Invariants for smooth conjugacy of hyperbolic dynamical systems. I

We prove that the Lyapunov exponents of periodic orbits are a total family of invariants forC conjugation of families of diffeomorphisms to a two-dimensional toral automorphism. In case of families of canonical diffeomorphisms, the corresponding conjugating diffeomorphisms are also canonical.Supported by CAICYT grant 397/84     

Invariants for smooth conjugacy of hyperbolic dynamical systems II

We show that the eigenvalues of the derivatives at periodic points form a complete set of invariants for smooth local conjugacy of Anosov diffeomorphisms ofT ².Supported in part by NSF grant # DMS 85-04984     

Invariants for smooth conjugacy of hyperbolic dynamical systems,III

We give a characterization of Anosov diffeomorphisms smoothly conjugated to a toral automorphism in dimension two in term of the Lyapunov exponents of periodic points. We also give necessary and sufficient conditions for the regularity of solutions of the vector cohomology equations associated to an Anosov flow in three dimensions. This allows us to prove a corresponding conjugation theorem.Supported by CAICyT grant #397/84     

Smooth conjugacy and S-R-B measures for uniformly and non-uniformly hyperbolic systems

We give a new proof of the fact that the eigenvalues at corresponding periodic orbits forms a complete set of invariants for the smooth conjugacy of low dimensional Anosov systems. We also show that, if a homeomorphism conjugating two smooth low dimensional Anosov systems is absolutely continuous, then it is as smooth as the maps. We furthermore prove generalizations of these facts for non-uniformly hyperbolic systems as well as extensions and counterexamples in higher dimensions.     

Fractal structure of hyperbolic Lipschitzian dynamical systems

In the paper, dynamical systems admitting no smooth structure are studied. A theorem on the semiconjugacy of a Lipschitzian dynamical system to the corresponding topological Markov chain is proved. A new approach to evaluating bounds for the Hausdorff and the Kolmogorov dimensions of the set of nonwandering points from a Markov partition is suggested.     

On comparing dynamical systems by defective conjugacy: A symbolic dynamics interpretation of commuter functions

While the field of dynamical systems has been focused on properties which are invariant to “good” change of variables, namely conjugacy, which is an equivalence relationship, when using dynamical systems methods in science and modeling, there lacks a dynamical way to compare dynamical systems, even when they are in some sense “close.” In Skufca and Bolt (2007) [7] and Skufca and Bolt (2008) [8], we introduced mathematics to support a philosophy that two dynamical systems should be compared through a change of coordinates between them, that is, a commuter between them which may fail to be a homeomorphism. The progressive degree to which the commuter fails to be a homeomorphism defines what we call a homeomorphic defect. However, at the time of publication of these papers, there were limits in the mathematical technology requiring that the transformations be one-dimensional mappings and flows which are well described, for construction of the commuters by fixed point iteration, and further, difficulties in numerically computing defects in the more complicated one-dimensional cases, and further limits to higher-dimensional problems. Therefore, here we extend the theory to allow for multivariate transformations, with construction methods separate from the fixed point iteration, and new methods to compute defect. In the course of this work, we introduce several new technical innovations in order to cope with much more general problems. We introduce assignment mappings to understand and illustrate commuters in a broader setting. We discuss the role of symbolic dynamics and coding as related to commuters as well as defect measure. Further, we discuss refinement and convergence of a nested refinement of commuter representations. This work represents a step forward in the possibility of using the commuter and defects to judge model quality in those dynamical systems for which a symbolic dynamics, and hence a generating partition may be available; while finding a generating partition is a problem in its own right, we offer this work as further perspective for interpretation of the meaning of commuters and defect measure.     

Fractal diffusion in smooth dynamical systems with virtual invariant curves

Preliminary results of extensive numerical experiments with a family of simple models specified by the smooth canonical strongly chaotic 2D map with global virtual invariant curves are presented. We focus on the statistics of the diffusion rate D of individual trajectories for various fixed values of the model perturbation parameters K and d. Our previous conjecture on the fractal statistics determined by the critical structure of both the phase space and the motion is confirmed and studied in some detail. In particular, we find additional characteristics of what we earlier termed the virtual invariant curve diffusion suppression, which is related to a new very specific type of critical structure. A surprising example of ergodic motion with a “hidden” critical structure strongly affecting the diffusion rate was also encountered. At a weak perturbation (K ? 1), we discovered a very peculiar diffusion regime with the diffusion rate D=K ²/3 as in the opposite limit of a strong (K ? 1) uncorrelated perturbation, but in contrast to the latter, the new regime involves strong correlations and exists for a very short time only. We have no definite explanation of such a controversial behavior.     

Some variational formulas for Hausdorff dimension,topological entropy,and SRB entropy for hyperbolic dynamical systems

We derive several variational formulas for the topological entropy and SRB entropy of Axiom A flows on compact manifolds and for the Hausdorff dimension of basic sets for Axiom A diffeomorphisms on compact surfaces.     

Invariants of chaotic Hamiltonian systems

The invariants of chaotic bounded Hamiltonian systems and their relation to the solutions of the first variational equations of the equations of motion are studied. We show that these invariants are characterized by the fact that they either lose the property of differentiability as functions on phase space or that a certain formal power series defined in terms of the derivatives of the invariants has zero radius of convergence. For a specific example, we show that the former possibility appears to apply.     

A theorem for time-dependent dynamical systems

A generalization of a theorem of Reid and Cullen is given. The limitations of using only point transformations are discussed and a procedure for the use of Noether's theorem with velocity-dependent transformations promoted.     

Locating resonances for AxiomA dynamical systems

For a class of differentiable dynamical systems (called Axiom A systems) it has been shown by Pollicott and the author that correlation functions have Fourier transforms which are meromorphic in a strip. The poles (or resonances) are, however, not easy to locate. This note reviews the results which are known and discusses a simple model where the position of resonances can be estimated. The effect of noise is also discussed.On leave of absence from Institut des Hautes Etudes Scientifiques, 35, route de Chartres, 91440 Buressur-Yvette, France.     

Coherent sets for nonautonomous dynamical systems

We describe a mathematical formalism and numerical algorithms for identifying and tracking slowly mixing objects in nonautonomous dynamical systems. In the autonomous setting, such objects are variously known as almost-invariant sets, metastable sets, persistent patterns, or strange eigenmodes, and have proved to be important in a variety of applications. In this current work, we explain how to extend existing autonomous approaches to the nonautonomous setting. We call the new time-dependent slowly mixing objects coherent sets as they represent regions of phase space that disperse very slowly and remain coherent. The new methods are illustrated via detailed examples in both discrete and continuous time.     

Fast time-evolution method for dynamical systems

A fast time-evolution method is developed for systems for which the dynamical behavior can be reduced to the eigenvector/eigenvalue problem. The method does not use the eigenvectors/eigenvalues themselves and is based on a polynominal expansion of the formal operator solution in the eigenfrequency domain. It is complementary to the standard time-integration approaches and allows one to calculate or simulate the state of a system at arbitrary times. The time evolution of, e.g., classical harmonic atomic systems and quantum systems described by linear Hamiltonians can be treated by this method.     

A nonequilibrium entropy for dynamical systems

It is proposed to define entropy for nonequilibrium ensembles using a method of coarse graining which partitions phase space into sets which typically have zero measure. These are chosen by considering the totality of future possibilities for observation on the system. It is shown that this entropy is necessarily a nondecreasing function of the timet. There is no contradiction with the reversibility of the laws of motion because this method of coarse graining is asymmetric under time reversal. Under suitable conditions (which are stated explicitly) this entropy approaches the equilibrium entropy ast+ and the fine-grained entropy ast–. In particular, the conditions can always be satisfied if the system is aK-system, as in the Sinai billiard models. Some theorems are given which give information about whether it is possible to generate the partition used here for coarse graining from time translates of a finite partition, and at the same time elucidate the connection between our concept of entropy and the entropy invariant of Kolmogorov and Sinai.Research supported in part by NSF grants PHY78-03816 and PHY78-15920.     

Hamilton's equations for constrained dynamical systems

We derive expressions for the conjugate momenta and the Hamiltonian for classical dynamical systems subject to holonomic constraints. We give an algorithm for correcting deviations of the constraints arising in numerical solution of the equations of motion. We obtain an explicit expression for the momentum integral for constrained systems.     

Invariants and radiation of some nonstationary systems

In this paper formulas are obtained by means of the coherent-state method for calculating the radiation power of a nonstationary quantum system ofN charged particles whose Hamiltonian is a general quadratic form with respect to coordinates and momenta. The transitions between the coherent states and the Fock states of this system are discussed. The radiation is calculated both in the dipole approximation and strictly. As an example, the radiation of a charge in homogeneous varying electric and magnetic fields is found. The classical limit is considered.     

Quantum dynamical entropies for classical stochastic systems

We compare two proposals for the dynamical entropy of quantum deterministic systems (CNT and AFL) by studying their extensions to classical stochastic systems. We show that the natural measurement procedure leads to a simple explicit expression for the stochastic dynamical entropy with a clear information-theoretical interpretation. Finally, we compare our construction with other recent proposals.     

A Hamiltonian approach for skew-product dynamical systems

The problem on the existence of Hamiltonian structures for (nonlinear) skew-product dynamical systems is studied via coupling Poisson structures. This research was partially supported by CONACYT under grant no. 55463.