topological conjugacy of linear endomorphisms of the 2 torus we describe two complete sets of numerical invariants of topological conjugacy for linear endomorphisms of the two dimensional torus i e continuous maps from the torus to itself which are covered by linear maps of the plane the trace and determinant are part of both complete sets and two candidates are proposed for a third and last invariant which in both cases can be understood from the topological point of view one of our invariants is in fact the ideal class of the latimer macduffee taussky theory reformulated in more elementary terms and interpreted as describing some topology merely one has to look at how closed curves on the torus intersect their image under the endomorphism part of the intersection information the intersection number counted with multiplicity can be captured by a binary quadratic form associated to the map so that we can use the classical theories initiated by lagrange and gauss to go beyond the intersection number and shortcut the classification theory for quadratic forms we use the rotation number of poincare

spectral behavior of contractive noise we study the behavior of the spectra corresponding to quantum systems subjected to a contractive noise i e the environment reduces the accessible phase space of the system but the total probability is conserved we find that the number of long lived resonances grows as a power law in h but surprisingly there is no relationship between the exponent of this power law and the fractal dimension of the corresponding classical attractor this is in disagreement with the predictions of the fractal weyl law which has been established for open systems where the probability is lost under the effect of a projective opening

quantum fractal eigenstates abstract we study quantum chaos in open dynamical systems and show that it is characterized by quantum fractal eigenstates located on the underlying classical strange repeller the states with longest life times typically reveal a scars structure on the classical fractal set

on bipartite drawings and the linear arrangement problem the bipartite crossing number problem is studied and a connection between this problem and the linear arrangement problem is established a lower bound and an upper bound for the optimal number of crossings are derived where the main terms are the optimal arrangement values two polynomial time approximation algorithms for the bipartite crossing number are obtained the performance guarantees are o log n and o log2 n times the optimal respectively for a large class of bipartite graphs on n vertices no polynomial time approximation algorithm which could generate a provably good solution had been known for a tree a formula is derived that expresses the optimal number of crossings in terms of the optimal value of the linear arrangement and the degrees resulting in an o n1 6 time algorithm for computing the bipartite crossing number the problem of computing a maximum weight biplanar subgraph of an acyclic graph is also studied and a linear time algorithm for solving it is derived no polynomial time algorithm for this problem was known and the unweighted version of the problem had been known to be np hard even for planar bipartite graphs of degree at most 3

weyl asymptotics from closed to open systems we present microwave experiments on the symmetry reduced five disk billiard studying the transition from a closed to an open system the measured microwave reflection signal is analyzed by means of the harmonic inversion and the counting function of the resulting resonances is studied for the closed system this counting function shows the weyl asymptotic with a leading exponent equal to 2 by opening the system successively this exponent decreases smoothly to a noninteger value for the open systems the extraction of resonances by the harmonic inversion becomes more challenging and the arising difficulties are discussed the results can be interpreted as a first experimental indication for the fractal weyl conjecture for resonances

role of short periodic orbits in quantum maps with continuous openings we apply a recently developed semiclassical theory of short periodic orbits to the continuously open quantum tribaker map in this paradigmatic system the trajectories are partially bounced back according to continuous reflectivity functions this is relevant in many situations that include optical microresonators and more complicated boundary conditions in a perturbative regime the shortest periodic orbits belonging to the classical repeller of the open map a cantor set given by a region of exactly zero reflectivity prove to be extremely robust in supporting a set of long lived resonances of the continuously open quantum maps moreover for steplike functions a significant reduction in the number needed is obtained similarly to the completely open situation this happens despite a strong change in the spectral properties when compared to the discontinuous reflectivity case in order to give a more realistic interpretation of these results we compare with a fresnel type reflectivity function

geometric bounds on the density of resonances for semiclassical problems in this paper we shall give upper bounds on the number of resonances in certain regions in the complex plane close to the real axis for semiclassical operators like h 2 v x when h is small

classical dynamics of a family of billiards with analytic boundaries the classical dynamics of a billiard which is a quadratic conformal image of the unit disc is investigated the author gives the stability analysis of major periodic orbits present the poincare maps demonstrate the mixing properties by following the evolution of a small element in phase space show the existence of homoclinic points and calculate the lyapunov exponent and the kolmogorov entropy h it turns out that the system becomes strongly chaotic positive h for sufficiently large deformations of the unit disc the system shows a generic stochastic transition the computations suggest that the system is mixing if the boundary is not convex

google matrix dynamical attractors and ulam networks we study the properties of the google matrix generated by a coarse grained perron frobenius operator of the chirikov typical map with dissipation the finite size matrix approximant of this operator is constructed by the ulam method this method applied to the simple dynamical model generates directed ulam networks with approximate scale free scaling and characteristics being in certain features similar to those of the world wide web with approximate scale free degree distributions as well as two characteristics similar to the web a power law decay in pagerank that mirrors the decay of pagerank on the world wide web and a sensitivity to the value ensuremath alpha in pagerank the simple dynamical attractors play here the role of popular websites with a strong concentration of pagerank a variation in the google parameter ensuremath alpha or other parameters of the dynamical map can drive the pagerank of the google matrix to a delocalized phase with a strange attractor where the google search becomes inefficient

fractal weyl law for quantum fractal eigenstates the properties of the resonant gamow states are studied numerically in the semiclassical limit for the quantum chirikov standard map with absorption it is shown that the number of such states is described by the fractal weyl law and their husimi distributions closely follow the strange repeller set formed by classical orbits nonescaping in future times for large matrices the distribution of escape rates converges to a fixed shape profile characterized by a spectral gap related to the classical escape rate

intermediate statistics in quantum maps we present a one parameter family of quantum maps whose spectral statistics are of the same intermediate type as observed in polygonal quantum billiards our central result is the evaluation of the spectral two point correlation form factor at small argument which in turn yields the asymptotic level compressibility for macroscopic correlation lengths

chaotic sets of continuous and discontinuous maps abstract this paper discusses li yorke chaotic sets of continuous and discontinuous maps with particular emphasis to shift and subshift maps scrambled sets and maximal scrambled sets are introduced to characterize li yorke chaotic sets the orbit invariant for a scrambled set is discussed some properties about maximality equivalence and uniqueness of maximal scrambled sets are also discussed it is shown that for shift maps the set of all scrambled pairs has full measure and chaotic sets of some discontinuous maps such as the gauss map interval exchange transformations and a class of planar piecewise isometries are studied finally some open problems on scrambled sets are listed and remarked

fractal weyl law for open quantum chaotic maps we study a semiclassical quantization of poincar e maps arising in scattering problems with fractal hyperbolic trapped sets the main application is the proof of a fractal weyl upper bound for the number of resonances scattering poles in small domains near the real axis this result includes the case of several convex hard obstacles satisfying a no eclipse condition

unstable periodic orbits and semiclassical quantisation the bohr sommerfeld quantisation condition has a meaningful extension to classically chaotic systems whose periodic unstable orbits are isolated it provides a semiclassical euler factorisation for the functional determinant of the quantal hamiltonian in contrast to the hadamard infinite product over the eigenvalues by which the exact determinant is defined

hecke theory and equidistribution for the quantization of linear maps of the torus 1 1 background one of the key issues of quantum chaos is the nature of the semiclassical limit of eigenstates of classically chaotic systems when the classical system is given by the geodesic flow on a compact riemannian manifoldm or rather on its cotangent bundle one can formulate the problem as follows the quantum hamiltonian is in suitable units represented by the positive laplacian on m to measure the distribution of its eigenstates we start with a smooth classical observable that is a smooth function on the unit cotangent bundle s m via some choice of quantization from symbols to pseudodifferential operators we form its quantization op f this is a zero order pseudodifferential operator with principal symbol f the expectation value of op f in the eigenstate is op f let j be a sequence of normalized eigenfunctions j j j 0 m j 2 1 the problem then is to understand the possible limits as j of the distributions f c s m op f j j 1 1

fractal weyl law for chaotic microcavities fresnel s laws imply multifractal scattering we demonstrate that the harmonic inversion technique is a powerful tool to analyze the spectral properties of optical microcavities as an interesting example we study the statistical properties of complex frequencies of the fully chaotic microstadium we show that the conjectured fractal weyl law for open chaotic systems lu et al phys rev lett 91 154101 2003 is valid for dielectric microcavities only if the concept of the chaotic repeller is extended to a multifractal by incorporating fresnel s laws

statistics of energy levels without time reversal symmetry aharonov bohm chaotic billiards a planar domain d contains a single line of magnetic flux phi switching on phi breaks time reversal symmetry t for quantal particles with charge q moving in d whilst preserving the geometry of classical billiard trajectories bouncing off the boundary delta d if delta d is such that these classical trajectories are chaotic the authors predict that t breaking will cause the local statistics of quantal energy levels to change their universality class from that of the gaussian orthogonal ensemble goe of random matrix theory to that of the gaussian unitary ensemble gue in the semiclassical limit this transition is abrupt for statistics involving the first n levels gue behaviour requires that the quantum flux alpha identical to q phi h 0 13n 14 the special flux alpha 1 2 corresponds to false t breaking and for this case goe statistics are predicted these predictions are confirmed by numerical computation of spectral statistics for a classically chaotic billiard without symmetry for which delta d is a cubic conformal image of the unit disc

a linear edge kernel for two layer crossing minimization the two layer crossing minimization problem tlcm given a bipartite graph g with n vertices and a positive integer k asks whether g has a two layer drawing with at most k crossings we consider a simple generalization of tlcm called leaf edge weighted tlcm lew tlcm where we allow positive weights on edges incident to leaves and show that this problem admits a kernel with o k edges provided that the given graph is connected as a straightforward consequence lew tlcm and hence tlcm has a fixed parameter algorithm that runs in 2 o k log k n o 1 time which improves on the previously best known algorithm with running time 2 o k 3 n

fractal weyl law behavior in an open hamiltonian system we numerically show fractal weyl law behavior in an open hamiltonian system that is described by a smooth potential and which supports numerous above barrier resonances this behavior holds even relatively far away from the classical limit the complex resonance wave functions are found to be localized on the fractal classical repeller

distribution of resonances for open quantum maps we analyze a simple model of quantum chaotic scattering system namely the quantized open baker s map this model provides a numerical confirmation of the fractal weyl law for the semiclassical density of quantum resonances the fractal exponent is related to the dimension of the classical repeller we also consider a variant of this model for which the full resonance spectrum can be rigorously computed and satisfies the fractal weyl law for that model we also compute the shot noise of the conductance through the system and obtain a value close to the prediction of random matrix theory

a linear time algorithm for finding tree decompositions of small treewidth in this paper we give for constant k a linear time algorithm that given a graph g v e determines whether the treewidth of g is at most k and if so finds a tree decomposition of g with treewidth at most k a consequence is that every minor closed class of graphs that does not contain all planar graphs has a linear time recognition algorithm another consequence is that a similar result holds when we look instead for path decompositions with pathwidth at most some constant k

arithmetical properties of strongly chaotic motions abstract the orbits of the generalized arnold sinai cat maps or hyperbolic automorphisms of the two dimensional torus typify purely chaotic anosov motion we transform the dynamics of the periodic orbits of these maps into modular arithmetic in suitable domains of quadratic integers classify all periodic orbits and show how to determine their periods and initial conditions the methods are based on ideal theory in quadratic fields which is reviewed it is shown that the structure of orbits rests upon some basic arithmetical notions such as unique factorization into prime ideals

lyapunov stability for a generalized henon heiles system in a rotating reference frame in this paper we focus on a generalized henon heiles system in a rotating reference frame in such a way that lagrangian like equilibrium points appear our goal is to study their nonlinear stability properties to better understand the dynamics around these points we show the conditions on the free parameters to have stability and we prove the superstable character of the origin for the classical case it is a stable equilibrium point regardless of the frequency value of the rotating frame

trace formula for dielectric cavities general properties the construction of the trace formula for open dielectric cavities is examined in detail using the krein formula it is shown that the sum over cavity resonances can be written as a sum over classical periodic orbits for the motion inside the cavity the contribution of each periodic orbit is the product of the two factors the first is the same as in the standard trace formula and the second is connected with the product of reflection coefficients for all points of reflection with the cavity boundary two asymptotic terms of the smooth resonance counting function related with the area and the perimeter of a convex cavity are derived the coefficient of the perimeter term differs from the one for closed cavities due to unusual high energy asymptotics of the s matrix for the scattering on the cavity corrections to the leading semi classical formula are briefly discussed obtained formulas agree well with numerical calculations for circular dielectric cavities

smoothed wave functions of chaotic quantum systems abstract it is shown that wave functions of quantum systems as 0 have an extra density near unstable periodic trajectories of the classical problem the averaged wave function square is represented as the sum over a finite number of periodic trajectories the contribution of each trajectory is expressed through the elements of the monodromy matrix of the trajectory the results are compared with the numerical calculations of the wave functions for the stadium billiard

symbolic dynamics for the modular surface and beyond in this expository article we describe the two main methods of representing geodesics on surfaces of constant negative curvature by symbolic sequences and their development a geometric method stems from a 1898 work of j hadamard and was developed by m morse in the 1920s it consists of recording the successive sides of a given fundamental region cut by the geodesic and may be applied to all finitely generated fuchsian groups another method of arithmetic nature uses continued fraction expansions of the end points of the geodesic at infinity and is even older it comes from the gauss reduction theory introduced to dynamics by e artin in a 1924 paper this method was used to exhibit dense geodesics on the modular surface for 80 years these classical works have provided inspiration for mathematicians and a testing ground for new methods in dynamics geometry and combinatorial group theory we present some of the ideas results old and recent and interpretations that illustrate the multiple facets of the subject

singular continuous spectra in a pseudointegrable billiard the pseudointegrable barrier billiard invented by hannay and mccraw j phys a 23 887 1990 a rectangular billiard with line segment barrier placed on a symmetry axis is generalized it is demonstrated that the fourier spectrum of a typical function on the phase space exhibits a singular continuous component

entropy production and information gain in axiom a systems anosov systems are mathematical models for chaotic systems in statistical mechanics and fluid dynamics most of these systems enjoy the property of positive entropy production we introduce the concept of specific information gain or specific relative entropy h in anosov systems and prove that it is identical to the entropy production rate e p defined by ruelle and gallavotti in anosov systems from this point of view the entropy production rate e p characterizes the degree of macroscopic irreversibility of the system

kam tori are very sticky rigorous lower bounds on the time to move away from an invariant lagrangian torus with linear flow abstract hamiltonian perturbation theory is used to derive rigorous nekhoroshev type lower bounds on the time it takes to move away from an invariant lagrangian torus when the flow on the torus is conjugate to a highly non resonant linear flow in particular these results give bounds on the time to move away from a kam torus if r is the distance from the torus then the lower bound on the time to double the distance from the torus is c r e k r where c and k are constants 1 n 2 n is the number of degrees of freedom and is a measure of the irrationality of the flow we thus prove that nearby trajectories take an extremely long time to move away from such an invariant torus and that the more irrational the linear flow the stickier the torus will be these estimates do not require the steepness or quasi convexity assumptions of nekherosev s theorem

fractal weyl law for three dimensional chaotic hard sphere scattering systems the fractal weyl law connects the asymptotic level number with the fractal dimension of the chaotic repeller we provide the first test for the fractal weyl law for a three dimensional open scattering system for the four sphere billiard we investigate the chaotic repeller and discuss the semiclassical quantization of the system by the method of cycle expansion with symmetry decomposition we test the fractal weyl law for various symmetry subspaces and sphere to sphere separations

the quantum perturbed cat map and symmetry we investigate a class of quantum symmetries of the perturbed cat map which exist only for a subset of possible values of planck s constant the effect of these symmetries is to change the spectral statistics along this positive density subset the symmetries are shown to be related to some simple classical symmetries of the map