Related papers: Symbolic dynamics for Lozi maps

The periodic orbits of the strongly chaotic cardioid billiard are studied by introducing a binary symbolic dynamics. The corresponding partition is mapped to a topological well-ordered symbol plane. In the symbol plane the pruning front is…

We prove the holding of chaos in the sense of Li-Yorke for a family of four-dimensional discrete dynamical systems that are naturally associated to ODE systems describing coupled oscillators subject to an external non-conservative force,…

For the family of Double Standard Maps $f_{a,b}=2x+a+\frac{b}{\pi} \sin2\pi x \quad\pmod{1}$ we investigate the structure of the space of parameters $a$ when $b=1$ and when $b\in[0,1)$. In the first case the maps have a critical point, but…

We construct classes in the motivic cohomology of certain 1-parameter families of Calabi-Yau hypersurfaces in toric Fano n-folds, with applications to local mirror symmetry (growth of genus 0 instanton numbers) and inhomogeneous…

We give an introduction to the "stable algebra of matrices" as related to certain problems in symbolic dynamics. We consider this stable algebra (especially, shift equivalence and strong shift equivalence) for matrices over general rings as…

Piecewise-linear maps describe dynamical phenomena that switch between distinct states and readily generate complex bifurcation structures due to their strong nonlinearity. We show that two-dimensional continuous piecewise-linear maps near…

We show that special perturbations of a particular holomorphic map on $\mathbf{P}^k$ give us examples of maps that possess chaotic nonalgebraic attractors. Furthermore, we study the dynamics of the maps on the attractors. In particular, we…

We develop dynamical theory for the family of holomorphic correspondences $\mathcal{F}_a$ proved by the current authors to be matings between the modular group and parabolic rational maps in the Milnor slice $Per_1(1)$ (in 'Mating quadratic…

We study the action of a relatively hyperbolic group on its boundary, by methods of symbolic dynamics. Under a condition on the parabolic subgroups, we show that this dynamical system is finitely presented. We give examples where this…

Given a proper, rational map of balls, D'Angelo and Xiao introduced five natural groups encoding properties of the map. We study these groups using a recently discovered normal form for rational maps of balls. Using this normal form, we…

We describe results on the dynamics of polynomial diffeomorphisms of ${\bf C^2}$ and draw connections with the dynamics of polynomial maps of ${\bf C}$ and the dynamics of polynomial diffeomorphisms of ${\bf R^2}$ such as the H\'enon…

We study the evolution of the probability density of ensembles of iterates of the logistic map that advance towards and finally remain at attractors of representative dynamical regimes. We consider the mirror families of superstable…

This paper surveys the recent advances in the interactions between symbolic dynamics and C*-algebras. We explain how conjugacies and orbit equivalences of both two-sided (invertible) and one-sided (noninvertible) symbolic systems may be…

We prove that almost every non-regular real quadratic map is Collet-Eckmann and has polynomial recurrence of the critical orbit (proving a conjecture by Sinai). It follows that typical quadratic maps have excellent ergodic properties, as…

We study the dynamics of the one-dimensional quasi-affine map $x\mapsto \left\lfloor \lambda x +\mu \right\rfloor$, providing a complete description of the map's periodic points, and of the limit points of every $x\in\mathbb{R}$ under the…

We develop an abstract model for the dynamics of an exponential map $z\mapsto \exp(z)+\kappa$ on its set of escaping points and, as an analog of Boettcher's theorem for polynomials, show that every exponential map is conjugate, on a…

We consider families of piecewise linear maps in which the moduli of the two slopes take different values. In some parameter regions, despite the variations in the dynamics, the Lyapunov exponent and the topological entropy remain constant.…

We study in this paper the behavior of a periodically driven nonlinear mechanical system. Bifurcation diagrams are found which locate regions of quasiperiodic, periodic and chaotic behavior within the parameter space of the system. We also…

This survey is a presentation of the arguments in the proof that Henon-like maps f_a(x,y)=(1-a x^2,0) + R(a,x,y) with |R(a,x,y)|< b have a "strange attractor", with positive Lebesgue probability in the parameter "a", if the perturbation…

Many of the 2-adic properties of the 3x+1 map generalize to the analogous mx+r map, where m and r are odd integers. We introduce the corresponding autoconjugacy map, prove some simple properties of it and make some further conjectures in…