horizontal visibility graphs exact results for random time series networks this procedure allows us to apply methods of complex network theory for characterizing time series in this work we present the horizontal visibility algorithm a geometrically simpler and analytically solvable version of our former algorithm focusing on the mapping of random series series of independent identically distributed random variables after presenting some properties of the algorithm we present exact results on the topological properties of graphs associated with random series namely the degree distribution the clustering coefficient and the mean path length we show that the horizontal visibility algorithm stands as a simple method to discriminate randomness in time series since any random series maps to a graph with an exponential degree distribution of the shape pk 1 32 3 k 2 independent of the probability distribution from which the series was generated accordingly visibility graphs with other pk are related to nonrandom series numerical simulations confirm the accuracy of the theorems for finite series in a second part we show that the method is able to distinguish chaotic series from independent and identically distributed i i d theory studying the following situations i noise free low dimensional chaotic series ii low dimensional noisy chaotic series even in the presence of large amounts of noise and iii high dimensional chaotic series coupled map lattice without needs for additional techniques such as surrogate data or noise reduction methods finally heuristic arguments are given to explain the topological properties of chaotic series and several sequences that are conjectured to be random are analyzed

the shortest possible return time of beta mixing processes we consider a stochastic process and a given n string we study the shortest possible return time or shortest return path of the string over all the realizations of process starting from this string for a beta mixing process having complete grammar and for each size n of the strings we approximate the distribution of this short return properly re scaled by a non degenerated distribution under mild conditions on the beta coefficients we prove the existence of the limit of this distribution to a non degenerated distribution we also prove that ergodicity is not enough to guaranty this convergence finally we present a connection between the shortest return and the shannon entropy showing that maximum of the re scaled variables grow as the matching function of wyner and ziv

entropy and data compression schemes some new ways of defining the entropy of a process by observing a single typical output sequence as well as a new kind of shannon mcmillan breiman theorem are presented this provides a new and conceptually very simple ways of estimating the entropy of an ergodic stationary source as well as new insight into the workings of such well known data compression schemes as the lempel ziv algorithm

a poisson limit theorem for a strongly ergodic non homogeneous markov chain a strongly ergodic non homogeneous markov chain is considered in the paper as an analog of the poisson limit theorem for a homogeneous markov chain recurring to small cylindrical sets a poisson limit theorem is given for the non homogeneous markov chain meanwhile some interesting results about approximation independence and probabilities of small cylindrical sets are given

elliptic surfaces and linear systems with fat points we investigate the expected dimensionality of linear systems with general fat points on certain surfaces using an approach by specialization to elliptic surfaces for the projectivization of the atiyah bundle over an elliptic curve with a certain polarization we observe that the special case of only one fat point implies the general case of arbitrarily many fat points as well as results concerning other surfaces we conjecture that this special case holds in characteristic 0 but prove that it fails in any positive characteristic

point processes of non stationary sequences generated by sequential and random dynamical systems we give general sufficient conditions to prove the convergence of marked point processes that keep record of the occurrence of rare events and of their impact for non autonomous dynamical systems we apply the results to sequential dynamical systems associated to both uniformly and non uniformly expanding maps and to random dynamical systems given by fibred lasota yorke maps

existence and nonexistence results of an optimal control problem by using relaxed control relaxed controls have proved to be very useful in studying the existence of optimal controls in optimal control theory many positive results have been obtained in the literature however negative results have also made their rare appearances the optimal control problem considered in this paper looks quite simple yet by treating such a problem we can get interesting results substantiating our idea as to whether an optimal control exists or not in our opinion the method used in the paper can be applied to more generalized cases

towards a generic constructive nonlinear control design tool using relaxed control in this technical note we revisit a control design approach for general non affine nonlinear systems using relaxed control using the notion of relaxed input where the ordinary real valued control input is replaced by a measure valued control input we are able to manipulate the original system such that the resulting relaxed system is amenable to control we show the applicability of using relaxed input for transforming a non affine nonlinear system into an affine one such that the relaxed system has desirable control properties finally we present a practical implementation of relaxed input via ordinary control input which is based on the sampled data of the relaxed input

entry times distribution for dynamical balls on metric spaces we show that the entry and return times for dynamical balls bowen balls is exponential for systems that have an alpha mixing invariant measure with certain regularities we also show that systems modeled by young s tower has exponential entry time distribution for dynamical balls we also apply the results to conformal maps and expanding maps on the interval

the extremal index hitting time statistics and periodicity abstract the extremal index appears as a parameter in extreme value laws for stochastic processes characterising the clustering of extreme events we apply this idea in a dynamical systems context to analyse the possible extreme value laws for the stochastic process generated by observations taken along dynamical orbits with respect to various measures we derive new easily checkable conditions which identify extreme value laws with particular extremal indices in the dynamical context we prove that the extremal index is associated with periodic behaviour the analogy of these laws in the context of hitting time statistics as studied in the authors previous works on this topic is explained and exploited extensively allowing us to prove for the first time the existence of hitting time statistics for balls around periodic points moreover for very well behaved systems uniformly expanding we completely characterise the extremal behaviour by proving that either we have an extremal index less than 1 at periodic points or equal to 1 at any other point this theory then also applies directly to general stochastic processes adding both useful tools to identify the extremal index and giving deeper insight into the periodic behaviour it suggests

poincare recurrences from the perspective of transient chaos we obtain a description of the poincare recurrences of chaotic systems in terms of the ergodic theory of transient chaos it is based on the equivalence between the recurrence time distribution and an escape time distribution obtained by leaking the system and taking a special initial ensemble this ensemble is atypical in terms of the natural measure of the leaked system the conditionally invariant measure accordingly for general initial ensembles the average recurrence and escape times are different however we show that the decay rate of these distributions is always the same our results remain valid for hamiltonian systems with mixed phase space and validate a split of the chaotic saddle in hyperbolic and nonhyperbolic components

weak chaos and poincare recurrences for area preserving maps the spectrum f t of poincare recurrence times for the standard map exhibits two distinct limits an integrable weak coupling limit with an inverse power law and a chaotic strong coupling limit with exponential decay in the domain where chaotic regions coexist with integrable structures the spectrum f t exhibits a superposition of exponential and power law decay such a law can be proved to occur in a model of area preserving map at the boundary of the mixing and integrable components

stickiness in generic low dimensional hamiltonian systems a recurrence time statistics approach we analyze the structure and stickiness in the chaotic components of generic hamiltonian systems with divided phase space following the method proposed recently in lozej and robnik phys rev e 98 022220 2018 2470 004510 1103 physreve 98 022220 the sticky regions are identified using the statistics of recurrence times of a single chaotic orbit into cells dividing the phase space into a grid we perform extensive numerical studies of three example systems the chirikov standard map the family of robnik billiards and the family of lemon billiards the filling of the cells is compared to the random model of chaotic diffusion introduced in robnik et al j phys a math gen 30 l803 1997 jphac50305 447010 1088 0305 4470 30 23 003 for the description of transport in the phase spaces of ergodic systems the model is based on the assumption of completely uncorrelated cell visits because of the strongly chaotic dynamics of the orbit and the distribution of recurrence times is exponential in generic systems the stickiness induces correlations in the cell visits the distribution of recurrence times exhibits a separation of timescales because of the dynamical trapping we model the recurrence time distributions to cells inside sticky areas as a mixture of exponential distributions with different decay times we introduce the variable s which is the ratio between the standard deviation and the mean of the recurrence times as a measure of stickiness we use s to globally assess the distributions of recurrence times we find that in the bulk of the chaotic sea s 1 while s 1 in areas of stickiness we present the results in the form of animated grayscale plots of the variable s in the largest chaotic component for the three example systems included as supplemental material to this paper

entry times distribution for mixing systems we consider the return times dynamics to bowen balls for continuous maps on metric spaces which have invariant probability measures with certain mixing properties these mixing properties are satisfied for instance by systems that allow young tower constructions we show that the higher order return times to bowen balls are in the limit poisson distributed we also provide a general result for the asymptotic behavior of the recurrence time for bowen balls for ergodic systems and those with specification

sunklodas approach to normal approximation for time dependent dynamical systems we consider time dependent dynamical systems arising as sequential compositions of self maps of a probability space we establish conditions under which the birkhoff sums for multivariate observations given a centering and a general normalizing sequence b n of invertible square matrices are approximated by a normal distribution with respect to a metric of regular test functions depending on the metric and the normalizing sequence b n the conditions imply that the error in the approximation decays either at the rate o n 1 2 or the rate o n 1 2 log n under the additional assumption that vert b n 1 vert lesssim n 1 2 the error comes with a multiplicative constant whose exact value can be computed directly from the conditions the proof is based on an observation due to sunklodas regarding stein s method of normal approximation we give applications to one dimensional random piecewise expanding maps and to sequential random and quasistatic intermittent systems

hitting times and periodicity in random dynamics we prove quenched laws of hitting time statistics for random subshifts of finite type in particular we prove a dichotomy between the law for periodic and for non periodic points we show that this applies to random gibbs measures

poisson and compound poisson approximations in conventional and nonconventional setups it was shown in kifer israel j math 2013 that for any subshift of finite type considered with a gibbs invariant measure the numbers of multiple recurrencies to shrinking cylindrical neighborhoods of almost all points are asymptotically poisson distributed here we not only extend this result to all psi mixing shifts with countable alphabet but actually show that for all points the distributions of these numbers are asymptotically close either to poisson or to compound poisson distributions it turns out that for all nonperiodic points a limiting distribution is always poisson while at the same time for periodic points there may be no limiting distribution at all unless the shift invariant measure is bernoulli in which case the limiting distribution always exists thus we describe essentially completely limiting distributions of multiple recurrence numbers in this setup as a corollary we obtain also that the first occurence time of the multiple recurrence event is asymptotically exponentially distributed most of the results are new also for the widely studied single recurrencies case see for instance haydn and vaienti discret contin dyn syst a 10 589 616 2004 probab theory relat fields 144 517 542 2009 abadi and saussol stoch process appl 121 314 323 2011 abadi and vergne nonlinearity 21 2871 2885 2008 as well

fluctuation bounds for entropy production estimators in gibbs measures we give bounds for the fluctuations of estimators of the mean entropy production in gibbsian sources these bounds are valid for every n where n denotes the size length of the sample we consider two estimators which are based on waiting and hitting times

the global statistics of return times return time dimensions versus generalized measure dimensions we investigate return times in dynamical systems i e the time required by a trajectory to complete a return journey to a neighborhood of the initial position in particular we study the relations holding between the scaling exponents of phase space moments of return times in balls of diminishing radius on the one side and the generalized dimensions of invariant measures on the other because of a heuristic use of kac theorem the former have been used in place of the latter in numerical and experimental investigations to mark the distinction we call them return time dimensions we derive a full set of inequalities linking generalized dimensions of invariant measures and return time dimensions we comment on their optimality with the aid of two maps due to von neumann kakutani and to gaspard wang we conjecture a formula for the return time dimensions in a typical system we only assume that the dynamical system under investigation is ergodic and that motion takes place in a compact finite dimensional space

on the noether lefschetz theorem and some remarks on codimension two cycles here of general moduli means that there is a countable union v of subvarieties of the space pn of surfaces of degree d in p3 such that the statement pie s z holds for s p n v noether it would seem stated this theorem but never completely proved it instead he gave a plausibility argument based on the following construction 1 let g be the hilbert scheme of curves of degree n and genus 9 in p3 l gr3 d l ip n the space of surfaces of degree d in p3 and zc t the incidence correspondence s o a c s c z s

the compound poisson limit ruling periodic extreme behaviour of non uniformly hyperbolic dynamics we prove that the distributional limit of the normalised number of returns to small neighbourhoods of periodic points of certain non uniformly hyperbolic dynamical systems is compound poisson the returns to small balls around a fixed point in the phase space correspond to the occurrence of rare events or exceedances of high thresholds so that there is a connection between the laws of return times statistics and extreme value laws the fact that the fixed point in the phase space is a repelling periodic point implies that there is a tendency for the exceedances to appear in clusters whose average sizes is given by the extremal index which depends on the expansion of the system at the periodic point we recall that for generic points the exceedances in the limit are singular and occur at poisson times however around periodic points the picture is different the respective point processes of exceedances converge to a compound poisson process so instead of single exceedances we have entire clusters of exceedances occurring at poisson times with a geometric distribution ruling its multiplicity the systems to which our results apply include general piecewise expanding maps of the interval rychlik maps maps with indifferent fixed points manneville pomeau maps and benedicks carleson quadratic maps

extreme value laws in dynamical systems for non smooth observations we prove the equivalence between the existence of a non trivial hitting time statistics law and extreme value laws in the case of dynamical systems with measures which are not absolutely continuous with respect to lebesgue this is a counterpart to the result of the authors in the absolutely continuous case moreover we prove an equivalent result for returns to dynamically defined cylinders this allows us to show that we have extreme value laws for various dynamical systems with equilibrium states with good mixing properties in order to achieve these goals we tailor our observables to the form of the measure at hand

degeneration of linear systems through fat points on k3 surfaces in this paper we introduce a technique to degenerate k3 surfaces and linear systems through fat points in general position on k3 surfaces using this degeneration we show that on generic k3 surfaces it is enough to prove that linear systems with one fat point are non special in order to obtain the non speciality of homogeneous linear systems through n 4 u 9 w fat points in general position moreover we use this degeneration to obtain a result for homogeneous linear systems through n 4 u 9 w fat points in general position on a general quartic surface in p 3

correlation decay and recurrence asymptotics for some robust nonuniformly hyperbolic maps we study a robust class of multidimensional non uniformly hyperbolic transformations considered by oliveira and viana ergod theory dyn syst 28 501 533 2008 for an open class of holder continuous potentials with small variation we show that the unique equilibrium state has exponential decay of correlations and that the distribution of hitting times is asymptotically exponential furthermore using that the equilibrium states satisfy a weak gibbs property we also prove log normal fluctuations of the return times around their average

timing variation in an analytically solvable chaotic system abstract we present analytic solutions for a chaotic dynamical system that do not have the regular timing characteristic of recently reported solvable chaotic systems the dynamical system can be viewed as a first order filter with binary feedback the feedback state may be switched only at instants defined by an external clock signal generalizing from a period one clock we show analytic solutions for period two and higher period clocks we show that even when the clock ticks randomly the chaotic system has an analytic solution these solutions can be visualized in a stroboscopic map whose complexity increases with the complexity of the clock we provide both analytic results as well as experimental data from an electronic circuit implementation of the system our findings bridge the gap between the irregular timing of well known chaotic systems such as lorenz and rossler and the well regulated oscillations of recently reported solvable chaotic systems

occurrence repetition and matching of patterns in the low temperature ising model we continue our study of the exponential law for occurrences and returns of patterns in the context of gibbsian random fields for the low temperature plus phase of the ising model we prove exponential laws with error bounds for occurrence return waiting and matching times moreover we obtain a poisson law for the number of occurrences of large cylindrical events and a gumbel law for the maximal overlap between two independent copies as a by product we derive precise fluctuation results for the logarithm of waiting and return times the main technical tool we use in order to control mixing is disagreement percolation

renyi entropies and large deviations for the first match function we define the first match function t n mathcal c n rightarrow 1 ldots n where mathcal c is a finite alphabet for two copies of x 1 n in mathcal c n this function gives the minimum number of steps one has to slide one copy of x 1 n to get a match with the other one for ergodic positive entropy processes saussol and coauthors proved the almost sure convergence of t n n we compute the large deviation properties of this function we prove that this limit is related to the renyi entropy function which is also proved to exist our results hold under a condition easy to check which defines a large class of processes we provide some examples

statistical properties of dynamical systems with some hyperbolicity this paper is about the ergodic theory of attractors and conservative dynamical systems with hyperbolic properties on large parts though not necessarily all of their phase spaces the main results are for discrete time systems to put this work into context recall that for axiom a attractors the picture has been fairly complete since the 1970 s see s1 b r2 since then much progress has been made on two fronts there is a general nonuniform theory that deals with properties common to all diffeomorphisms with nonzero lyapunov exponents o p1 ka ly and there are detailed analyses of specific kinds of dynamical systems including for example billiards 1 dimensional and henon type maps s2 bsc hk j bc2 by1 statistical properties such as exponential decay of correlations are not enjoyed by all diffeomorphisms with nonzero lyapunov exponents the goal of this paper is a systematic understanding of these and other properties for a class of dynamical systems larger than axiom a this class will not be defined explicitly but it includes some of the much studied examples by looking at regular returns to sets with good hyperbolic properties one could give systems in this class a simple dynamical representation conditions for the existence of natural invariant measures exponential mixing and central limit theorems are given in terms of the return times these conditions can be checked in concrete situations giving a unified way of proving a number of results some new and some old among the new results are the exponential decay of correlations for a class of scattering billiards and for a positive measure set of henon type maps

on the computation of the extremal index for time series the extremal index is a quantity introduced in extreme value theory to measure the presence of clusters of exceedances in the dynamical systems framework it provides important information about the dynamics of the underlying systems in this paper we provide a review of the meaning of the extremal index in dynamical systems depending on the observables used this quantity can inform on local properties of attractors such as periodicity stability and persistence in phase space or on global properties such as the lyapunov exponents we also introduce a new estimator of the extremal index and show its relation with those previously introduced in the statistical literature we reserve a particular focus to the systems perturbed with noise as they are a good paradigm of many natural phenomena different kind of noises are investigated in the annealed and quenched situations applications to climate data are also presented

gibbs measures for fibred systems abstract we consider a topological dynamical system t y y on a metric space y which forms a fibre bundle over another dynamical system if t is fibrewise expanding and exact along fibres and if is a holder continuous function we prove the existence of a system of conditional measures called a family of gibbs measures where the jacobian is determined by this theorem reduces to ruelle s perron frobenius theorem when the base of the fibred system consists of a single point the method of proof does not use any form of symbolic representation we also study continuity properties of a family of gibbs measures over the base and give applications to the equilibrium theory of higher dimensional complex dynamics

back to balls in billiards we consider a billiard in the plane with periodic configuration of convex scatterers this system is recurrent in the sense that almost every orbit comes back arbitrarily close to the initial point in this paper we study the time needed to get back in an e ball about the initial point in the phase space and also for the position in the limit when e 0 we establish the existence of an almost sure convergence rate and prove a convergence in distribution for the rescaled return times

hitting time statistics and extreme value theory we consider discrete time dynamical systems and show the link between hitting time statistics the distribution of the first time points land in asymptotically small sets and extreme value theory distribution properties of the partial maximum of stochastic processes this relation allows to study hitting time statistics with tools from extreme value theory and vice versa we apply these results to non uniformly hyperbolic systems and prove that a multimodal map with an absolutely continuous invariant measure must satisfy the classical extreme value laws with no extra condition on the speed of mixing for example we also give applications of our theory to higher dimensional examples for which we also obtain classical extreme value laws and exponential hitting time statistics for balls we extend these ideas to the subsequent returns to asymptotically small sets linking the poisson statistics of both processes

potential well spectrum and hitting time in renewal processes the potential well of a state can be interpreted physically as the energy that a stationary process needs to leave the state we prove that for discrete time renewal processes the potential well is the right scaling for the hitting and return time distributions of the state we further detail the potential well spectrum of these processes by giving a complete classification of the states according to their potential well

a multifractal analysis of equilibrium measures for conformal expanding maps and moran like geometric constructions in this paper we establish the complete multifractal formalism for equilibrium measures for holder continuous conformal expanding maps andexpanding markov moran like geometric constructions examples include markov maps of an interval beta transformations of an interval rational maps with hyperbolic julia sets and conformal toral endomorphisms we also construct a holder continuous homeomorphism of a compact metric space with an ergodic invariant measure of positive entropy for which the dimension spectrum is not convex and hence the multifractal formalism fails

on match lengths zero entropy and large deviations with application to sliding window lempel ziv algorithm the sliding window lempel ziv swlz algorithm that makes use of recurrence times and match lengths has been studied from various perspectives in information theory literature in this paper we undertake a finer study of these quantities under two different scenarios 1 zero entropy sources that are characterized by strong long term memory and 2 the processes with weak memory as described through various mixing conditions for zero entropy sources a general statement on match length is obtained it is used in the proof of almost sure optimality of fixed shift variant of lempel ziv fslz and swlz algorithms given in literature through an example of stationary and ergodic processes generated by an irrational rotation we establish that for a window of size n w a compression ratio given by o log n w n w a where a depends on n w and approaches 1 as n w is obtained under the application of fslz and swlz algorithms in addition we give a general expression for the compression ratio for a class of stationary and ergodic processes with zero entropy next we extend the study of ornstein and weiss on the asymptotic behavior of the normalized version of recurrence times and establish the large deviation property for a class of mixing processes in addition an estimator of entropy based on recurrence times is proposed for which large deviation principle is proved for sources satisfying similar mixing conditions

poincare recurrence for observations a high dimensional dynamical system is often studied by experimentalists through the measurement of a relatively low number of different quantities called an observation following this idea and in the continuity of boshernitzan s work for a measure preserving system we study poincare recurrence for the observation the link between the return time for the observation and the hausdorff dimension of the image of the invariant measure is considered we prove that when the decay of correlations is super polynomial the recurrence rates for the observations and the pointwise dimensions relative to the push forward are equal

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

limiting entry and return times distribution for arbitrary null sets we describe an approach that allows us to deduce the limiting return times distribution for arbitrary sets to be compound poisson distributed we establish a relation between the limiting return times distribution and the probability of the cluster sizes where clusters consist of the portion of points that have finite return times in the limit where random return times go to infinity in the special case of periodic points we recover the known polya aeppli distribution which is associated with geometrically distributed cluster sizes we apply this method to several examples the most important of which is synchronisation of coupled map lattices for the invariant absolutely continuous measure we establish that the returns to the diagonal is compound poisson distributed where the coefficients are given by certain integrals along the diagonal

linear systems of plane curves with base points of equal multiplicity in this article we address the problem of computing the dimenlsion of the space of plane curves of degree d with n general points of multiplicity m a conjecture of harbourne and hirschowitz implies that when d 3m the dimension is equal to the expected dimension given by the riemann roch theorem also systems for which the dimension is larger than expected should have a fixed part containing a multiple 1 curve we reformulate this conjecture by explicitly listing those systems which have unexpected dimension then we use a degeneration technique developed to show that the conjecture holds for all m 12

clustering of extreme events created by multiple correlated maxima abstract we consider stochastic processes arising from dynamical systems by evaluating an observable function along the orbits of the system the novelty is that we will consider observables achieving a global maximum value possible infinite at multiple points with special emphasis for the case where these maximal points are correlated or bound by belonging to the same orbit of a certain chosen point these multiple correlated maxima can be seen as a new mechanism creating clustering of extreme observations i e the occurrence of several extreme observations concentrated in the time frame we recall that clustering was intimately connected with periodicity when the maximum was achieved at a single point we will study this mechanism for creating clustering and will address the existence of limiting extreme value laws the repercussions on the value of the extremal index the impact on the limit of rare events points processes the influence on clustering patterns and the competition of domains of attraction we also consider briefly and for comparison purposes multiple uncorrelated maxima the systems considered include expanding maps of the interval such as rychlik maps but also maps with an indifferent fixed point such as manneville pomeau maps

sharp error terms and neccessary conditions for exponential hitting times in mixing processes we prove an upper bound for the error in the exponential approximation of the hitting time law of a rare event in mixing processes with exponential decay mixing processes with a summable function and for general mixing processes with a finite alphabet in the first case the bound is uniform as a function of the measure of the event in the last two cases the bound depends also on the time scale t this allows us to get further statistical properties as the ratio convergence of the expected hitting time and the expected return time a uniform bound is a consequence we present an example that shows that this bound is sharp we also prove that second moments are not necessary for having the exponential law moreover we prove a necessary condition for having the exponential limit law

meeting time distributions in bernoulli systems meeting time is defined as the time for which two orbits approach each other within the distance in phase space we show that the distribution of the meeting time is exponential in p1 pk bernoulli systems in the limit of 0 the distribution converges to exp where is the meeting time normalized by the average the exponent is shown to be kl 1pl 1 pl for the bernoulli systems

poincare recurrences and transient chaos in systems with leaks in order to simulate observational and experimental situations we consider a leak in the phase space of a chaotic dynamical system we obtain an expression for the escape rate of the survival probability by applying the theory of transient chaos this expression improves previous estimates based on the properties of the closed system and explains dependencies on the position and size of the leak and on the initial ensemble with a subtle choice of the initial ensemble we obtain an equivalence to the classical problem of poincare recurrences in closed systems which is treated in the same framework finally we show how our results apply to weakly chaotic systems and justify a split of the invariant saddle into hyperbolic and nonhyperbolic components related respectively to the intermediate exponential and asymptotic power law decays of the survival probability

rates of convergence in the clt for two dimensional dispersive billiards we consider billiards in the two dimensional torus with convex obstacles central limit theorems have been established for regular functions for the billiard transformation in 2 1 and 14 we are interested here in the problem of the rate of convergence in this paper we establish a rate in for any 0 for the billiard transformation by adapting the proof of 7 6 8 in our proof we use a strong decorrelation result obtained by the method developped in 14 for the study of general hyperbolic systems moreover we establish a rate of convergence in in the central limit theorem for the billiard flow

deterministic bak sneppen model lyapunov spectrum and avalanches as return times abstract a deterministic version of the bak sneppen model is studied the role of the lyapunov spectrum in the onset of scale free behavior is established as well as the measure theoretic nature of the bak sneppen self organized state avalanches are interpreted as return times to a small measure set and the problem of accurate determination of the scaling exponents near the critical barrier is addressed

exponential decay of correlations for random lasota yorke maps we consider random piecewise smooth piecewise invertible maps mainly on the interval but also in higher dimensions we assume that on the average and possibly without any stochastic uniformity i the maps expand distances ii do not have too many pieces iii do not have too large a distortion and iv are strongly mixing we assume it no markov property we prove that as in the classical case of the iteration of a fixed piecewise expanding map of the interval we have it exponential decay of random correlations our proof builds on the one given by c liverani for deterministic mixing and piecewise expanding interval maps

extreme value theory for non uniformly expanding dynamical systems we establish extreme value statistics for functions with multiple maxima and some degree of regularity on certain non uniformly expanding dynamical systems we also establish extreme value statistics for time series of observations on discrete and continuous suspensions of certain non uniformly expanding dynamical systems via a general lifting theorem the main result is that a broad class of observations on these systems exhibit the same extreme value statistics as i i d processes with the same distribution function

the distribution of the first return time for rational maps we obtain exponential error estimates for the approximation of the zeroth return time to the poisson distribution for rational maps which might have critical points within the julia set

return time entropies for a class of circle homeomorphisms poincare recurrence for a class of circle maps is used to study the properties of the corresponding invariant measures in the subcritical case when the map is a diffeomorphism the return time measure is smooth and in the critical case when the map is only a homeomorphism the measure is only continuous furthermore in the considered class of critical maps the behaviour of the return time entropy depends only on the tail in the continued fraction expansion of the rotation number

poincare recurrences and multifractal properties of genomic sequences we propose the return times spectra as a tool to analyze the genomic sequences the spectra discriminate between regular chaotic and mixed dynamical systems since generally their decay follows respectively an exponential law a power law or is a linear combination of them the analysis of words of n bases from a dna sequence exhibits the same exponential behavior for the coding and noncoding component the multifractal analysis shows a denite dierence which suggests a less uniform structure for the noncoding part

hausdorff dimension of measures via poincare recurrence we study the quantitative behavior of poincare recurrence in particular for an equilibrium measure on a locally maximal hyperbolic set of a c1 diffeomorphism f we show that the recurrence rate to each point coincides almost everywhere with the hausdorff dimension d of the measure that is inf k 0 fkx b x r r d this result is a non trivial generalization of work of boshernitzan concerning the quantitative behavior of recurrence and is a dimensional version of work of ornstein and weiss for the entropy we stress that our approach uses different techniques furthermore our results motivate the introduction of a new method to compute the hausdorff dimension of measures

recurrence time analysis long term correlations and extreme events the recurrence times between extreme events have been the central point of statistical analyses in many different areas of science simultaneously the poincar e recurrence time has been extensively used to characterize nonlinear dynamical systems we compare the main properties of these statistical methods pointing out their consequences for the recurrence analysis performed in time series in particular we analyze the dependence of the mean recurrence time and of the recurrence time statistics on the probability density function on the interval whereto the recurrences are observed and on the temporal correlations of time series in the case of long term correlations we verify the validity of the stretched exponential distribution which is uniquely defined by the exponent ensuremath gamma at the same time showing that it is restricted to the class of linear long term correlated processes simple transformations are able to modify the correlations of time series leading to stretched exponentials recurrence time statistics with different ensuremath gamma which shows a lack of invariance under the change of observables

asymptotics for hitting times essentially decreasing sequences of balls in ametric space x and identi ed to be the distribution function of the posi tive exponential law with parameter 1 in many classes of mixing systems in abadi and galves 2001 collet and galves 1993 hirata saussol andvaienti 1999 saussol 1998 and young 1999 nonexponential asymp totics have been obtained in coelho and de faria 1990 for instance theliterature on the subject is rather important and our list is incomplete for further information we refer the reader to the surveys abadi 2004 orcoelho 2000 there exists a connection between return time asymptotics and hittingtime asymptotics indeed as shown in hirata saussol and vaienti 1999 when the asymptotics for return times is exponential with parameter 1 thenso is the one for hitting times possible asymptotics for return times that is weak limits for f

limit theorems for partially hyperbolic systems we consider a large class of partially hyperbolic systems containing among others affine maps frame flows on negatively curved manifolds and mostly contracting diffeomorphisms if the rate of mixing is sufficiently high the system satisfies many classical limit theorems of probability theory

extreme value laws for dynamical systems with countable extremal sets we consider stationary stochastic processes arising from dynamical systems by evaluating a given observable along the orbits of the system we focus on the extremal behaviour of the process which is related to the entrance in certain regions of the phase space which correspond to neighbourhoods of the maximal set mathcal m i e the set of points where the observable is maximised the main novelty here is the fact that we consider that the set mathcal m may have a countable number of points which are associated by belonging to the orbit of a certain point and may have accumulation points in order to prove the existence of distributional limits and study the intensity of clustering given by the extremal index we generalise the conditions previously introduced in freitas adv math 231 5 2626 2665 2012 stoch process appl 125 4 1653 1687 2015

structure size and statistical properties of chaotic components in a mixed type hamiltonian system we perform a detailed study of the chaotic component in mixed type hamiltonian systems on the example of a family of billiards introduced by robnik in j phys a math gen 16 3971 1983 jphac50305 447010 1088 0305 4470 16 17 014 the phase space is divided into a grid of cells and a chaotic orbit is iterated a large number of times the structure of the chaotic component is discerned from the cells visited by the chaotic orbit the fractal dimension of the border of the chaotic component for various values of the billiard shape parameter is determined with the box counting method the cell filling dynamics is compared to a model of uncorrelated motion the so called random model robnik et al j phys a math gen 30 l803 1997 jphac50305 447010 1088 0305 4470 30 23 003 and deviations attributed to sticky objects in the phase space are found the statistics of the number of orbit visits to the cells is analyzed and found to be in agreement with the random model in the long run the stickiness of the various structures in the phase space is quantified in terms of the cell recurrence times the recurrence time distributions in a few selected cells as well as the mean and standard deviation of recurrence times for all cells are analyzed the standard deviation of cell recurrence time is found to be a good quantifier of stickiness on a global scale three methods for determining the measure of the chaotic component are compared and the measure is calculated for various values of the billiard shape parameter lastly the decay of correlations and the diffusion of momenta is analyzed

maximal clusters in non critical percolation and related models we investigate the maximal non critical cluster in a big box in various percolation type models we investigate its typical size and the fluctuations around this typical size the limit law of these fluctuations is related to maxima of independent random variables with law described by a single cluster

the optimal sink and the best source in a markov chain it is well known that the distributions of hitting times in markov chains are quite irregular unless the limit as time tends to infinity is considered we show that nevertheless for a typical finite irreducible markov chain and for nondegenerate initial distributions the tails of the distributions of the hitting times for the states of a markov chain can be ordered i e they do not overlap after a certain finite moment of time if one considers instead each state of a markov chain as a source rather than a sink then again the states can generically be ordered according to their efficiency the mechanisms underlying these two orderings are essentially different though our results can be used e g for a choice of the initial distribution in numerical experiments with the fastest convergence to equilibrium stationary distribution for characterization of the elements of a dynamical network according to their ability to absorb and transmit the substance information that is circulated over the network for determining optimal stopping moments stopping signals words when dealing with sequences of symbols etc

degenerations of planar linear systems fixing n general points pi in the plane what is the dimension of the space of plane curves of degree d having multiplicity mi at pi for each i in this article we propose an approach to attack this problem and demonstrate it by successfully computing this dimension for all n and for mi constant at most 3 this application while previously known see hi1 demonstrates the utility of our approach which is based on an analysis of the corresponding linear system on a degeneration of the plane itself leading to a simple recursion for these dimensions we also obtain results in the quasi homogeneous case when all the multiplicities are equal except one this is the natural family to consider in the recursion

statistics of close visits to the indifferent fixed point of an interval map we study a dynamical system defined by a map of the interval 0 1 which has 0 as an indifferent fixed point but is otherwise expanding we prove that the sequence of successive entrance times in a small neighborhood 0 a converges in law when suitably normalized to a homogeneous poisson point process

conditions for almost global attractivity of a synchronous generator connected to an infinite bus conditions for existence and global attractivity of the equilibria of a realistic model of a synchronous generator with constant field current connected to an infinite bus are derived first necessary and sufficient conditions for existence and uniqueness of equilibrium points are provided then sufficient conditions for local asymptotic stability and almost global attractivity of one of these equilibria are given the analysis is carried out by employing a new lyapunov like function to establish convergence of bounded trajectories while the latter is proven using the powerful theoretical framework of cell structures pioneered by leonov and noldus the efficiency of the derived sufficient conditions is illustrated via extensive numerical experiments based on two benchmark examples taken from the literature

the compound poisson distribution and return times in dynamical systems previously it has been shown that some classes of mixing dynamical systems have limiting return times distributions that are almost everywhere poissonian here we study the behaviour of return times at periodic points and show that the limiting distribution is a compound poissonian distribution we also derive error terms for the convergence to the limiting distribution we also prove a very general theorem that can be used to establish compound poisson distributions in many other settings

laws of rare events for deterministic and random dynamical systems the object of this paper is twofold from one side we study the dichotomy in terms of the extremal index of the possible extreme value laws when the rare events are centred around periodic or non periodic points then we build a general theory of extreme value laws for randomly perturbed dynamical systems we also address in both situations the convergence of rare events point processes decay of correlations against l 1 observables will play a central role in our investigations

compound poisson approximation for unbounded functions on a group with application to large deviations a second order error bound is obtained for approximating h d tilde q by h d tilde q where tilde q is a convolution of measures andq a compound poisson measure on a measurable abelian group and the functionh is not necessarily bounded this error bound is more refined than the usual total variation bound in the sense that it contains the functionh the method used is inspired by stein s method and hinges on bounding radon nikodym derivatives related to d tilde q mathord left vphantom d tilde q dq right kern nulldelimiterspace dq the approximation theorem is then applied to obtain a large deviation result on groups which in turn is applied to multivariate poisson approximation

fast time varying phasor analysis in the balanced three phase large electric power system traditional phasor representation of sinusoidal signals the standard analytical tool for power system stability analysis is limited by the quasistationary assumption on the speeds of the phasor states this paper provides a rigorous formulation of a time varying phasor representation for the balanced three phase large power system with no restrictions on the speeds power balance equations become a set of differential equations in the phasor dynamic states and singularly perturbed behavior of the resulting dynamics is explored

compound poisson approximation for nonnegative random variables via stein s method the aim of this paper is to extend stein s method to a compound poisson distribution setting the compound poisson distributions of concern here are those of the form pois nu where nu is a finite positive measure on 0 infty a number of results related to these distributions are established these in turn are used in a number of examples to give bounds for the error in the compound poisson approximation to the distribution of a sum of random variables

the loss of tightness of time distributions for homeomorphisms of the circle for a minimal circle homeomorphism f we study convergence in law of rescaled hitting time point process of an interval of length epsilon 0 although the point process in the natural time scale never converges in law we study all possible limits under a subsequence the new feature is the fact that for rotation numbers of unbounded type there is a sequence epsilon n going to zero exhibiting coexistence of two non trivial asymptotic limit point processes depending on the choice of time scales used when rescaling the point process the phenomenon of loss of tightness of the first hitting time distribution is an indication of this coexistence behaviour moreover tightness occurs if and only if the rotation number is of bounded type therefore tightness of time distributions is an intrinsic property of badly approximable irrational rotation numbers

recurrence dimensions and lyapunov exponents we show that the poincare return time of a typical cylinder is at least its length for one dimensional maps we express the lyapunov exponent and dimension via return times

weakly defective varieties a projective variety x is k weakly defective when its intersection with a general k 1 tangent hyperplane has no isolated singularities at the k 1 points of tangency if x is k defective i e if the k secant variety of x has dimension smaller than expected then x is also k weakly defective the converse does not hold in general a classification of weakly defective varieties seems to be a basic step in the study of defective varieties of higher dimension we start this classification here describing all weakly defective irreducible surfaces our method also provides a new proof of the classical terracini s classification of k defective surfaces

hitting and return times in ergodic dynamical systems given an ergodic dynamical system x t and u c x measurable with u 0 let u u x denote the normalized hitting time of x x to u we prove that given a sequence u n with u n 0 the distribution function of the normalized hitting times to u n converges weakly to some subprobability distribution f if and only if the distribution function of the normalized return time converges weakly to some distribution function f and that in the converging case f t 0 t 1 f s ds t 0 this in particular characterizes asymptotics for hitting times and shows that the asymptotics for return times is exponential if and only if the one for hitting times is also

statistics of return times a general framework and new applications in this paper we provide general estimates for the errors between the distribution of the first and more generally the k th return time suitably rescaled and the poisson law for measurable dynamical systems in the case that the system exhibits strong mixing properties these bounds are explicitly expressed in terms of the speed of mixing using these approximations the poisson law is finally proved to hold for a large class of non hyperbolic systems on the interval

mixing properties in the advection of passive tracers via recurrences and extreme value theory in this paper we characterize the mixing properties in the advection of passive tracers by exploiting the extreme value theory for dynamical systems with respect to classical techniques directly related to the poincare recurrences analysis our method provides reliable estimations of the characteristic mixing times and distinguishes between barriers and unstable fixed points the method is based on a check of convergence for extreme value laws on finite datasets we define the mixing times in terms of the shortest time intervals such that extremes converge to the asymptotic known parameters of the generalized extreme value distribution our technique is suitable for applications in the analysis of other systems where mixing time scales need to be determined and limited datasets are available