We introduce index systems, a tool for studying isolated invariant sets of dynamical systems that are not necessarily hyperbolic. The mapping of the index systems mimics the expansion and contraction of hyperbolic maps on the tangent space, and they may be used like Markov partitions to generate symbolic dynamics. Every continuous dynamical system satisfying a weak form of expansiveness possesses an index system. Because of their topological robustness, they can be used to obtain rigorous results from computer approximations of a dynamical system.
We study dynamics in a neighborhood of a nonhyperbolic fixed point or an irreducible homoclinic tangent point. General type conditions for the existence of infinite sets of periodic points are obtained. A new method, based on the study of the dynamics of center disks, is introduced. Some results on shadowing near a non-hyperbolic fixed point of a homeomorphism are obtained.
The main purpose of this article is to study box dimension of orbits near hyperbolic and nonhyperbolic fixed points of discrete dynamical systems in higher dimensions. We generalize the known results for one-dimensional systems, that is, the orbits near the hyperbolic fixed point in one-dimensional discrete dynamical system has the box dimension equal to zero and the orbits near nonhyperbolic fixed point has positive box dimension. In the process of studying box dimensions, we use the stable, unstable and center manifolds and appropriate system restrictions. The main result is that the box dim…
We study transitive step skew-product maps modeled over a complete shift of $k$, $k\ge2$, symbols whose fiber maps are defined on the circle and have intermingled contracting and expanding regions. These dynamics are genuinely nonhyperbolic and exhibit simultaneously ergodic measures with positive, negative, and zero exponents. We introduce a set of axioms for the fiber maps and study the dynamics of the resulting skew-product. These axioms turn out to capture the key mechanisms of the dynamics of nonhyperbolic robustly transitive maps with compact central leaves. Focusing on the nonhyperbolic…
We show that diffeomorphisms with a dominated splitting of the form $E^s\oplus E^c\oplus E^u$, where $E^c$ is a nonhyperbolic central bundle that splits in a dominated way into 1-dimensional subbundles, are entropy-expansive. In particular, they have a principal symbolic extension and equilibrium states.
We provide examples of transitive partially hyperbolic dynamics (specific but paradigmatic examples of homoclinic classes) which blend different types of hyperbolicity in the one-dimensional center direction. These homoclinic classes have two disjoint parts: an "exposed" piece which is poorly homoclinically related with the rest and a "core" with rich homoclinic relations. There is an associated natural division of the space of ergodic measures which are either supported on the exposed piece or on the core. We describe the topology of these two parts and show that they glue along nonhyperbolic…
In dynamical systems theory, a fixed point of the dynamics is called nonhyperbolic if the linearization of the system around the fixed point has at least one eigenvalue with zero real part. The center manifold existence theorem guarantees the local existence of an invariant subspace of the dynamics, known as a center manifold, around such nonhyperbolic fixed points. A growing number of theoretical and experimental studies suggest that some neural systems utilize nonhyperbolic fixed points and corresponding center manifolds to display complex, nonlinear dynamics and to flexibly adapt to wide-ra…
In this paper we study several dynamical properties of the riemannian $1$-dimensional foliation $\mathcal{L}$ on an oriented closed 3-manifold $M$. Carriere classified such pairs $(M,\mathcal{L})$. Using the classification we prove the nonhyperbolicity of $(M,\mathcal{L})$. Also we describe in detail recurrence points, $ω$-limit sets and attractors.
We introduce some tools of symbolic dynamics to study the hyperbolic directions of partially hyperbolic diffeomorphisms, emulating the well known methods available for uniformly hyperbolic systems.
Necessary and sufficient conditions for the symbolic dynamics of a Lorenz map to be fully embedded in the symbolic dynamics of a piecewise continuous interval map are given. As an application of this result, we describe a new algorithm for calculating the topological entropy of a Lorenz map.
We discuss Devaney chaos on compact metric spaces using a decomposition space characterized by topological nature of symbolic dynamics. A chaotic map obtained here is defined as a topologically conjugate of the chaotic map on a decomposition space which is induced by a chaotic map of symbolic dynamics. In particular, the chaotic character of the tent map and the baker map on [0,1] are reconsidered based on decomposition dynamics involving symbolic dynamics with different two chaotic maps. As an example of compact metric space we exhibit a chaotic map existing on any given finite graph.
We define canonical subshift of finite type cover for Williams' 1-dimensional generalized solenoids, and use resulting invariants to distinguish some closely related solenoids.
The rotation of Hyperion is often modelled by equations of motion of an ellipsoidal satellite. The model is expected to be chaotic for large range of parameters. The paper contains a rigorous computer-assisted proof of the existence of symbolic dynamics in its dynamics by the use of CAPD C++ library.
A major motivation for the development of semigroup theory was, and still is, its applications to the study of formal languages. Therefore, it is not surprising that the correspondence $\mathcal X\mapsto B(\mathcal X)$, associating to each symbolic dynamical system $\mathcal X$ the formal language $B(\mathcal X)$ of its blocks, entails a connection between symbolic dynamics and semigroup theory. In this article we survey some developments on this connection, since when it was noticed in an article by Almeida, published in the CIM bulletin, in 2003.
The decimal expansion real numbers, familiar to us all, has a dramatic generalization to representation of dynamical system orbits by symbolic sequences. The natural way to associate a symbolic sequence with an orbit is to track its history through a partition. But in order to get a useful symbolism, one needs to construct a partition with special properties. In this work we develop a general theory of representing dynamical systems by symbolic systems by means of so-called Markov partitions. We apply the results to one of the more tractable examples: namely hyperbolic automorphisms of the two…
In this paper we study the family of the Lozi maps $L_{a,b} : {\mathbb R}^2 \to {\mathbb R}^2$, $L_{a,b} = (1 + y - a|x|, bx)$, and their strange attractors $Λ_{a,b}$. We introduce the set of kneading sequences for the Lozi map and prove that it determines the symbolic dynamics for that map. We also introduce two other equivalent approaches.
Iterations of odd piecewise continuous maps with two discontinuities, i.e., symmetric discontinuous bimodal maps, are studied. Symbolic dynamics is introduced. The tools of kneading theory are used to study the homology of the discrete dynamical systems generated by the iterations of that type of maps. When there is a Markov matrix, the spectral radius of this matrix is the inverse of the least root of the kneading determinant.
A probability measure is a characteristic measure of a topological dynamical system if it is invariant to the automorphism group of the system. We show that zero entropy shifts always admit characteristic measures. We use similar techniques to show that automorphism groups of minimal zero entropy shifts are sofic.
We consider a specific piecewise rotation of the plane that is continuous on two half-planes, as studied by some authors like Boshernitzan, Goetz and Quas. If the angle belongs to the set $\{\fracπ{2},\frac{2π}{3},\fracπ{4}\}$, we give a complete description of the symbolic dynamics of this map in the non symmetric bijective case.
In this paper we discuss a coding and the associated symbolic dynamics for the geodesic flow on Hecke triangle surfaces. We construct an explicit cross section for which the first return map factors through a simple (explicit) map given in terms of the generating map of a particular continued fraction expansion closely related to the Hecke triangle groups. We also obtain explicit expressions for the associated first return times.
This paper aims at completing and clarifying a delicate step in the proof of Theorem 5.3 of our paper \cite{ST}, where it was used the differentiability of a function $F$, which a priori can appear not necessarily differentiable.
The thesis is devoted to relations between algebra and symbolic dynamics. Various generalisations of sturmian sequences are discoursed. Let $W$ be an infinite word over a finite alphabet $A$. The combinatorial criteria of existence of interval exchange transformations that generate the word $W$ is obtained in full generality -- no IDOC condition assumed, case of orientation non-preserving also considered.
We consider the planar $N$-centre problem, with homogeneous potentials of degree $-\a<0$, $\a \in [1,2)$. We prove the existence of infinitely many collisions-free periodic solutions with negative and small energy, for any distribution of the centres inside a compact set. The proof is based upon topological, variational and geometric arguments. The existence result allows to characterize the associated dynamical system with a symbolic dynamics, where the symbols are the partitions of the $N$ centres in two non-empty sets.
We develop some aspects of a general theory of presentations of subshifts by labelled directed graphs, in particular by compact graphs. Also considered are synchronization properties of subshifts that lead to presentations by countable graphs.
Given a piecewise $C^{1+β}$ map of the interval, possibly with critical points and discontinuities, we construct a symbolic model for invariant probability measures with nonuniform expansion that do not approach the critical points and discontinuities exponentially fast almost surely. More specifically, we code the lift of these measures in the natural extension of the map.
We provide special cross sections for the Weyl chamber flow on a sample class of Riemannian locally symmetric spaces of higher rank, namely the direct product spaces of Schottky surfaces. We further present multi-parameter transfer operator families for the discrete dynamical systems on Furstenberg boundary that are related to these cross sections.
We construct countable Markov partitions for non-uniformly hyperbolic diffeomorphisms on compact manifolds of any dimension, extending earlier work of O. Sarig for surfaces. These partitions allow us to obtain symbolic coding on invariant sets of full measure for all hyperbolic measures whose Lyapunov exponents are bounded away from zero by a constant. Applications include counting results for hyperbolic periodic orbits, and structure of hyperbolic measures of maximal entropy.
This survey describes the recent advances in the construction of Markov partitions for nonuniformly hyperbolic systems. One important feature of this development comes from a finer theory of nonuniformly hyperbolic systems, which we also describe. The Markov partition defines a symbolic extension that is finite-to-one and onto a non-uniformly hyperbolic locus, and this provides dynamical and statistical consequences such as estimates on the number of closed orbits and properties of equilibrium measures. The class of systems includes diffeomorphisms, flows, and maps with singularities.
Heaviness refers to a sequence of partial sums maintaining a certain lower bound and was recently introduced and studied in "Heaviness: and Extension of a Lemma of Y. Peres." After a review of basic properties to familiarize the reader with the ideas of heaviness, general principles of heaviness in symbolic dynamics are introduced. The classical Morse sequence is used to study a specific example of heaviness in a system with nontrivial rational eigenvalues. To contrast, Sturmian sequences are examined, including a new condition for a sequence to be Sturmian.
This work constructs symbolic dynamics for non-uniformly hyperbolic surface maps with a set of discontinuities $D$. We allow the derivative of points nearby $D$ to be unbounded, of the order of a negative power of the distance to $D$. Under natural geometrical assumptions on the underlying space $M$, we code a set of non-uniformly hyperbolic orbits that do not converge exponentially fast to $D$. The results apply to non-uniformly hyperbolic planar billiards, e.g. Bunimovich stadia.
We consider shift spaces in which elements of the alphabet may overlap nontransitively. We define a notion of entropy for such spaces, give several techniques for computing lower bounds for it, and show that it is equal to a limit of entropies of (standard) full shifts. When a shift space with overlaps arises as a model for a discrete dynamical system with a finite set of overlapping neighborhoods, the entropy gives a lower bound for the topological entropy of the dynamical system.
We afford the problem of counting the blocks of a given length made with symbols drawn from an alphabet and relate this number to Fibonacci-like recurrent relations. The recurrence polynomia allows to calculate the limit ratio of two adjacent terms and this information is used to determine the topological entropy of a discrete numbers of associate shift spaces. We then describe a scheme to build a shift space with a pre-selected entropy.
We survey the connections between entropy, chaos, and independence in topological dynamics. We present extensions of two classical results placing the following notions in the context of symbolic dynamics: 1. Equivalence of positive entropy and the existence of a large (in terms of asymptotic and Shnirelman densities) set of combinatorial independence for shift spaces. 2. Existence of a mixing shift space with a dense set of periodic points with topological entropy zero and without ergodic measure with full support, nor any distributionally chaotic pair. Our proofs are new and yield conclusion…
We construct cross sections for the geodesic flow on the orbifolds $Γ\backslash H$ which are tailor-made for the requirements of transfer operator approaches to Maass cusp forms and Selberg zeta functions. Here, $H$ denotes the hyperbolic plane and $Γ$ is a nonuniform geometrically finite Fuchsian group (not necessarily a lattice, not necessarily arithmetic) which satisfies an additional condition of geometric nature. The construction of the cross sections is uniform, geometric, explicit and algorithmic.
Path sets are spaces of one-sided infinite symbol sequences associated to pointed graphs (G_v_0), which are edge-labeled directed graphs G with a distinguished vertex v_0. Such sets arise naturally as address labels in geometric fractal constructions and in other contexts. The resulting set of symbol sequences need not be closed under the one-sided shift. this paper establishes basic properties of the structure and symbolic dynamics of path sets, and shows they are a strict generalization of one-sided sofic shifts.
Let $K$ be the Cantor set. We prove that arbitrarily close to a homeomorphism $T:K\rightarrow K$ there exists a homeomorphism $\widetilde T:K\rightarrow K$ such that the $α$-limit and the $ω$-limit of every orbit is a periodic orbit. We also prove that arbitrarily close to an endomorphism $T:K\rightarrow K$ there exists an endomorphism $\widetilde T:K\rightarrow K$ close to $T$ such that every orbit is finally periodic.
An adequate characterization of the dynamics of Hamiltonian systems at physically relevant scales has been largely lacking. Here we investigate this fundamental problem and we show that the finite-scale Hamiltonian dynamics is governed by effective dynamical invariants, which are significantly different from the dynamical invariants that describe the asymptotic Hamiltonian dynamics. The effective invariants depend both on the scale of resolution and the region of the phase space under consideration, and they are naturally interpreted within a framework in which the nonhyperbolic dynamics of th…
In this paper, we mainly elucidate a close relationship between the topological entropy and mean dimension theory for actions of polynomial growth groups. We show that metric mean dimension and mean Hausdorff dimension of subshifts with respect to the lower rank subgroup are equal to its topological entropy multiplied by the growth rate of the subgroup. Meanwhile, we also prove the above result holds for the rate distortion dimension of subshifts with respect to the lower rank subgroup and measure entropy. Furthermore, some relevant examples are indicated.
The planar $N$-centre problem describes the motion of a particle moving in the plane under the action of the force fields of $N$ fixed attractive centres: \[ \ddot{x}(t)=\sum_{j=1}^N\nabla V_j(x-c_j). \] In this paper we prove symbolic dynamics at slightly negative energy for an $N$-centre problem where the potentials $V_j$ are positive, anisotropic and homogeneous of degree $-α_j$: \[ V_j(x)=|x|^{-α_j}V_j\left(\frac{x}{|x|}\right). \] The proof is based on a broken geodesics argument and trajectories are extremals of the Maupertuis' functional. Compared with the classical $N$-centre problem w…
A `symbolic dynamical system' is a continuous transformation F:X-->X of a closed perfect subset X of A^V, where A is a finite set and V is countable. (Examples include subshifts, odometers, cellular automata, and automaton networks.) The function F induces a directed graph structure on V, whose geometry reveals information about the dynamical system (X,F). The `dimension' dim(V) is an exponent describing the growth rate of balls in the digraph as a function of their radius. We show: if X has positive entropy and dim(V)>1, and the system (A^V,X,F) satisfies minimal symmetry and mixing condition…
WE trace the beginning of symbolic dynamics--the study of the shift dynamical system--as it arose from the use of coding to study recurrence and transitivity properties of geodesics. The normal citations for the first appearance of symbolic dynamics, Hadamard's 1898 paper and the 1938 and 1940 papers of Morse and Hedlund, don't truly represent the abstract point of view associated with the subject today. Based in part on the evidence of a 1941 letter from Hedlund to Morse, we place the beginning of symbolic dynamics in a 1944 paper by Hedlund.
We present a strategy for a geometric construction of cross sections for the geodesic flow on locally symmetric orbifolds of rank one. We work it out in detail for $Γ\backslash H$, where $H$ is the upper half plane and $Γ=\PGamma_0(p)$, $p$ prime. Its associated discrete dynamical system naturally induces a symbolic dynamics on $\R$. The transfer operator produced from this symbolic dynamics has a particularly simple structure.
We discuss a synchronization property for subshifts, that we call $λ$-synchronization. Under an irreducibility assumption we associate to a $λ$-synchronizing subshift a simple and purely infinite $C^*$-algebra.
We study classes of musical scales obtained from shift spaces in symbolic dynamics through the first symbol rule, which yields scales in any $n$-TET tuning system. The modes are thought as elements of orbit equivalence classes of cyclic shift actions on languages, and we study their orbitals and transversals. We present explicit formulations of the generating functions that allow us to deduce the orbital and transversal dimensions of classes of musical scales generated by vertex shifts, for all $n$, in particular for the 12-TET tuning system.
In an effort to aid communication among different fields and perhaps facilitate progress on problems common to all of them, this article discusses hidden Markov processes from several viewpoints, especially that of symbolic dynamics, where they are known as sofic measures, or continuous shift-commuting images of Markov measures. It provides background, describes known tools and methods, surveys some of the literature, and proposes several open problems.
We consider a specific %piecewise rotation of the plane that is continuous on two half-planes, class of piecewise rotations of the plane that are continuous on two half-planes, as studied in \cite{Bosh.Goet.03}, \cite{Goet.Quas.09} and \cite{Che.Goe.Qua.12}. %If Assuming that the angle belongs to the set $\{\fracπ{2},\fracπ{3},\fracπ{6},\fracπ{4}\} %$ $, we give a description of the symbolic dynamics of this map in the bijective symmetric case.
We study the central objects of symbolic dynamics, that is, subshifts and block maps, from the perspective of basic category theory, and present several natural categories with subshifts as objects and block maps as morphisms. Our main goals are to find universal objects in these symbolic categories, to classify their block maps based on their category theoretic properties, and to establish as many natural properties (finite completeness, regularity etc.) as possible. Existing definitions in category theory suggest interesting new problems for block maps. Our main technical contributions are t…
We construct symbolic dynamics on sets of full measure (w.r.t. an ergodic measure of positive entropy) for $C^{1+ε}$ flows on compact smooth three-dimensional manifolds. One consequence is that the geodesic flow on the unit tangent bundle of a compact $C^\infty$ surface has at least const $\times(e^{hT}/T)$ simple closed orbits of period less than $T$, whenever the topological entropy $h$ is positive -- and without further assumptions on the curvature.
For n convex magnetic bumps in the plane, whose boundary has a curvature somewhat smaller than the absolute value of the constant magnetic field inside the bump, we construct a complete symbolic dynamics of a classical particle moving with speed one.
