Absolutely continuous invariant measures for expanding piecewise linear maps

Abstract

We prove the existence of absolutely continuous invariant measures for arbitrary expanding piecewise linear maps on bounded polyhedral domains in Euclidean spaces R d .

Less Is More I: A Pessimistic View of Piecewise Smooth Bifurcation Theory

“…Using some beautiful technical results of Buzzi and Tsujii [1,14] it is possible to prove bifurcations from fixed points to n-dimensional attractors in these maps.…”

Switched flow systems: pseudo billiard dynamics

“…Observe that the "cut" system is again piecewise expanding and is piecewise linear on domains consisting of a finite number of simplexes. It is known (Tsujii 2001) that under these assumptions Figure 5: Existence of a stable periodic trajectory under "cutting" the map has an absolutely continuous (with respect to Lebesgue measure) invariant measure, i.e. it is strongly chaotic.…”

Spectrum and Statistical Properties of Chaotic Dynamics

“…Buzzi and Keller [28] successfully carried through the analysis of the dynamical zeta function ζ 1/| det Df | (z) (2) when f is a piecewise affine and expanding surface transformation (Jérôme Buzzi informed us that this can be extended to arbitrary dimensions by using Tsujii's [95] work.) They prove that the zeta function is analytic in the open unit disc and meromorphic in a disc of larger radius, where its poles are the inverse eigenvalues of the associated transfer operator acting on functions of higher-dimensional bounded variation.…”