New development of entropy theory: I: Regularity
Entropy is usually used to measure the complicity of a system, it was introduced into dynamical system first by Kolmogrov.
After the work of Sinai, Ruelle, Bowen, Katok, Ledrappier, Young, people began to realize that understanding the entropy of a system is a key problem for ergodic theory.
The regularity of entropy is an important and complicated problem: which is closely associated to the regularity of the maps. There are C^r (r<∞) counter example to show that, the entropy is neither upper, nor lower semi-continuous. And by Yomdin's work, the entropy varies in a upper semi-continuous manner respect to the maps.
In this talk, we will show several new results on the continuity of entropy for maps with lower regularity (only  C¹). We will also show some applications.
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The second talk:
New development of entropy theory: II: Partial entropy and invariant principle
The partial entropy was defined in the Ledrappier-Young's entropy formula for C² diffeomorphisms, which denotes the entropy along the Pesin unstable lamination. We generalize this definition to expanding foliation of any C¹ diffeomorphism.
In this talk, we will show the regularity of the partial entropy: which varies upper semi-continuously with respect to the
measures and diffeomorphisms. We will also show some surprising corollaries of this regularity: The Gibbs u-states of $C¹⁺$ partially hyperbolic diffeomorphisms vary continuously in the C¹ topology.
With Tahzibi, we build the relation of partial entropy with the Avila-Viana's invariant principle, and provide a different proof. We provide an application to the measures with larger entropy for a kind of skew product map.