Large derivatives, backward contraction and invariant densities for interval maps

Synthesizing Chaotic Maps with Prescribed Invariant Densities

The Inverse Frobenius-Perron problem (IFPP) concerns the creation of discrete chaotic mappings with arbitrary invariant densities. In this note, we present a new and elegant solution to the IFPP, based on positive matrix theory. Our method allows chaotic maps with arbitrary piecewise-constant invariant densities, and with arbitrary mixing properties, to be synthesized.

Orthonormal Expansions of Invariant Densities for Expanding Maps

Invariant Measures for Interval Maps with Critical Points and Singularities

We prove that, under a mild summability condition on the growth of the derivative on critical orbits any piecewise monotone interval map possibly containing discontinuities and singularities with infinite derivative (cusp map) admits an ergodic invariant probability measures which is absolutely continuous with respect to Lebesgue measure.

Invariant densities of delayed maps in the limit of large time delay.

The marginal invariant density of chaotic attractors of scalar systems with time delayed feedback has an asymptotic form in the limit of large delay. It is well known that the dimension and the entropy of such attractors obey interesting scaling laws in this limit, but very little has been said about properties of the invariant density. We present general considerations, detailed analytical res...

Scaling properties of invariant densities of coupled Chebyshev maps

We study 1-dimensional coupled map lattices consisting of diffusively coupled Chebyshev maps of N -th order. For small coupling constants a we determine the invariant 1-point and 2-point densities of these nonhyperbolic systems in a perturbative way. For arbitrarily small couplings a > 0 the densities exhibit a selfsimilar cascade of patterns, which we analyse in detail. We prove that there are...