On the approach to the final aperiodic regime in maps of the interval

We present a global approach to the final aperiodic regime in maps of the interval displaying a simple pattern with similarities to the Feigenbaum scheme.     

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.     

Invariant measures for Markov maps of the interval

There is a theorem in ergodic theory which gives three conditions sufficient for a piecewise smooth mapping on the interval to admit a finite invariant ergodic measure equivalent to Lebesgue. When the hypotheses fail in certain ways, this work shows that the same conclusion can still be gotten by applying the theorem mentioned to another transformation related to the original one by the method of inducing.Partially supported by NSF MCS74-19388. A01     

Universal properties of maps on an interval

We consider itcrates of maps of an interval to itself and their stable periodic orbits. When these maps depend on a parameter, one can observe period doubling bifurcations as the parameter is varied. We investigate rigorously those aspects of these bifurcations which are universal, i.e. independent of the choice of a particular one-parameter family. We point out that this universality extends to many other situations such as certain chaotic regimes. We describe the ergodic properties of the maps for which the parameter value equals the limit of the bifurcation points.     

Singular perturbations of piecewise monotonic maps of the interval

Let t: [0, 1] [0, 1] be a piecewise monotonic, C₂, and expanding map. In computing an orbit { ⁱ (x ₀)} _i=0 , we model the roundoff error at each iteration by a singular perturbation; i.e.,X _n+1=(X _n )+W , whereW is a random variable taking on discrete values in an interval (-ε, ). The main result proves that this process admits an absolutely continuous invariant measure which approaches the absolutely continuous measure invariant under the deterministic map t as the precision of computation 0.     

Critical-point dependence of universality in maps of the interval

The variation of the Feigenbaum ratios with the degree of degeneracy of the critical point is studied using analytical approximations and numerical calculations.     

Universal scaling behaviour for iterated maps in the complex plane

According to the theory of Schröder and Siegel, certain complex analytic maps possess a family of closed invariant curves in the complex plane. We have made a numerical study of these curves by iterating the map, and have found that the largest curve is a fractal. When the winding number of the map is the golden mean, the fractal curve has universal scaling properties, and the scaling parameter differs from those found for other types of maps. Also, for this winding number, there are universal scaling functions which describe the behaviour asn→∞ of theQ _n ^th iterates of the map, whereQ _n is then ^th Fibonacci number.     

Convergence of the natural approximations of piecewise monotone interval maps

We consider piecewise monotone interval mappings which are topologically mixing and satisfy the Markov property. It has previously been shown that the invariant densities of the natural approximations converge exponentially fast in uniform pointwise topology to the invariant density of the given map provided its derivative is piecewise Lipshitz continuous. We provide an example of a map which is Lipshitz continuous and for which the densities converge in the bounded variation norm at a logarithmic rate. This shows that in general one cannot expect exponential convergence in the bounded variation norm. Here we prove that if the derivative of the interval map is Holder continuous and its variation is well approximable (gamma-uniform variation for gamma>0), then the densities converge exponentially fast in the norm.     

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.     

On the fixed points for circle maps

Renormalization group transformations have been developed to study the critical behavior of circle maps. When the winding number equals the golden mean, the fixed point functions must satisfy two functional equations. However, it turns out that one of these equations already determines the fixed point solutions. It is shown that under certain conditions the second functional equation is automatically satisfied.     

Universal behaviour in families of area-preserving maps

We have investigated numerically the behaviour, as a perturbation parameter is varied, of periodic orbits of some reversible area-preserving maps of the plane. Typically, an initially stable periodic orbit loses its stability at some parameter value and gives birth to a stable orbit of twice the period. An infinite sequence of such bifurcations is accomplished in a finite parameter range. This period-doubling sequence has a universal limiting behaviour: the intervals in parameter between successive bifurcations tend to a geometric progression with a ratio of $\text{1}\text{δ}\text{=}\text{1}\text{8.721097200}$…, and when examined in the proper coordinates, the pattern of periodic points reproduces itself, asymptotically, from one bifurcation to the next when the scale is expanded by α = ?4.018076704… in one direction, and by β = 16.363896879… in another. Indeed, the whole map, including its dependence on the parameter, reproduces itself on squaring and rescaling by the three factors α, β and δ above. In the limit we obtain a universal one-parameter, area-preserving map of the plane. The period-doubling sequence is found to be connected with the destruction of closed invariant curves, leading to irregular motion almost everywhere in a neighbourhood.     

On universality for area-preserving maps of the plane

We present detailed evidence that one-parameter families of area-preserving maps exhibit cascades of period doubling with universal geometric scaling in the parameter. We relate this behaviour to a fixed point equation of the form

$\text{Λ}{}^{\mathrm{?1}}\text{°Φ°Φ°Λ = Φ}$

and

$\text{det}\text{DΦ = 1}$

, Φ:$\text{R}$²→$\text{R}$². In particular we argue that the scaling transformation Λ:$\text{R}$²→$\text{R}$² is conjugate to the transformation Λ₀:(x, y)→(λx, μy), with λ² ≠ μ, and in fact λ² >μ. We present some numerical evidence that

$\text{δ = 8.721}$

…,

$\text{?}\text{1}\text{λ}\text{= 4.018}$

…,

$\text{1}\text{μ}\text{= 16.36}$

…, where δ is the asymptotic ratio of the differences of the parameter values corresponding to the successive periods 2^k described above.     

A parameter renormalization for orbit-splittings and band-mergings of iterated maps of an interval

A renormalization scheme based directly on a spatial scaling is presented for a parameter which shows the location of the maximum in the “confining square” of iterated maps of an interval into itself. The cause and the properties of band-mergings are made especially clear in the geometrical set-up which allows one to see the symmetry between the forward and the reverse bifurcations (the orbit-splittings and the band-mergings) in an obvious way and yields an economical estimate of Feigenbaum's ratios. The method is applicable to any sequences and to arbitrary maxima of the mapping function. If the maximum is a cusp, a pseudo fixed “point” causes a finite number of pairwise bandmergings which follow the same number of foregoingband-splittings.     

Computing the topological entropy of maps of the interval with three monotone pieces

An algorithm is presented for computing the topological entropy of a piecewise monotone map of the interval having three monotone pieces. The accuracy of the algorithm is discussed and some graphs of the topological entropy obtained using the algorithm are displayed. Some of the ideas behind the algorithm have application to piecewise monotone functions with more than three monotone pieces.     

Algorithmic information for interval maps with an indifferent fixed point and infinite invariant measure

Measuring the average information that is necessary to describe the behavior of a dynamical system leads to a generalization of the Kolmogorov-Sinai entropy. This is particularly interesting when the system has null entropy and the information increases less than linearly with respect to time. We consider a class of maps of the interval with an indifferent fixed point at the origin and an infinite natural invariant measure. We show that the average information that is necessary to describe the behavior of the orbits increases with time n approximately as nalpha, where alpha < 1 depends only on the asymptotic behavior of the map near the origin.     

On the transition to aperiodic motion in linear systems with strong relaxation

The Bloch equations are shown to possess purely aperiodic solutions at intermediate values of the transverse relaxation coefficient only. We present also a simple example of a mechanical system where the regions of aperiodic motion are separated by the region of damped oscillations.     

On the essential spectrum of the transfer operator for expanding markov maps

The essential spectrum of the transfer operator for expanding markov maps of the interval is studied in detail. To this end we construct explicityly an infinite set of eigenfunctions which allows us to prove that the essential spectrum inC ^k is a disk whose radius is related to the free energy of the Liapunov exponent.