave reflection due to changes in the cell properties along the structure and boundary conditions such computational schemes enables one to overcome numerical difficulties arising in the transfer matrix formulation for structures with a large number of periodic units

a new proof of m herman s theorem a new proof of the m herman theorem on the smooth conjugacy of a circle map is presented here it is based on the thermodynamic representation of dynamical systems and the study of the ergodic properties for the corresponding radom variables

universal properties of the transition from quasi periodicity to chaos in dissipative systems abstract an exact renormalization group transformation is developed for dissipative systems which describes how the transition to chaos may occur in a continuous and universal manner if the frequency ratio in the quasi periodic regime is held at a fixed irrational value our approach is a natural extension of k a m theory to strong coupling most of our analysis is for analytic circle maps we have found a strong coupling fixed point where invertibility is lost which describes the universal features of the transition to chaos we find numerically that any two such critical maps with the same winding number are c 1 conjugate it follows that the low frequency peaks in an experimental spectrum are universal and we determine how their envelope scales with frequency when the winding number has a periodic continued fraction our renormalization transform has a fixed point and spectra are self similar in addition for a set of non periodic winding numbers with full measure our renormalization transformation yields an ergodic trajectory in a sub space of all critical maps physically one finds singular and universal spectra that do not scale

quasiperiodicity in dissipative systems a renormalization group analysis abstract dynamical systems with quasiperiodic behavior i e two incommensurate frequencies may be studied via discrete maps which show smooth continuous invariant curves with irrational winding number in this paper these curves are followed using renormalization group techniques which are applied to a one dimensional system circle and also to an area contracting map of an annulus two fixed points are found representing different types of universal behavior a trivial fixed point for smooth motion and a nontrivial fixed point the latter representsthe incipient breakup of a quasiperiodic motion with frequency ratio the golden mean into a more chaotic flow fixed point functions are determined numerically and via an e expansion and eigenvalues are calculated

free vibrations of a mono coupled periodic system abstract vibration problems of periodic systems can be analyzed efficiently by means of the transfer matrix method the frequency equation for the whole system is shown to be obtained in terms of the eigenvalues or their natural logarithms which are often called propagation constants of the transfer matrix for a single periodic subsystem in case of a mono coupled system this frequency equation may be solved graphically by using the propagation constant curve thereby saving a great deal of computational effort two types of mono coupled systems are considered as numerical examples a spring mass oscillating system and a continuous timoshenko beam resting on regularly spaced knife edge supports depending on whether the transfer matrix is derived by an analytical procedure or by the finite element method the numerical solutions become either exact or approximate

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

computation of derivatives of the rotation number for parametric families of circle diffeomorphisms in this paper we present a numerical method to compute derivatives of the rotation number for parametric families of circle diffeomorphisms with high accuracy our methodology is an extension of a recently developed approach to compute rotation numbers based on suitable averages of iterates of the map and richardson extrapolation we focus on analytic circle diffeomorphisms but the method also works if the maps are differentiable enough in order to justify the method we also require the family of maps to be differentiable with respect to the parameters and the rotation number to be diophantine in particular the method turns out to be very efficient for computing taylor expansions of arnold tongues of families of circle maps finally we adapt these ideas to study invariant curves for parametric families of planar twist maps

differentiability at the tip of arnold tongues for diophantine rotations numerical studies and renormalization group explanations we study numerically the regularity of arnold tongues corresponding to diophantine rotation numbers of circle maps at the edge of validity of kam theorem this serves as a good test for the numerical stability of two different algorithms we find empirically that arnold tongues are only finitely differentiable at the tip we also find several scaling properties of the sobolev norms of the conjugacy near the breakdown we also provide a renormalization group explanation of the regularity at the tip and the scaling behaviors of the sobolev regularity we also uncover empirically some other patterns which require explanation

random f expansions we consider the asymptotic distribution properties of f expansion digits in particular if x 1 varphi 0 x 1 varphi 1 x cdots etc then frac 1 n sum n 1 k 0 varphi k rightarrow 3 text in measure

renormalization horseshoe for critical circle maps we prove that the invariant horseshoe for the renormalization of critical circle maps is globally uniformly hyperbolic with one dimensional unstable direction this extends our earlier result on hyperbolicity of periodic orbits of the renormalization and completes the proof of lanford s program

phase transitions for uniformly expanding maps given a uniformly expanding map of two intervals we describe a large class of potentials admitting unique equilibrium measures this class includes all holder continuous potentials but goes far beyond them we also construct a family of continuous but not holder continuous potentials for which we observe phase transitions this provides a version of the example in 9 for uniformly expanding maps

on the derivative of the hausdorff dimension of the julia sets for z2 c c r at parabolic parameters with two petals abstract let d c denote the hausdorff dimension of the julia set j c of the polynomial f c z z 2 c let c 0 r be such that f c 0 has a parabolic cycle with two petals we investigate in this paper how the bifurcation that occurs as c r crosses c 0 reflects on the variations of d c

subexponential decay of correlations we describe a method for proving subexponential lower bounds for correlations functions and apply it to study decay of correlations for maps with countable markov partitions one result is that ls young s upper estimates y2 are optimal in many situations our method is based on a general result concerning the asymptotics of renewal sequences of bounded operators acting on banach spaces which we apply to the iterates of the transfer operator

uniqueness of minimizer for countable markov shifts and equidistribution of periodic points for a finitely irreducible countable markov shift and a potential with summable variations we provide a condition on the associated pressure function which ensures that bowen s gibbs state the equilibrium state and the minimizer of the level 2 large deviations rate function are all unique and they coincide from this we deduce that the set of periodic points weighted with the potential equidistributes with respect to the gibbs equilibrium state as the periods tend to infinity applications are given to the gauss map and the bowen series map associated with a finitely generated free fuchsian group with parabolic elements

bloch theorem with revised boundary conditions applied to glide screw and rotational symmetric structures abstract wave propagation in complex periodic systems is often addressed with the bloch theorem and consists in applying periodic boundary conditions to a discretized unit cell while this method has been developed for structures periodic by translation in a recent work for quasi one dimensional wave propagation it has been shown that screw translation plus rotation and glide translation plus reflection periodicities can be accounted for as well keeping the cartesian coordinate system but revisiting the periodic boundary conditions the goal of the present paper is to generalize this concept to quasi two dimensional wave propagation two dimensional waves propagating in three dimensional structures dispersion relations for a set of reduced problems are then compared to results from the classical method when available by considering a smaller periodicity the computational cost is decreased and the number of folding curves and non interacting intersecting curves is reduced improving their interpretability while the size of a unit cell is divided by a factor two when glide symmetries are considered this ratio is significantly increased for screw or rotational symmetries moreover the proposed revisited bloch method is applicable to screw symmetric structures that do not possess purely translational symmetries and for which the classical method cannot be used e g chiral nanotubes longitudinally wrinkled helicoids

continuous phase transitions for dynamical systems we study the asymptotic expansion of the topological pressure of one parameter families of potentials at a point of non analyticity the singularity is related qualitatively and quantitatively to non gaussian limit laws and to slow decay of correlations with respect to the equilibrium measure

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

renewal sequences and intermittency in this paper we examine the generating function z of a renewal sequence arising from the distribution of return times in the turbulent region for a class of piecewise affine interval maps introduced by gaspard and wang and studied by several authors we prove that it admits a meromorphic continuation to the entire complex z plane with a branch cut along the ray 1 moreover we compute the asymptotic behavior of the coefficients of its taylor expansion at z 0 from this we obtain the exact polynomial asympotics for the rate of mixing when the invariant measure is finite and of the scaling rate when it is infinite

spectral gap and transience for ruelle operators on countable markov shifts we find a necessary and sufficient condition for the ruelle operator of a weakly holder continuous potential on a topologically mixing countable markov shift to act with spectral gap on some rich banach space we show that the set of potentials satisfying this condition is open and dense for a variety of topologies we then analyze the complement of this set in a finer topology and show that among the three known obstructions to spectral gap weak positive recurrence null recurrence transience transience is open and dense and null recurrence and weak positive recurrence have empty interior

equilibriums of some non holder potentials we consider one sided subshifts with some potential functions which satisfy the holder condition everywhere except at a fixed point and its preimages we prove that the systems have conformal measures and invariant measures absolutely continuous with respect to where may be finite or infinite we show that the systems are exact and are weak gibbs measures and equilibriums for we also discuss uniqueness of equilibriums and phase transition these results can be applied to some expanding dynamical systems with an indifferent fixed point

exact solutions for dynamic response of a periodic spring and mass structure this paper derives exact analytical solutions for the dynamic response of a periodic structure for which the unit cell consists of one mass and one spring the solutions cover arbitrary initial conditions and both polynomial and harmonic external forces they involve bessel lommel weber and hypergeometric functions

real wave vectors for dynamic analysis of periodic structures a modified version of the traditional wave vector computational scheme for the dynamic analysis of long undamped periodic structures is presented first the consistency of the complex wave vector mathematical formulation is discussed placing particular emphasis on the real or complex nature of the resulting characteristic equation from which the natural frequencies are derived it is shown that the rearrangement in terms of complex waves entering the domain devised to overcome ill conditioning arising in the transfer matrix formulation entails as a side effect an ill posed problem as it leads to a complex characteristic equation in a real unknown next the proposed approach is described it provides for transformation of frequency dependent real transfer matrices for state vectors to real transfer matrices for wave vectors thus avoiding an ill conditioned and ill posed problem finally applications to mono coupled and bi coupled structures are illustrated aiming at comparing the proposed method with the transfer matrix and complex wave vector approaches

theory of circle maps and the problem of one dimensional optical resonator with a periodically moving wall we consider the electromagnetic field in a cavity with a periodically oscillating perfectly reflecting boundary and show that the mathematical theory of circle maps leads to several physical predictions notably well known results in the theory of circle maps which we review briefly imply that there are intervals of parameters where the waves in the cavity get concentrated in wave packets whose energy grows exponentially even if these intervals are dense for typical motions of the reflecting boundary in the complement there is a positive measure set of parameters where the energy remains bounded

the parameterization method for invariant manifolds iii overview and applications abstract we describe a method to establish existence and regularity of invariant manifolds and at the same time to find simple maps which are conjugated to the dynamics on them the method establishes several invariant manifold theorems for instance it reduces the proof of the usual stable manifold theorem near hyperbolic points to an application of the implicit function theorem in banach spaces we also present several other applications of the method

dynamic response analysis of truss type structural networks a wave propagation approach abstract a typical space structure is composed of piece wise periodic arrays of connected truss units these arrays can intersect each other and form multiple closed loops in this paper a detailed step by step numerical procedure is provided for computing the response of such a structure to external excitations the basic idea is to treat an array of connected truss units as a multi channel waveguide capable of transmitting several types of wave motions each waveguide or each junction where several waveguides intersect can be characterized by a wave scattering matrix which is composed of the transmission and reflection submatrices a reflection submatrix reverses the direction of incoming waves whereas a transmission submatrix permits the incoming waves to enter or to be directed to other waveguides by use of wave scattering matrices in the formulation the numerical computation always follows the directions of propagation of various waves thus numerical stability is assured since the amplitude of wave motion decays as it propagates along a path due to structural damping the numerical procedure described herein represents an extension of the author s previous works by 1 describing for the first time the dynamics of individual truss arrays in terms of scattering matrices 2 adopting a new bookkeeping system to accommodate more complicated configurations of structural networks and 3 developing an efficient procedure to eliminate step by step superfluous unknowns thus simplifying the calculation combination of the three advances makes it possible to treat for the first time such complicated configurations as multiple closed loops since the actual truss type construction is retained when obtaining the wave scattering matrices the procedure is considered more accurate than those based on substitution for truss construction of timoshenko beam elements application of the procedure is illustrated by an example

group extensions of gibbs markov maps let be an aperiodic cocycles with values in a locally compact abelian second countable group defined on an exact gibbs markov map t x x we show that the group extension t x g t x g x x x g is exact equivalent conditions for exactness are found

on the existence of conformal measures a general notion of conformal measure is introduced and some basic properties are studied sufficient conditions for the existence of these measures are obtained using a general construction principle the geometric properties of conformal measures relate equilibrium states and hausdorff measures this is shown for invariant subsets of s under expanding maps

propagation of decaying waves in periodic and piecewise periodic structures of finite length abstract a new method is described for the analysis of long and complicated structures which are composed of spatially periodic units or sections of spatially periodic units the response of such a structure to external excitations is treated as a superposition of wave motions with account taken of the effects of wave reflection due to change in the construction pattern along the structure and boundary conditions since damping usually exists in a real structure a wave motion always decays as it propagates along the structure however this can cause numerical difficulties if the structure is rather long because a decaying wave appears as a growing wave in the opposite direction and since propagations in both directions must be accounted for in the analysis computational errors associated with the apparent growing waves also grow and the numerical results become meaningless in the proposed new scheme use is made of wave reflection and transmission matrices making it possible for all computations to proceed in the direction of wave propagation so that the results are always stable numerically furthermore the required computing time is greatly reduced in comparison with that of an all encompassing finite element analysis for the entire structure application of the new method is illustrated by examples

invariant manifolds associated to nonresonant spectral subspaces we show that if the linearization of a map at a fixed point leaves invariant a spectral subspace which satisfies certain nonresonance conditions the map leaves invariant a smooth manifold tangent to this subspace this manifold is as smooth as the map when the smoothness is measured in appropriate scales but is unique amongcl invariant manifolds wherel depends only on the spectrum of the linearization or on some more general smoothness classes that we detail we show that if the nonresonance conditions are not satisfied a smooth invariant manifold need not exist and we also establish smooth dependence on parameters we also discuss some applications of these invariant manifolds and briefly survey related work

aging generates regular motions in weakly chaotic systems using intermittent maps with infinite invariant measures we investigate the universality of time averaged observables under aging conditions according to aaronson s darling kac theorem in non aged dynamical systems with infinite invariant measures the distribution of the normalized time averages of integrable functions converges to the mittag leffler distribution this well known theorem holds when the start of observations coincides with the start of the dynamical processes introducing a concept of the aging limit where the aging time t a and the total measurement time t goes to infinity while the aging ratio t a t is kept constant we obtain a novel distributional limit theorem of time averaged observables integrable with respect to the infinite invariant density applying the theorem to the lyapunov exponent in intermittent maps we find that regular motions and a weakly chaotic behavior coexist in the aging limit this mixed type of dynamics is controlled by the aging ratio and hence is very different from the usual scenario of regular and chaotic motions in hamiltonian systems the probability of finding regular motions in non aged processes is zero while in the aging regime it is finite and it increases when system ages

thermodynamic formalism for countable to one markov systems for countable to one transitive markov systems we establish thermodynamic formalism for non holder potentials in nonhyperbolic situations we present a new method for the construction of conformal measures that satisfy the weak gibbs property for potentials of weak bounded variation and show the existence of equilibrium states equivalent to the weak gibbs measures we see that certain periodic orbits cause a phase transition non gibbsianness and force the decay of correlations to be slow we apply our results to higher dimensional maps with indifferent periodic points

weak gibbs measures for intermittent systems and weakly gibbsian states in statistical mechanics in this paper we associate weak gibbs measures for intermittent maps with non gibbsian weakly gibbsian states in statistical mechanics in the sense of dobrushin 4 5 we show a higher dimensional intermittent map of which the sinai bowen ruelle measure is a weak gibbs equilibrium state and a weakly gibbsian state in the sense of dobrushin admitting essential discontinuities in its conditional probabilities

complex analytic dynamics on the riemann sphere dichotomie dynamique de fatou et julia points periodiques consequences du theoreme de montel l ensemble de julia est la fermeture de l ensemble des points periodiques repulseurs resultats classiques sur l ensemble de fatou classification de sullivan de l ensemble de fatou une condition pour le developpement sur l ensemble de julia la dynamique des polynomes l ensemble de mandelbrot et le travail de douady et hubbard le theoreme d application de riemann mesurable et la dynamique analytique

multifractal analysis of weak gibbs measures for intermittent systems in this paper we establish a multifractal formalism of weak gibbs measures associated to potentials of weak bounded variation for certain nonhyperbolic systems we apply our results to manneville pomeau type maps and a piecewise conformal two dimensional countable markov map with indifferent periodic points which is related to a complex continued fraction