Abstract: For an expanding circle map of degree at least 2, its Lyapunov spectrum is 
defined as the set of all Lyapunov exponents (multipliers) at periodic orbits. This set 
is analogous to the unmarked length spectrum of negatively curved metrics on surfaces 
of genus at least 2. Motivated by the unmarked length spectral rigidity conjecture for 
metrics, in the talk we will discuss whether the Lyapunov spectrum defines a smooth 
conjugacy class of the corresponding expanding circle map. In general this is false 
(we will present a counterexample). However, the following local rigidity holds: every 
C^r smooth expanding circle map f has a neighborhood (in C^r topology) such that any 
perturbation of f within this neighborhood that keeps the Lyapunov spectrum must be 
smoothly conjugate to f (subject to some sparsity assumption on the spectrum on f). 
The proof uses a novel iterative scheme which we will outline in the talk. 
This is joint work in progress with Vadim Kaloshin.