# Cantor sets and Cantor measures

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A subset of the real line is called a Cantor set if it is compact,

perfect, and nowhere dense. Cantor sets arise in many areas; in this

talk we will discuss their relevance in the spectral theory of

Schr\"odinger operators. We discuss several results showing that the

spectrum of such an operator is a Cantor set, from the discovery of the

first example by Moser to a genericity result by Avila, Bochi, and

Damanik. A Cantor measure is a probability measure on the real line

whose topological support is a Cantor set. A primary example in the

spectral theory context is the density of states measure in situations

where the spectrum is a Cantor set. A conjecture of Simon claims a

strict inequality between the dimensions of the set and the measure for

the Fibonacci potential. If time permits, we will discuss a recent

result of Damanik, Gorodetski, and Yessen, which establishes this

conjecture in full generality.