Global invertibility of expanding maps

A Survey of Invertibility and Spectrum Preserving Linear Maps

Invertibility Preserving Linear Maps of Banach Algebras

This talk discusses a conjecture of R. V. Kadison and myself. Our conjecture is that each one-to-one linear map of one unital C*-algebra onto another that preserves the identity is a Jordan isomorphism if it maps the invertible elements of the first C*-algebra onto the invertible elements of the other C*-algebra. Connections are shown between this conjecture and Cartan’s uniqueness theorem. 1. ...

Nonuniformly Expanding 1D Maps

This paper attempts to make accessible a body of ideas surrounding the following result: Typical families of (possibly multi-model) 1-dimensional maps passing through “Misiurewicz points” have invariant densities for positive measure sets of parameters. The simplest paradigms of chaotic behavior in dynamical systems are found in uniformly expanding and uniformly hyperbolic (or Anosov) maps. All...

a survey of invertibility and spectrum preserving linear maps

Invertibility-preserving Maps of C∗-algebras with Real Rank Zero

In 1996, Harris and Kadison posed the following problem: show that a linear bijection between C∗-algebras that preserves the identity and the set of invertible elements is a Jordan isomorphism. In this paper, we show that if A and B are semisimple Banach algebras andΦ : A→ B is a linear map onto B that preserves the spectrum of elements, thenΦ is a Jordan isomorphism if either A or B is a C∗-al...