Statistical stability of interval maps with critical points and singularities

TL;DR

This paper establishes strong statistical stability for a broad class of one-dimensional maps with critical points, singularities, and discontinuities, extending previous results and introducing a new metric for analyzing such maps.

Contribution

It generalizes known stability results to maps with multiple discontinuities and singularities, and introduces a novel metric on the space of such maps.

Findings

Proves statistical stability for maps with critical points and singularities.

Extends stability results to maps with multiple discontinuities.

Introduces a new metric on the space of maps with discontinuities.

Abstract

We prove strong statistical stability of a large class of one-dimensional maps which may have an arbitrary finite number of discontinuities and of non-degenerate critical points and/or singular points with infinite derivative, and satisfy some expansivity and bounded recurrence conditions. This generalizes known results for maps with critical points and bounded derivatives and in particular proves statistical stability of Lorenz-like maps with critical points and singularities studied in [S. Luzzatto and W. Tucker. Non-uniformly expanding dynamics in maps with singularities and criticalities. Inst. Hautes Etudes Sci. Publ. Math., (89):179-226, 1999]. We introduce a natural metric on the space of maps with discontinuities which does not seem to have been used in the literature before.