Measure Of Maximal Entropy Research Articles

Semi-rigidity of horocycle flows over compact surfaces of variable negative curvature

AbstractLet g be the geodesic flow on the unit tangent bundle of a C3 compact surface of negative curvature. Let μ be the g-invariant measure of maximal entropy. Let h be a uniformly parametrized flow along the horocycle foliation, i.e., such a flow exists, leaves μ invariant, and is unique up to constant scaling of the parameter (Margulis). We show that any measure-theoretic conjugacy: (h, μ) → (h′, μ′) is a.e. of the form θ, where θ is a homeomorphic conjugacy: g → g′. Furthermore, any homeomorphic conjugacy g → g′; must be a C1 diffeomorphism.

Equilibrium measures for rational maps

AbstractFor a polynomial map the measure of maximal entropy is the equilibrium measure for the logarithm potential in the Julia set [1], [4].Here we will show that in the case where f is a rational map such that f(∞) = ∞ and the Julia set is bounded, then the two measures mentioned above are equal if and only if f is a polynomial.

Synchronisation of canonical measures for hyperbolic attractors

Under suitable conditions it is shown how to change the velocity of aC 2 AxiomA attractor so that the Sinai-Ruelle-Bowen measure coincides with the measure of maximal entropy. These measures are obtained as limits of certain closed orbital measures.

Information flow and maximum entropy measures for 1-D maps

The invariant measures of maximal metric entropy are constructed explicitly for some maps of the interval, by iterating the maps backward. The construction illustrates in a particularly clear way the information flow in simple systems, as well as recently conjectured relationships between dimensions of invariant measures, Lyapunov exponents, and entropies. maps, it is conjectured that the natural measure is the invariant measure with strongest mixing.

Entropy properties of rational endomorphisms of the Riemann sphere

AbstractIn this paper the existence of a unique measure of maximal entropy for rational endomorphisms of the Riemann sphere is established. The equidistribution of pre-images and periodic points with respect to this measure is proved.

The maximum-entropy measure of a rational endomorphism of the Riemann sphere

The maximum-entropy measure of a rational endomorphism of the Riemann sphere

Topological Entropy for Geodesic Flows

There have been many results in recent years showing that the topological entropy of a diffeomorphism [5], [16], [23], or a continuous map [11], [12], [17], is at least the logarithm of certain eigenvalues of the map it induces in real homology. These relationships, conjectured by Shub [21, p. 37], are part of an effort to find connections between the dynamics of a discrete system and the topology of the underlying manifold [20], [22]. There do not seem to have been any results of this type for flows because the time one map of any flow is isotopic to the identity, making the natural generalization of Shub's conjecture vacuous. In this paper we present two results linking the topological entropy h(q) of the geodesic flow qS on a compact Riemannian manifold M with the exponential growth rate X of volume in the universal cover M: (1) h(o) ? X and (2) h(q) = X if M has non-positive curvature. The technique is to look for separated and spanning sets of geodesics using the Hopf-Rinow theorem for (1) and the way geodesics spread out in the case of non-positive curvature for (2). Our first result may be regarded as a strengthening of Section 4 of [8] where Dinaburg showed that h(q) > 0 if w1(M) has exponential growth while the second simplifies [13] where Margulis began by construcing the measure of maximal entropy [14]. I should like to thank the staff at IMPA Rio de Janeiro, where this work was done, for their hospitality during the year 1977-78 and Manfredo do Carmo, Ricardo Mahe6 and Floris Takens for helpful remarks.

On the distribution of periodic points for β-shifts

It is shown that for β-shifts the periodic points are uniformly distributed with respect to the unique measure of maximal entropy, and that the invariant measures with support on a single periodic orbit are dense in the space of all invariant measures.

An Entropy Measure for Partially Polarized Radiation and its Application to Estimating Radio Sky Polarization Distributions from Incomplete 'Aperture Synthesis' Data by the Maximum Entropy Method

To apply the maximum entropy method (for obtaining an estimate of a power spectrum from an incomplete estiniate of its autocorrelation function) to the task of mapping a radio sky polarization distribution from incomplete aperture synthesis'' data, it is necessary to have a measure of the relative entropy in a polarization map. Such a measure is derived. It allows the change in entropy measure due to a variation in intensity to be compared with that due to a change in the degree of polarization. Using the new measure the variational problem for choosing the maximum entropy estimate of a polarized sky distribution is set up, and some progress is made towards its solution. A field of given mean intensity is found to have maximum entropy measure when it is unpolarized and of uniform brightness. (auth)