Dynamical Systems Seminar

New development of entropy theory 1,2

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 Cr (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.