Near-critical behavior for one-parameter families of circle maps

Past studies of systems showing mixed-mode oscillations have revealed behavior along arbitrarily chosen parameter paths similar to that on the critical surface marking the break-up of invariant tori. Observations of this behavior in a model of the Belousov-Zhabotinskii reaction is presented. Using the theory of circle maps, it is shown that near-critical behavior can arise along one-parameter paths.     

Induced maps,Markov extensions and invariant measures in one-dimensional dynamics

A way to study ergodic and measure theoretic aspects of interval maps is by means of the Markov extension. This tool, which ties interval maps to the theory of Markov chains, was introduced by Hofbauer and Keller. More generally known are induced maps, i.e. maps that, restricted to an element of an interval partition, coincide with an iterate of the original map.We will discuss the relation between the Markov extension and induced maps. The main idea is that an induced map of an interval map often appears as a first return map in the Markov extension. For S-unimodal maps, we derive a necessary condition for the existence of invariant probability measures which are absolutely continuous with respect to Lebesgue measure. Two corollaries are given.     

Polynomial invariant measures for maps on the unit interval

A class of polynomial solutions is found for a functional equation which certain invariant measures must satisfy. These solutions exist only for specific values of the parameter of the triangular map on the unit interval. Using this fact, a method is proposed for approximating the invariant measures for the standard quadratic map.     

Subexponential instability in one-dimensional maps implies infinite invariant measure

We characterize dynamical instability of weak chaos as subexponential instability. We show that a one-dimensional, conservative, ergodic measure preserving map with subexponential instability has an infinite invariant measure, and then we present a generalized Lyapunov exponent to characterize subexponential instability.     

Noise-amplitude dependence of the invariant density for noisy, fully chaotic one-dimensional maps

We present some analytic, nonperturbative results for the invariant density rho(x) for noisy one-dimensional maps at fully developed chaos. Under periodic boundary conditions, the Fourier expansion method is used to show precisely how noise makes rho(x) absolutely continuous and smooths it out. Simple solvable models are used to illustrate the explicit dependence of rho(x) on the amplitude eta of the noise distribution, all the way from the case of zero noise (eta-->0) to the completely noise-dominated limit (eta=1).     

Global attractors and invariant measures for non-invertible planar piecewise isometric maps

The authors investigate dynamical behaviors of discrete systems defined by iterating non-invertible planar piecewise isometries, which are piecewisely defined maps that preserve Euclidean distance. After discussing subtleties for these kind of dynamical systems, they have characterized global attractors via invariant measures and via positive continuous functions on phase space. The main result of this Letter is that a compact set A is the global attractor for a piecewise isometry if and only if the Lebesgue measure restricted to A is invariant, while it is not invariant restricted to any measurable set B which contains A and whose Lebesgue measure is strictly larger than that of A.     

The projection method for computing multidimensional absolutely continuous invariant measures

We present an algorithm for numerically computing an absolutely continuous invariant measure associated with a piecewiseC ² expanding mappingS: on a bounded region R ^N. The method is based on the Galerkin projection principle for solving an operator equation in a Banach space. With the help of the modern notion of functions of bounded variation in multidimension, we prove the convergence of the algorithm.     

Percival variational principle for invariant measures and commensurate-incommensurate phase transitions in one-dimensional chains

We prove for a one-dimensional system of classical particles with potential energy, $$U_{\alpha ,\gamma } = \sum\limits_n {\left[ {\alpha V(x_n ) + F(x_{n + 1} - x_n - \gamma )} \right]} $$ , the existence of such a smooth function γ(α), 0≦α≦α₀(ω) that the system with potential energyU _αγ(α) has the equilibrium state at the temperatureT=0. This is the incommensurate phase with the ratio of periods equal to the prescribed irrational number ω, badly approximated by rational ones. A simple geometric condition for the invariant curve of the corresponding dynamical system is established under which it is the support of the invariant measure minimizing Percival's energy functional.     

Hierarchy of rational order families of chaotic maps with an invariant measure

We introduce an interesting hierarchy of rational order chaotic maps that possess an invariant measure. In contrast to the previously introduced hierarchy of chaotic maps [1–5], with merely entropy production, the rational order chaotic maps can simultaneously produce and consume entropy. We compute the Kolmogorov-Sinai entropy of these maps analytically and also their Lyapunov exponent numerically, where the obtained numerical results support the analytical calculations.     

On the existence of invariant,absolutely continuous measures

Let (, , ) be a measure space with normalized measure,f: a nonsingular transformation. We prove: there exists anf-invariant normalized measure which is absolutely continuous with respect to if and only if there exist >0, and , 0<<1, such that (E)< implies (f ^–k(E))< for allk0.     

Hyperbolicity and invariant measures for generalC 2 interval maps satisfying the Misiurewicz condition

In this paper we will show that piecewiseC ² mappingsf on [0,1] orS ¹ satisfying the so-called Misiurewicz conditions are globally expanding (in the sense defined below) and have absolute continuous invariant probability measures of positive entropy. We do not need assumptions on the Schwarzian derivative of these maps. Instead we need the conditions thatf is piecewiseC ², that all critical points off are non-flat, and thatf has no periodic attractors. Our proof gives an algorithm to verify this last condition. Our result implies the result of Misiurewicz in [Mi] (where only maps with negative Schwarzian derivatives are considered). Moreover, as a byproduct, the present paper implies (and simplifies the proof of) the results of Mañé in [Ma], who considers generalC ² maps (without conditions on the Schwarzian derivative), and restricts attention to points whose forward orbit stay away from the critical points. One of the main complications will be that in this paper we want to prove the existence of invariant measures and therefore have to consider points whose iterations come arbitrarily close to critical points. Misiurewicz deals with this problem using an assumption on the Schwarzian derivative of the map. This assumption implies very good control of the non-linearity off ⁿ, even for highn. In order to deal with this non-linearity, without an assumption on the Schwarzian derivative, we use the tools of [M.S.]. It will turn out that the estimates we obtain are so precise that the existence of invariant measures can be proved in a very simple way (in some sense much simpler than in [Mi]). The existence of these invariant measures under such general conditions was already conjectured a decade ago.     

Regularity and other properties of absolutely continuous invariant measures for the quadratic family

In the current paper we study in more detail some properties of the absolutely continuous invariant measures constructed in the course of the proof of Jakobson's Theorem. In particular, we show that the density of the invariant measure is continuous at Misiurewicz points. From this we deduce that the Lyapunov exponent is also continuous at these points (our considerations apply just to the parameters constructed in the proof of Jakobson's Theorem). Other properties, like the positivity of the Lyapunov exponent, uniqueness of the absolutely continuous invariant measure and exactness of the corresponding dynamical system, are also proved.This paper was written during the author's stay at the IAS while supported by NSF grant DMS-860 1978     

On continuity of Bowen-Ruelle-Sinai measures in families of one dimensional maps

Let us consider a family of mapsQ _a (x)=ax(1?x) from the unit interval [0,1] to itself, wherea∈[0,4] is the parameter. We show that, for any β<2, there exists a subsetE?4 in [0,4] with the properties

1. Leb([4??,4]?E) < ?^β for sufficiently small ?>0,
2. Q _a admits an absolutely continuous BRS measure µ_a whena∈E, and
3. µ_a converges to the measure µ₄ asa tends to 4 on the setE. Also we give some generalization of this results.

     

Bifurcations from an invariant circle for two-parameter families of maps of the plane: A computer-assisted study

We consider a two-parameter family of maps of the plane to itself. Each map has a fixed point in the first quadrant and is a diffeomorphism in a neighborhood of this point. For certain parameter values there is a Hopf bifurcation to an invariant circle, which is smooth for parameter values in a neighborhood of the bifurcation point. However, computer simulations show that the corresponding invariant set fails to be even topologically a circle for parameter values far from the bifurcation point. This paper is an attempt to elucidate some of the mechanisms involved in this loss of smoothness and alteration of topological type.     

The invariant densities for maps modeling intermittency

For a one-parameter family of maps modeling intermittency the explicit formula of the invariant density is presented.     

On infinite direct products of continuous unitary one-parameter groups

We discuss the infinite product of unitary operators in an incomplete direct product of Hilbert spaces. Necessary and sufficient conditions are derived under which this infinite product leads to a continuous unitary one-parameter group provided each factor is assumed to have this property. A certain minimal condition guarantees the existence of a renormalized unitary group. An application is made to product representations of the canonical commutation relations in order to determine the admissible test functions.     

Geometry of geometrically finite one-dimensional maps

We study the geometry of certain one-dimensional maps as dynamical systems. We prove the property of bounded and bounded nearby geometry of certainC ¹⁺ one-dimensional maps with finitely many critical points. This property enables us to give the quasisymmetric classification of geometrically finite one-dimensional maps.     

New graphical method for the iteration of one-dimensional maps

We present the “inverse-curve method” which (1) simplifies a graphical iteration of the logistic map by avoiding the use of the diagonal; (2) naturally demonstrates the first period-doubling bifurcation; and (3) allows one to read immediately the stable pair of the two-point cycle off from the graph of the recursive function. Cycles of order m 2 are reduced to graphs of k- and l-iterate functions with k + l = m. The method can easily be applied to other one-dimensional maps with more complicated recursion relations.