An effective numerical method of the word-lifting technique in one-dimensional multimodal maps

For general algebraic maps of orders higher than or equal to 5, the word-lifting technique for calculating parameters of symbolic sequences meets with a difficulty of lacking explicit expressions of the inverse functions of the maps. This Letter presents an effective numerical method to solve such a problem. Parameters of some quadruply superstable kneading sequences in quadrumodal quintic maps are calculated as examples.     

A note on chaotic unimodal maps and applications

Based on the word-lift technique of symbolic dynamics of one-dimensional unimodal maps, we investigate the relation between chaotic kneading sequences and linear maximum-length shift-register sequences. Theoretical and numerical evidence that the set of the maximum-length shift-register sequences is a subset of the set of the universal sequence of one-dimensional chaotic unimodal maps is given. By stabilizing unstable periodic orbits on superstable periodic orbits, we also develop techniques to control the generation of long binary sequences.     

Some flesh on the skeleton: The bifurcation structure of bimodal maps

One of the main tools in the numerical study of two-parameter families of one-dimensional maps is the drawing of curves in parameter space corresponding to the existence of superstable periodic orbits. We use kneading theory to describe the structure of these sets of curves for the case of maps with at most two turning points. Then we explain how the bifurcation structure hangs on this “skeleton”.     

The Generalized Milnor–Thurston Conjecture and Equal Topological Entropy Class in Symbolic Dynamics of Order Topological Space of Three Letters

This paper presents an answer to an open problem in the dynamical systems of three letters: the generalized Milnor–Thurston conjecture on the existence of infinitely many plateaus of topological entropy in the two-dimensional parameter plane. The concept of equal topological entropy class is introduced by the dual star product which is a generalization of the Derrida–Gervois–Pomeau star product to the symbolic dynamics of three letters for the endomorphisms on the interval. The algebraic rules established by the dual star products for the doubly superstable kneading sequences are equivalent to the normal factorization of the Milnor–Thurston characteristic polynomials. Moreover, the classification theory of symbolic primitive and compound sequences based on the topological conjugacy in the meaning of equal entropy is completed in the topological space Σ₃ of three letters. Received: 4 February 1998 / Accepted: 1 March 2000     

Cyclic star products and universalities in symbolic dynamics of trimodal maps

Cyclic star products for the triple superstable kneading (TSSK) sequences are presented for symbolic dynamics of trimodal maps of endomorphisms on the interval. Feigenbaum’s metric universalities in unimodal maps are generalized to trimodal maps. Four equal-value universal convergent rates {δ_a,δ_c,δ_η,δ_a,c,η} and three universal scaling factors {_C,_D,_E} are first obtained.     

Binary tree approach to scaling in unimodal maps

Ge, Rusjan, and Zweifel introduced a binary tree which represents all the periodic windows in the chaotic regime of iterated one-dimensional unimodal maps. We consider the scaling behavior in a modified tree which takes into account the self-similarity of the window structure. A nonuniversal geometric convergence of the associated superstable parameter values towards a Misiurewicz point is observed for almost all binary sequences with periodic tails. For these sequences the window period grows arithmetically down the binary tree. There are an infinite number of exceptional sequences, however, for which the growth of the window period is faster. Numerical studies with a quadratic maximum suggest more rapid than geometric scaling of the superstable parameter values for such sequences.     

Dynamics of Infinitely Many Particles Mutually Interacting in Three Dimensions via a Bounded Superstable Long-Range Potential

We show existence and uniqueness for the solutions to the Newton equations relative to a system of infinitely many particles moving in the three-dimensional space and mutually interacting via a bounded superstable long-range potential. The present paper complete an analogous result obtained for positive short-range interaction.     

Coupled trivial maps

The first nontrivial example of coupled map lattices that admits a rigorous analysis in the whole range of the strength of space interactions is considered. This class is generated by one-dimensional maps with a globally attracting superstable periodic trajectory that are coupled by a diffusive nearest-neighbor interaction.     

Some remarks on nonequilibrium dynamics of infinite particle systems

Classical mechanics of infinitely many particles in dimensions one and two is considered, particles interacting by a superstable pair potential of finite range. The group of motion generated by Newton's equations is constructed in the space of locally finite configurations with a logarithmic order of energy fluctuations at infinity. A core of the Liouville operator is also described. Results of Dobrushin and the author and of Marchioro-Pellegrinotti-Pulvirenti are improved.     

A metric space of interactions and the thermodynamic limit

A metric space of interactions is formed for classical continuous systems and for quantum and classical lattice systems. It is shown that the thermodynamic limit of the grand canonical pressure exists on an extended class of potentials. In each neighborhood of each superstable lower regular, weakly tempered pair interaction and for each of a countable number of test functions there is an interaction for which the Fisher thermodynamic limit of the correlation functionals applied to the test function exists.     

Bifurcation structures in two-dimensional maps: The endoskeletons of shrimps

Bifurcation structures for nonlinear dynamical systems in a space of two parameters often display geometric shapes resembling shrimps. For one-dimensional maps with two parameters and multiple extrema, the underlying structure of the shrimps can be elucidated by computing the locus of superstable cycles which form a “skeleton” that supports the shrimps. Here we use continuation methods to identify and compute structures in two-dimensional maps that play the same role as the skeleton in one-dimensional maps. This facilitates determining the complex geometries for situations in which there is multistability, and for which the regions of parameter space supporting stable orbits get vanishingly small.     

Local Gaussian domination and Bose condensation in lattice boson model

We use infrared bounds method to study the existence of a Bose condensation phase transition in a three-dimensional lattice model of interacting bosons. Upper bound (local Gaussian domination) is presented on the Bogolyubov inner product of creation and annihilation bosonic operators in momentum space. We focus on the situation with a non-negative chemical potential. The presence of a Bose condensation is established for sufficiently small superstable interaction potential. A special case of the model considered is the Bose–Hubbard model.     

A characterization of Gibbs states of lattice boson systems

We consider lattice boson systems interacting via potentials which are superstable and regular. By using the Wiener integral formalism and the concept of conditional reduced density matrices we are able to give a characterization of Gibbs (equilibrium) states. It turns out that the space of Gibbs states is nonempty, convex, and also weak-compact if the interactions are of finite range. We give a brief discussion on the uniqueness of Gibbs states and the existence of phase transitions in our formalism.     

On non-equilibrium dynamics of multidimensional infinite particle systems in the translation invariant case

It is shown that for a set of full measure with respect to any translation invariant probability distribution on the space of initial configurarations of classical particle systems on ^d with interaction given by a smooth superstable potential of finite range there is a solution to the Newtonian equations of motion, provided that the specific energy and the particle density of the initial configuration exist a.s.     

Universality and self-similarity in the bifurcations of circle maps

The bifurcation structure in a two-parameter family of circle maps is considered. These maps have a (topological) degree that may be different from one. A generalization of the rotation number is given and symmetries of the bifurcations in parameter space are described. Continuity arguments are used to establish the existence of periodic orbits. By plotting the locus of parameter values associated with superstable cycles, self-similar bifurcations are found. These bifurcations are a generalization of the familiar period-doubling cascade in maps with one extrema, to two-parameter maps with two extrema. Finally, a scheme for the global organization of bifurcation in these maps is proposed.     

Probability estimates for continuous spin systems

Probability estimates for classical systems of particles with superstable interactions [1] are extended to continuous spin systems.     

Kneading Theory: A Functorial Approach

We study the homology theory of ? - modal maps of the interval. We give another proof of the Milnor and Thurston results about zeta-functions and we give a functorial approach to kneading theory. Our results give explicit methods for computing the sequences of lap numbers ? (f ^k ) and the sequences of numbers of periodic points in an arbitrary interval [x,y]. Received: 25 February 1998 / Accepted: 15 January 1999     

Quadratic maps without asymptotic measure

An interval map is said to have an asymptotic measure if the time averages of the iterates of Lebesgue measure converge weakly. We construct quadratic maps which have no asymptotic measure. We also find examples of quadratic maps which have an asymptotic measure with very unexpected properties, e.g. a map with the point mass on an unstable fix point as asymptotic measure. The key to our construction is a new characterization of kneading sequences.     

Complexity of routes to chaos and global regularity of fractal dimensions in bimodal maps

The dual-star composition rule of doubly superstable (DSS) sequences presents a complete renormalizable algebraic structure for studying Feigenbaum's metric universality and self-similar classification of DSS sequences in symbolic dynamics of bimodal maps of the interval. Here an important feature is that the complete combinations of up- and down-star products create all the generalized Feigenbaum's routes of transitions to chaos. These routes can be classified into two types: one consists of countably infinitely many regular routes which preserve Feigenbaum's metric universality; another consists of uncountably infinitely many universal nonscaling routes described by the irregularly mixed dual-star products, which break Feigenbaum's asymptotically convergent metric universality although they are structurally universal. The combinatorial complexity of dual-star products may increase the grammatical complexity of languages of symbolic dynamics. Moreover, it is found that there exists a global regularity between the fractal dimensions d and the scaling factors [alpha(C),alpha(D)] for Feigenbaum-type attractors: d(Z)log(/Z/)/alpha(C)(Z)alpha(D)(Z)/=beta((2)), where beta((2)) is independent of the concrete DSS sequences Z.