Invariant Densities for Random Maps of the Interval

Mechanisms for Phase Transitions in the Multifractal Analysis of Invariant Densities of Correlated Random Maps *

für Naturforschung in cooperation with the Max Planck Society for the Advancement of Science under a Creative Commons Attribution 4.0 International License. Dieses Werk wurde im Jahr 2013 vom Verlag Zeitschrift für Naturforschung in Zusammenarbeit mit der Max-Planck-Gesellschaft zur Förderung der Wissenschaften e.V. digitalisiert und unter folgender Lizenz veröffentlicht: Creative Commons Namen...

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

Ulam's method for random interval maps

We consider the approximation of absolutely continuous invariant measures (ACIMs) of systems defined by random compositions of piecewise monotonic transformations. Convergence of Ulam’s finite approximation scheme in the case of a single transformation was dealt with by Li (1976 J. Approx. Theory 17 177–86). We extend Ulam’s construction to the situation where a family of piecewise monotonic tr...

Regularity of Invariant Densities for 1d-systems with Random Switching

This is a detailed analysis of invariant measures for one-dimensional dynamical systems with random switching. In particular, we prove smoothness of the invariant densities away from critical points and describe the asymptotics of the invariant densities at critical points.