stability of peakons the peakons are peaked solitary wave solutions of a certain nonlinear dispersive equation that is a model in shallow water theory and the theory of hyperelastic rods we give a very simple proof of the orbital stability of the peakons in the h1 norm 2000 john wiley sons inc

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

algebraic multigrid amg experiences and comparisons algebraic multigrid methods are designed for the solution of sparse linear systems of equations using multigrid principles in contrast to standard multigrid methods amg does not take advantage of the origin of a particular system of equations at hand nor does it exploit any underlying geometrical situation fully automatically and based solely on algebraic information contained in the given matrix amg constructs a sequence of grids and corresponding operators a special amg algorithm will be presented for a wide range of problems including certain problems which do not have a continuous background this algorithm yields an iterative method which exhibits a convergence behavior typical for multigrid methods

a finite difference method on layer adapted meshes for an elliptic reaction diffusion system in two dimensions an elliptic system of m 2 singularly perturbed linear reaction diffusion equations coupled through their zero order terms is considered on the unit square this system does not in general satisfy a maximum principle it is solved numerically using a standard difference scheme on tensor product bakhvalov and shishkin meshes an error analysis for these numerical methods shows that one obtains nodal o n 2 convergence on the bakhvalov mesh and o n 2 ln 2 n convergence on the shishkin mesh where n mesh intervals are used in each coordinate direction and the convergence is uniform in the singular perturbation parameter the analysis is much simpler than previous analyses of similar problems even in the case of a single reaction diffusion equation as it does not require the construction of an elaborate decomposition of the solution numerical results are presented to confirm our theoretical error estimates

a note on well posedness for camassa holm equation in this note we investigate the problem of well posedness for a shallow water equation with data having critical regularity our results are based on the use of besov spaces b2 rs which generalize the sobolev spaces hs with critical index s 3 2

the cauchy problem for a generalized two component short pulse system with high order nonlinearities abstract this paper is concerned with the cauchy problem of a generalized short pulse system in the periodic setting local well posedness of this system is established in the besov space with s max 3 2 1 1 p it is then shown that the data to solution map is not uniformly continuous from b l r s b l r s to c 0 t b l r s c 0 t b l r s finally we provide a blow up result of the solution

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

well posedness of the fornberg whitham equation on the circle abstract in this paper we show that the fornberg whitham equation is well posed in sobolev spaces h s for s 3 2 and in the periodic case we then show that the well posedness is sharp in the sense that the continuity of the data to solution map is not better than continuous by using the method of approximate solutions however we also show that the solution map is holder continuous in a weaker topology these results are based on the well posedness result as well as the solution size and lifespan estimates

wave breaking analysis for the fornberg whitham equation abstract in this paper the wave breaking of the fornberg whitham equation is studied we first investigate the local well posedness and show a blow up scenario by local in time a priori estimates next we present a sufficient condition on the initial data to ensure wave breaking moreover the estimates of life span and blow up rate are obtained

renormalization of cocycles and linear ode with almost periodic coefficients in the current paper we address the problem of classification of cocycles over an irrational rotation we use the renormalization group approach

a note on iterative methods for solving singularly perturbed problems using non monotone methods on shishkin meshes abstract non monotone methods with shishkin meshes are employed in obtaining finite difference schemes for solving a linear two dimensional steady state convection diffusion problem preconditioners are used that significantly reduce the number of iterations of the linear solver computational results for a galerkin method are presented which indicate parameter robust super linear orders of convergence

a normal form theorem for brjuno skew systems through renormalization abstract we develop a dynamical renormalization method for the problem of local reducibility of analytic linear quasi periodic skew product flows on t 2 sl 2 r this approach is based on the continued fraction expansion of the linear base flow and deals with small divisors by turning them into large divisors we use this method to give a new proof of a normal form theorem for a brjuno base flow

robust solution of singularly perturbed problems using multigrid methods we consider the problem of solving linear systems of equations that arise in the numerical solution of singularly perturbed ordinary and partial differential equations of reaction diffusion type standard discretization techniques are not suitable for such problems and so specially tailored methods are required usually involving adapted or fitted meshes that resolve important features such as boundary and or interior layers in this study we consider classical finite difference schemes on the layer adapted meshes of shishkin and bakhvalov we show that standard direct solvers exhibit poor scaling behavior with respect to the perturbation parameter when solving the resulting linear systems we propose and prove optimality of a new block structured preconditioning approach that is robust for small values of the perturbation parameter and compares favorably with standard robust multigrid preconditioners for these linear systems we also derive stopping criteria which ensure that the potential accuracy

discontinuous traveling waves as weak solutions to the fornberg whitham equation abstract we analyze the weak solution concept for the fornberg whitham equation in case of traveling waves with a piecewise smooth profile function the existence of discontinuous weak traveling wave solutions is shown by means of analysis of a corresponding planar dynamical system and appropriate patching of disconnected orbits

renormalizations and rigidity theory for circle homeomorphisms with singularities of the break type circle homeomorphisms with singularities of the break type are considered in the case when rotation numbers have periodic continued fraction expansion we establish hyperbolicity for renormalizations and then use it in order to prove the following rigidity result namely we show that any two homeomorphisms with a single break point are smoothly conjugate to each other provided they have the same quadratic irrational rotation number and the same size of a break

black box multigrid one major problem with the multigrid method has been that each new grid configuration has required a major programming effort to develop a code that specifically handles that grid configuration such a penalty is not required for methods like sor iccg etc in these methods one need only specify the matrix problem no matter what the grid configuration in this paper we investigate such a situation for the multigrid method the end result is code boxmg in which one need only specify the logically rectangular positive definite matrix problem boxmg does everything else necessary to set up the auxilliary coarser problems to achieve a multigrid solution

the inverse spectral transform for the conservative camassa holm flow with decaying initial data we establish the inverse spectral transform for the conservative camassa holm flow with decaying initial data in particular it is employed to prove the existence of weak solutions for the corresponding cauchy problem

convergence estimates for multigrid algorithms without regularity assumptions a new technique for proving rate of convergence estimates of multi grid algorithms for symmetric positive definite problems will be given in this paper the standard multigrid theory requires a regularity and approxima tion assumption in contrast the new theory requires only an easily verified approximation assumption this leads to convergence results for multigrid re finement applications problems with irregular coefficients and problems whose coefficients have large jumps in addition the new theory shows why it suffices to smooth only in the regions where new nodes are being added in multigrid refinement applications

rigidity for piecewise smooth homeomorphisms on the circle abstract we find conditions for two piecewise c 2 homeomorphisms f and g of the circle to be c 1 conjugate besides the restrictions on the combinatorics of the maps we assume that the maps have bounded combinatorics and necessary conditions on the one side derivatives of points where f and g are not differentiable we also assume zero mean nonlinearity for f and g

iterative parameter choice and multigrid methods for anisotropic diffusion denoising anisotropic diffusion methods are well known for giving good qualitative results for image denoising this paper gives a review of the anisotropic diffusion methodology and its application to image denoising we propose a fixed point iteration using a multigrid solver to solve a regularized anisotropic diffusion equation which is not only well posed but also has a nontrivial steady state solution a new regularization parameter choice method brent ncp combining brent s method and the normalized cumulative periodogram information of the misfit is also introduced we test our algorithm on several common test images with different noise levels the experimental results demonstrate the effectiveness of the anisotropic diffusion with a multigrid approach and the broad applicability of the brent ncp parameter choice algorithm

on c2 diffeomorphisms of the circle which are of type iii1 for every irrational number 0 1 which is not of constant type we construct ac 2 diffeomorphism of the circle with rotation number which is of type iii1 this diffeomorphism can be chosen arbitrarily close to the rotationr our methods also allow us to construct for every liouville number 0 1 ac diffeomorphism of the circle with rotation number which is of type iii1

on the existence of feigenbaum s fixed point we give a proof of the existence of ac2 even solution of feigenbaum s functional equation g text x lambda 0 1 g text g lambda 0 x g text 0 1 whereg is a map of 1 1 into itself it extends to a real analytic function over

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

stability of peakons for the degasperis procesi equation the degasperis procesi equation can be derived as a member of a oneparameter family of asymptotic shallow water approximations to the euler equations with the same asymptotic accuracy as that of the camassaholm equation in this paper we study the orbital stability problem of the peaked solitons to the degasperis procesi equation on the line by constructing a liapunov function we prove that the shapes of these peakon solitons are stable under small perturbations

on the distribution of long term time averages on symbolic space the pressure was studied in a rather abstract theory as an important notion of the thermodynamic formalism the present paper gives a more concrete account in the case of symbolic spaces including subshifts of finite type we relate the pressure of an interaction function to its long term time averages through the hausdorff and packing dimensions of the subsets on which has prescribed long term time average values functions with values in d are considered for those depending only on finitely many symbols we get complete results unifying and completing many partial results

new proofs of the existence of the feigenbaum functions a new proof of the existence of analytic unimodal solutions of the cvitanovic feigenbaum functional equation g x g g x g x 1 const x r at 0 valid for all in 0 1 is given and the existence of the eckmann wittwer functions 8 is recovered the method also provides the existence of solutions for certain given values ofr and in particular forr 2 a proof requiring no computer

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

on the existence of fixed points of the composition operator for circle maps in the theory of circle maps with golden ratio rotation number formulated by feigenbaum kadanoff and shenker fks and by ostlund rand sethna and siggia orss a central role is played by fixed points of a certain composition operator in map space we define a common setting for the problem of proving the existence of these fixed points and of those occurring in the theory of maps of the interval we give a proof of the existence of the fixed points for a wide range of the parameters on which they depend

some numerical experiments with multigrid methods on shishkin meshes piecewise uniform meshes introduced by shishkin are a very useful tool to construct robust and efficient numerical methods to approximate the solution of singularly perturbed problems for small values of the diffusion coefficient the step size ratios in this kind of grids can be very large in this case standard multigrid methods are not convergent to avoid this troublesome in this paper we propose a modified multigrid algorithm which works fine on shishkin meshes we show some numerical experiments confirming that the proposed multigrid method is convergent and it has similar properties that standard multigrid for classical elliptic problems

smooth linearization of commuting circle diffeomorphisms we show that a finite number of commuting diffeomorphisms with simultane ously diophantine rotation numbers are smoothly conjugated to rotations this solves a problem raised by moser

analyticity of the cauchy problem for an integrable evolution equation in this paper we consider the periodic cauchy problem for the camassa holm equation with analytic initial data and prove that its solutions are analytic in both variables globally in space and locally in time

a parameter robust numerical method for a two dimensional reaction diffusion problem in this paper a singularly perturbed reaction diffusion partial differential equation in two space dimensions is examined by means of an appropriate decomposition we describe the asymptotic behaviour of the solution of problems of this kind a central finite difference scheme is constructed for this problem which involves an appropriate shishkin mesh we prove that the numerical approximations are almost second order uniformly convergent in the maximum norm with respect to the singular perturbation parameter some numerical experiments are given that illustrate in practice the theoretical order of convergence established for the numerical method

well posedness and continuity properties of the fornberg whitham equation in besov spaces abstract in this paper we prove well posedness of the fornberg whitham equation in besov spaces b 2 r s in both the periodic and non periodic cases this will imply the existence and uniqueness of solutions in the aforementioned spaces along with the continuity of the data to solution map provided that the initial data belongs to b 2 r s we also establish sharpness of continuity on the data to solution map by showing that it is not uniformly continuous from any bounded subset of b 2 r s to c t t b 2 r s furthermore we prove a cauchy kowalevski type theorem for this equation that establishes the existence and uniqueness of real analytic solutions and also provide blow up criterion for solutions

numerical determination of the continued fraction expansion of the rotation number abstract the rotation number is an important invariant of homeomorphisms on the circle in this paper an algorithm is described which computes the continued fraction expansion of this number it therefore detects rational rotation numbers and in general determines a sequence of intervals of decreasing length containing the rotation number it will be shown that for almost all rotation numbers the convergence is of third order also error analyses of some other algorithms and numerical examples are given including a computation of resonance tongues in the arnold family

on the indefinite helmholtz equation complex stretched absorbing boundary layers iterative analysis and preconditioning this paper studies and analyzes a preconditioned krylov solver for helmholtz problems that are formulated with absorbing boundary layers based on complex coordinate stretching the preconditioner problem is a helmholtz problem where not only the coordinates in the absorbing layer have an imaginary part but also the coordinates in the interior region this results into a preconditioner problem that is invertible with a multigrid cycle we give a numerical analysis based on the eigenvalues and evaluate the performance with several numerical experiments the method is an alternative to the complex shifted laplacian and it gives a comparable performance for the studied model problems

an efficient finite difference scheme for electromagnetic logging in 3d anisotropic inhomogeneous media we consider a problem of computing the electromagnetic field in 3d anisotropic media for electromagnetic logging the proposed finite difference scheme for maxwell equations has the following new features based on some recent and not so recent developments in numerical analysis coercivity i e the complete discrete analogy of all continuous equations in every grid cell even for nondiagonal conductivity tensors a special conductivity averaging that does not require the grid to be small compared to layering or fractures and a spectrally optimal grid refinement minimizing the error at the receiver locations and optimizing the approximation of the boundary conditions at infinity all of these features significantly reduce the grid size and accelerate the computation of electromagnetic logs in 3d geometries without sacrificing accuracy

a robust adaptive method for a quasi linear one dimensional convection diffusion problem a quasi linear conservative convection diffusion two point boundary value problem is considered to solve it numerically an upwind finite difference scheme is applied the mesh used has a fixed number n 1 of nodes and is initially uniform but its nodes are moved adaptively using a simple algorithm of de boor based on equidistribution of the arc length of the current computed piecewise linear solution it is proved for the first time that a mesh exists that equidistributes the arc length along the polygonal solution curve and that the corresponding computed solution is first order accurate uniformly in varepsilon where varepsilon is the diffusion coefficient in the case when the boundary value problem is linear if n is sufficiently large independently of varepsilon it is shown that after o rm ln 1 varepsilon rm ln n iterations of the algorithm the piecewise linear interpolant of the computed solution achieves first order accuracy in the l infty 0 1 norm uniformly in varepsilon numerical experiments are presented that support our theoretical results

uniform partitions and a dimensions spectrum for lacunar measures a multifractal analysis has been proposed to investigate complex systems which exhibit a nonuniform local structure multifractal measures are normalized mass distributions inhomogeneous at every scale to every point p is associated the exponent of the power law r describing the asymptotic behaviour as r 0 of the mass of a sphere of radius r and centre p the measure properties are specified by the dimension spectrum f of the set of points p with the same scaling exponent this spectrum is nontrivial even when the mass distribution has no holes measure with euclidean support and the box counting dimension is an integer in this case it can be evaluated by considering a sequence of uniform grids whose spacing decreases as 2 n known as uniform partitions we consider here a family of iterated function system fractal measures with support on the unit square with random defects specified by three parameters a scale a probability p and an attenuation factor given a partition with a diameter less than any cell is chosen with probability p and its mass is multiplied by the corresponding pre measure is obtained after normalization in all the examples examined the f spectrum appears to widen monotonically as decreases this result suggests the possibility of using the method to detect the presence of defects in sponge like structures such as cancellous bone sponge like by analysing their radiographs since the grey intensity is related to the mass density of the structure a widening of the f spectrum is indeed observed in osteoporotic bone radiographs compared to normal bone radiographs this may be attributed to the rarefaction of the cancellous bone

the dimension spectrum of some dynamical systems we analyze the dimension spectrum previously introduced and measured experimentally by jensen kadanoff and libchaber using large deviation theory we prove for some invariant measures of expanding markov maps that the hausdorff dimensionf of the set on which the measure has a singularity is a well defined concave and regular function in particular we show that this is the case for the accumulation of period doubling and critical mappings of the circle with golden rotation number we also show in these particular cases that the functionf is universal

a perfectly matched layer for the helmholtz equation in a semi infinite strip the perfectly matched layer pml has become a widespread technique for preventing reflections from far field boundaries for wave propagation problems in both the time dependent and frequency domains we develop a discretization to solve the helmholtz equation in an infinite two dimensional strip we solve the interior equation using high order finite differences schemes the combined helmholtz pml problem is then analyzed for the parameters that give the best performance we show that the use of local high order methods in the physical domain coupled with a specific second order approximation in the pml yields global high order accuracy in the physical domain we discuss the impact of the parameters on the effectiveness of the pml numerical results are presented to support the analysis

generalizations of the camassa holm equation we classify generalized camassa holm type equations which possess infinite hierarchies of higher symmetries we show that the obtained equations can be treated as negative flows of integrable quasi linear scalar evolution equations of orders 2 3 and 5 we present the corresponding lax representations or linearization transformations for these equations some of the obtained equations seem to be new

the trajectories of particles in stokes waves analyzing a free boundary problem for harmonic functions we show that there are no closed particle paths in an irrotational inviscid traveling wave propagating at the surface of water over a flat bed within a period each particle experiences a backward forward motion with a slight forward drift

statistical description of chaotic attractors the dimension function a method for the investigation of fractal attractors is developed based on statistical properties of the distributionp n of nearest neighbor distances between points on the attractor a continuous infinity of dimensions called dimension function is defined through the moments ofp n in particular for the case of self similar sets we prove that the dimension function df yields in suitable points capacity information dimension and all other renyi dimensions an algorithm to compute df is derived and applied to several attractors as a consequence an estimate of nonuniformity in dynamical systems can be performed either by direct calculation of the uniformity factor or by comparison among various dimensions finally an analytical study of the distributionp n is carried out in some simple meaningful examples

multigrid for high dimensional elliptic partial differential equations on non equidistant grids this work presents techniques theory and numbers for multigrid in a general d dimensional setting the main focus of this paper is the multigrid convergence for high dimensional partial differential equations on non equidistant grids such as may be encountered in a sparse grid solution as a model problem we have chosen the anisotropic stationary diffusion equation on a rectangular hypercube we present some techniques for building the general d dimensional adaptations of the multigrid components and propose grid coarsening strategies to handle anisotropies that are induced due to discretization on a non equidistant grid apart from the practical formulas and techniques we present in detail the smoothing analysis of the point omega red black jacobi method for a general multidimensional case we show how relaxation parameters may be evaluated efficiently and used for better convergence this analysis incorporates full and partial doubling and quadrupling coarsening strategies as well as the second and the fourth order finite difference operators finally we present some results derived from numerical experiments based on the test problem

some results on the behavior and estimation of the fractal dimensions of distributions on attractors the strong interest in recent years in analyzing chaotic dynamical systems according to their asymptotic behavior has led to various definitions of fractal dimension and corresponding methods of statistical estimation in this paper we first provide a rigorous mathematical framework for the study of dimension focusing on pointwise dimension x and the generalized renyi dimensionsd q and give a rigorous proof of inequalities first derived by grassberger and procaccia and hentschel and procaccia we then specialize to the problem of statistical estimation of the correlation dimension and information dimension it has been recognized for some time that the error estimates accompanying the usual procedures which generally involve least squares methods and nearest neighbor calculations grossly underestimate the true statistical error involved in least squares analyses of and we identify sources of error not previously discussed in the literature and address the problem of obtaining accurate error estimates we then develop an estimation procedure for which corrects for an important bias term the local measure density and provides confidence intervals for the general applicability of this method is illustrated with various numerical examples

symplectic structures their backlund transformations and hereditary symmetries abstract it is shown that compatible symplectic structures lead in a natural way to hereditary symmetries we recall that a hereditary symmetry is an operator valued function which immediately yields a hierarchy of evolution equations each having infinitely many commuting symmetries all generated by this hereditary symmetry furthermore this hereditary symmetry usually describes completely the soliton structure and the conservation laws of these equations this result then provide us with a method for constructing hereditary symmetries and hence exactly solvable evolution equations in addition we show how symplectic structures transform under backlund transformations this leads to a method for generating a whole class of symplectic structures from a given one several examples and applications are given illustrating the above results also the connection of our results with those of gelfand and dikii and of magri is briefly pointed out

fast and robust solvers for pressure correction in bubbly flow problems we consider the numerical simulation of two phase fluid flow where bubbles or droplets of one phase move against a background of the other phase such flows are governed by the navier stokes equations the solution of which may be approximated using a pressure correction approach within such an approach the computational cost is often dominated by the solution of a linear system corresponding to a discrete poisson equation with discontinuous coefficients in this paper we explore the efficient solution of these linear systems using robust multilevel solvers such as deflated variants of the preconditioned conjugate gradient method or robust multigrid techniques we consider these families of methods in more detail and compare their performance in the simulation of bubbly flows some of these methods turn out to be very effective and reduce the amount of work to solve the pressure correction system substantially resulting in efficient calculations for two phase flows on highly resolved grids

a robust hierarchical basis preconditioner on general meshes in this paper we introduce a multi level direct sum space decomposition of general possibly locally refined linear or multi linear finite element spaces the resulting additive schwarz preconditioner is optimal for symmetric second order elliptic problems moreover it turns out to be robust with respect to coefficient jumps over edges in the coarsest mesh perturbations with positive zeroth order terms and after a further decomposition of the spaces also with respect to anisotropy along the grid lines important for an efficient implementation is that stable bases of the subspaces defining our decomposition consisting of functions having small supports can be easily constructed

fractal properties of critical invariant curves we examine the dimension of the invariant measure for some singular circle homeomorphisms for a variety of rotation numbers through both the thermodynamic formalism and numerical computation the maps we consider include those induced by the action of the standard map on an invariant curve at the critical parameter value beyond which the curve is destroyed our results indicate that the dimension is universal for a given type of singularity and rotation number and that among all rotation numbers the golden mean produces the largest dimension

on locking and robustness in the finite element method a numerical scheme for the approximation of a parameter dependent problem is said to exhibit locking if the accuracy of the approximations deteriorates as the parameter tends to a limiting value a robust numerical scheme for the problem is one that is essentially uniformly convergent for all values of the parameter precise mathematical definitions for these terms are developed their quantitative characterization is given and some general theorems involving locking and robustness are proven a model problem involving heat transfer is analyzed in detail using this mathematical framework and various related computational results are described applications to some different problems involving locking are presented

rigidity of some abelian by cyclic solvable group actions on mathbb t n in this paper we study a natural class of groups that act as affine transformations of mathbb t n we investigate whether these solvable abelian by cyclic groups can act smoothly and nonaffinely on mathbb t n while remaining homotopic to the affine actions in the affine actions elliptic and hyperbolic dynamics coexist forcing a priori complicated dynamics in nonaffine perturbations we first show using the kam method that any small and sufficiently smooth perturbation of such an affine action can be conjugated smoothly to an affine action provided certain diophantine conditions on the action are met in dimension two under natural dynamical hypotheses we get a complete classification of such actions namely any such group action by c r diffeomorphisms can be conjugated to the affine action by c r varepsilon conjugacy next we show that in any dimension c 1 small perturbations can be conjugated to an affine action via c 1 varepsilon conjugacy the method is a generalization of the herman theory for circle diffeomorphisms to higher dimensions in the presence of a foliation structure provided by the hyperbolic dynamics

multifractality of the feigenbaum attractor and fractional derivatives it is shown that fractional derivatives of the integrated invariant measure of the feigenbaum map at the onset of chaos have power law tails in their cumulative distributions whose exponents can be related to the spectrum of singularities f this is a new way of characterizing multifractality in dynamical systems so far applied only to multifractal random functions frisch and matsumoto j stat phys 108 1181 2002 the relation between the thermodynamic approach vul sinai and khanin russian math surveys 39 1 1984 and that based on singularities of the invariant measures is also examined the theory for fractional derivatives is developed from a heuristic point view and tested by very accurate simulations

generalized dimensions entropies and liapunov exponents from the pressure function for strange sets for conformal mixing repellers such as julia sets and nonlinear one dimensional cantor sets we connect the pressure of a smooth transformation on the repeller with its generalized dimensions entropies and liapunov exponents computed with respect to a set of equilibrium gibbs measures this allows us to compute the pressure by means of simple numerical algorithms our results are then extended to axiom a attractors and to a nonhyperbolic invariant set of the line in this last case we show that a first order phase transition appears in the pressure

robust rigidity for circle diffeomorphisms with singularities we prove that under certain regularity conditions imposed on the renormalizations of two circle diffeomorphisms with singularities their c 1 smooth equivalence follows from exponential convergence of those renormalizations as an easy corollary any two analytical critical circle maps with the same order of critical points and the same irrational rotation number are c 1 smoothly conjugate