the metric entropy of endomorphisms we prove that for a c 2 non invertible but non degenerate map on a compact riemannian manifold without boundary an invariant measure satisfies an equality relating entropy folding entropy and negative lyapunov exponents this generalizes ledrappier young s entropy formula 5 for negative lyapunov exponents of diffeomorphisms to the case of endomorphisms

arithmetic monodromy groups pinkham 12 showed that the monodromy groups of the 14 exceptional unimodular singularities can be characterized arithmetically using results of m kneser we proved in 4 that this is true for all singularities of arnold s list with certain exceptions see below the object of this note is to show that this is even true for almost all isolated hypersurface singularities more precisely we prove the following theorem

ground states are generically a periodic orbit we prove that for an expanding transformation the maximizing measures of a generic lipschitz function are supported on a single periodic orbit

cobordism of algebraic knots the isolated singularities of complex hypersurfaces are studied by considering the topology of the highly connected submanifolds of spheres determined by the singularity by introducing the notion of the link of a perturbation of the singularity and using techniques of surgery theory we are able to describe which invariants associated to a singularity can be used to determine the cobordism type of the singularity

the metric entropy of diffeomorphisms part ii relations between entropy exponents and dimension on considere f m m m m un c 2 diffeomorphisme f d une variete de riemann compacte m preservant une mesure de probabilite de borel m soit hm f l entropie metrique de f et 1 x r x x les exposants de lyapunov distincts en x on etablit une formule qui relie h m f et les r x x

ruelle inequality of folding type for c 1 alpha maps for all c 1 alpha maps we prove an inequality conjectured by ruelle that the metric entropy is bounded from above by the folding entropy minus the sum of the negative lyapunov exponents

the metric entropy of diffeomorphisms part i characterization of measures satisfying pesin s entropy formula soit m une variete de riemann compacte soit f m m un diffeomorphisme et soit m une mesure de probabilite de borel f invariante sur m on identifie les mesures pour lesquelles l inegalite de margulis et ruelle qui relie l entropie aux exposants de lyapunov atteint l egalite on demontre une formule valable pour toutes les mesures invariantes

fast algorithms for discrete and continuous wavelet transforms several algorithms are reviewed for computing various types of wavelet transforms the mallat algorithm 1989 the a trous algorithm and their generalizations by shensa the goal of this work is to develop guidelines for implementing discrete and continuous wavelet transforms efficiently and to compare the various algorithms obtained and give an idea of possible gains by providing operation counts most wavelet transform algorithms compute sampled coefficients of the continuous wavelet transform using the filter bank structure of the discrete wavelet transform although this general method is already efficient it is shown that noticeable computational savings can be obtained by applying known fast convolution techniques such as the fft fast fourier transform in a suitable manner the modified algorithms are termed fast because of their ability to reduce the computational complexity per computed coefficient from l to log l within a small constant factor for large filter lengths l for short filters smaller gains are obtained fast running fir finite impulse response filtering techniques allow one to achieve typically 30 savings in computations

the discrete wavelet transform wedding the a trous and mallat algorithms two separately motivated implementations of the wavelet transform are brought together it is observed that these algorithms are both special cases of a single filter bank structure the discrete wavelet transform the behavior of which is governed by the choice of filters in fact the a trous algorithm is more properly viewed as a nonorthonormal multiresolution algorithm for which the discrete wavelet transform is exact moreover it is shown that the commonly used lagrange a trous filters are in one to one correspondence with the convolutional squares of the daubechies filters for orthonormal wavelets of compact support a systematic framework for the discrete wavelet transform is provided and conditions are derived under which it computes the continuous wavelet transform exactly suitable filter constraints for finite energy and boundedness of the discrete transform are also derived relevant signal processing parameters are examined and it is observed that orthonormality is balanced by restrictions on resolution

invariant measures satisfying an equality relating entropy folding entropy and negative lyapunov exponents in this paper we prove that for a c2 non invertible but non degenerate map on a compact manifold an invariant measure satisfies an equality relating entropy folding entropy and negative lyapunov exponents if and under a condition on the jacobian of the map only if the measure has absolutely continuous conditional measures on the stable manifolds

preimage pressure stable pressure and equilibrium states abstract in this paper the pressure of a potential via the dynamics of preimage structure are investigated for noninvertible systems and the properties for the related equilibrium states are considered for a continuous map on a compact metric space a notion of preimage pressure via the preimages of single points is introduced and then some fundamental properties are considered and a variational principle is established for the system with uniform separation of preimages for a c 1 partially hyperbolic endomorphism on a closed riemannian manifold a notion of stable pressure via the preimages of local stable manifolds is introduced and a corresponding variational principle is established moreover equilibrium states and tangent functionals for pointwise preimage pressures and stable pressures are also considered by describing the set of equilibrium states and tangent functionals and a sense of continuous dependence of equilibrium states with respect to the potential we prove that the continuous potentials with unique equilibrium state form a dense g set in addition for any finite collection of ergodic measures we can find a continuous potential such that its set of equilibrium states contains the given set

ruelle inequality relating entropy folding entropy and negative lyapunov exponents in this paper we prove an inequality conjectured by ruelle relating the entropy folding entropy and negative lyapunov exponents of a differentiable map on a compact manifold under a set of conditions on degenerate points of the map

wave propagation and sampling theory part ii sampling theory and complex waves morlet et al 1982 this issue showed the advantages of using complex values for both waves and characteristics of the media we simulated the theoretical tools we present here using the goupillaud kunetz algorithm now we present sampling methods for complex signals or traces corresponding to received waves and sampling methods for complex characterization of multilayered or heterogeneous media regarding the complex signals we present a twodimecsional 2 d method of sampling in the time frequency domain using a special or extended gabor expansion on a set of basic wavelets adapted to phase preservation such a 2 d expansion permits us to handle in a proper manner instantaneous frequency spectra we show the differences between wavelet resolution and sampling grid resolution we also show the importance of phase preservation in high resolution seismic regarding the media we show how analytical studies of wave propagation in periodic structured layers could help when trying to characterize the physical properties of the layers and their large scale granularity as a result of complex deconvolution analytical studies of wave propagation in periodic structures are well known in solid state physics and lead to the so called bloch waves the introduction of complex waves leads to replacing the classical wave equation by a schriidinger equation finally we show that complex wave equations gabor expansion and bloch waves are three different ways of introducing the tools of quantum mechanics in highresolution seismic gabor 1946 kittel 1976 morlet 1975 and conversely the goupillaud kunetz algorithm and an extended gabor expansion may be of some use in solid state physics

orthonormal bases of compactly supported wavelets we construct orthonormal bases of compactly supported wavelets with arbitrarily high regularity the order of regularity increases linearly with the support width we start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction the construction then follows from a synthesis of these different approaches

continuous and discrete wavelet transforms this paper is an expository survey of results on integral representations and discrete sum expansions of functions in l 2 bf r in terms of coherent states two types of coherent states are considered weyl heisenberg coherent states which arise from translations and modulations of a single function and affine coherent states called wavelets which arise as translations and dilations of a single function in each case it is shown how to represent any function in l 2 bf r as a sum or integral of these states most of the paper is a survey of literature most notably the work of i daubechies a grossmann and j morlet a few results of the authors are included

wave propagation and sampling theory part i complex signal and scattering in multilayered media from experimental studies in digital processing of seismic reflection data geophysicists know that a seismic signal does vary in amplitude shape frequency and phase versus propagation time to enhance the resolution of the seismic reflection method we must investigate these variations in more detail we present quantitative results of theoretical studies on propagation of plane waves for normal incidence through perfectly elastic multilayered media as wavelet shapes we use zero phase cosine wavelets modulated by a gaussian envelope and the corresponding complex wavelets a finite set of such wavelets for an appropriate sampling of the frequency domain may be taken as the basic wavelets for a gabor expansion of any signal or trace in a two dimensional 2 d domain time and frequency we can then compute the wave propagation using complex functions and thereby obtain quantitative results including energy and phase of the propagating signals these results appear as complex 2 d functions of time and frequency i e as instantaneous frequency spectra choosing a constant sampling rate on the logarithmic scale in the frequency domain leads to an appropriate sampling method for phase preservation of the complex signals or traces for this purpose we developed a gabor expansion involving basic wavelets with a constant time duration mean period ratio for layered media as found in sedimentary basins

the wavelet transform time frequency localization and signal analysis two different procedures for effecting a frequency analysis of a time dependent signal locally in time are studied the first procedure is the short time or windowed fourier transform the second is the wavelet transform in which high frequency components are studied with sharper time resolution than low frequency components the similarities and the differences between these two methods are discussed for both schemes a detailed study is made of the reconstruction method and its stability as a function of the chosen time frequency density finally the notion of time frequency localization is made precise within this framework by two localization theorems

on the geometry of principal homogeneous spaces let b be a curve defined over an algebraically closed field k and let pi colon x to b be an elliptic surface with base curve b we investigate the geometry of everywhere locally trivial principal homogeneous spaces for x i e elements of the tate shafarevich group of the generic fiber of pi if y is such a principal homogeneous space of order n we find strong restrictions on the bbb p n 1 bundle over b into which y embeds examples for small values of n show that at least in some cases these restrictions are sharp finally we determine these bundles in case k has characteristic zero b bbb p 1 and x is generic in a suitable sense

unstable entropies and variational principle for partially hyperbolic diffeomorphisms abstract we study entropies caused by the unstable part of partially hyperbolic systems we define unstable metric entropy and unstable topological entropy and establish a variational principle for partially hyperbolic diffeomorphisms which states that the unstable topological entropy is the supremum of the unstable metric entropy taken over all invariant measures the unstable metric entropy for an invariant measure is defined as a conditional entropy along unstable manifolds and it turns out to be the same as that given by ledrappier young though we do not use increasing partitions the unstable topological entropy is defined equivalently via separated sets spanning sets and open covers along a piece of unstable leaf and it coincides with the unstable volume growth along unstable foliation we also obtain some properties for the unstable metric entropy such as affineness upper semi continuity and a version of shannon mcmillan breiman theorem