| Abstract: |
| Given an entropy preserving factor map between two irreducible shifts of finite type $X$ and $Y$, there is a number $d$, called the degree of $f$, such that almost all points in $Y$ have $d$ preimages in $X$. The factor map naturally induces a factor map between the sets of invariant measures of $X$ and $Y$, and it is folklore that each fully supported ergodic measure on $Y$ has at most $d$ ergodic preimage measures on $X$. This number may vary: For example, if a given measure is Markov, then it lifts to a unique measure which is also Markov (Boyle and Tuncel). Recently, Yoo defined the notion of the multiplicity of each lifted measures and showed that the degree is the sum of the multiplicity of all ergodic measures over it. In this talk after explaining the history and known results regarding the multiplicity structure of invariant measures, we show that there is a residual set of ergodic invariant measures on $Y$ number of whose ergodic invariant measures is exactly $d$. |
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