Related papers: Newhouse phenomena in the Fibonacci trace map
This paper will study topological, geometrical and measure-theoretical properties of the real Fibonacci map. Our goal was to figure out if this type of recurrence really gives any pathological examples and to compare it with the infinitely…
Trace maps of two-letter substitution rules are investigated with special emphasis on the underlying algebraic structure and on the existence of invariants. We illustrate the results with the generalized Fibonacci chains and show that the…
We consider the trace map associated with the Fibonacci Hamiltonian as a diffeomorphism on the invariant surface associated with a given coupling constant and prove that the non-wandering set of this map is hyperbolic if the coupling is…
The only recurrence sequence satisfying the Fibonacci recurrence and realizable as the number of periodic points of a map is (a multiple of) the Lucas sequence.
Properties of the phase space of the standard map with memory are investigated. This map was obtained from a kicked fractional differential equation. Depending on the value of the parameter of the map and the fractional order of the…
We study the dynamics of a family of continued fraction maps parametrized by the unit interval. This family contains as special instances the Gauss continued fraction map and the Fibonacci map. We determine the transfer operators of these…
We show that the combinatorial Lefschetz number is a topological invariant. This is an important result in itself; in order to point it out, we will also work here several relevant consequences in different directions. The first of them is…
A relevant problem in dynamics is to characterize how deterministic systems may exhibit features typically associated to stochastic processes. A widely studied example is the study of (normal or anomalous) transport properties for…
We consider diffeomorphisms of compact Riemmanian manifolds which have a Gibbs-Markov-Young structures, consisting of a reference set $\Lambda$ with a hyperbolic product structure and a countable Markov partition. We assume polynomial…
We study the topological dynamics by iterations of a piecewise continuous, non linear and locally contractive map in a real finite dimensional compact ball. We consider those maps satisfying the "separation property": different continuity…
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…
Invariant tori are prominent features of symplectic and volume preserving maps. From the point of view of chaotic transport the most relevant tori are those that are barriers, and thus have codimension one. For an $n$-dimensional…
We describe the statistical properties of the dynamics of the quadratic polynomials P_a(z):=e^{2\pi a i} z+z^2 on the complex plane, with a of high return times. In particular, we show that these maps are uniquely ergodic on their measure…
We study the dynamics of the five-parameter quadratic family of volume-preserving diffeomorphisms of R^3. This family is the unfolded normal form for a bifurcation of a fixed point with a triple-one multiplier and also is the general form…
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…
The purpose of this paper is to study some properties of the Newton maps associated to real quintic polynomials. First using the Tschirnhaus transformation, we reduce the study of Newton's method for general quintic polynomials to the case…
Completely positive trace preserving maps are widely used in quantum information theory. These are mostly studied using the master equation perspective. A central part in this theory is to study whether a given system of dynamical maps…
We consider the homotopical dynamics on compact orientable surfaces of positive genus g. We establish a sufficient and necessary algebraic criterion for homotopy classes with infinitely many periodic points of maps on such surfaces in terms…
Dynamical fluctuations or rare events associated with atypical trajectories in chaotic maps due to specific initial conditions can crucially determine their fate, as the may lead to stability islands or regions in phase space otherwise…
We prove recurrence relations and modulo periodic properties of multiple derivatives of Fibonacci polynomials. We apply the obtained results to present the dynamic structures of Fibonacci polynomials over the ring of 2-adic integers by…