$$\boldsymbol{L}^{\boldsymbol{p}}$$ and Hölder Estimates for Cauchy–Riemann Equations on Convex Domain of Finite/Infinite Type with Piecewise Smooth Boundary in $$\boldsymbol{\mathbb{C}}^{\mathbf{2}}$$

2021 ◽  
Vol 56 (4) ◽  
pp. 225-233
Author(s):  
L. K. Ha ◽  
T. K. An
Keyword(s):  
Convex Domain ◽  
Smooth Boundary ◽  
Piecewise Smooth ◽  
Piecewise Smooth Boundary ◽  
Infinite Type ◽  
Cauchy Riemann ◽  
Hölder Estimates

Conditionally invariant measures for Anosov maps with small holes

1998 ◽  
Vol 18 (5) ◽  
pp. 1049-1073 ◽  
Author(s):  
N. CHERNOV ◽  
R. MARKARIAN ◽  
S. TROUBETZKOY
Keyword(s):  
Invariant Measure ◽  
Existence And Uniqueness ◽  
Invariant Measures ◽  
Smooth Boundary ◽  
Piecewise Smooth ◽  
Piecewise Smooth Boundary ◽  
Conditional Distributions ◽  
The Past ◽  
Set Of Points ◽  
Small Holes

We study Anosov diffeomorphisms on surfaces in which some small ‘holes’ are cut. The points that are mapped into those holes disappear and never return. We assume that the holes are arbitrary open domains with piecewise smooth boundary, and their sizes are small enough. The set of points whose trajectories never enter holes under the past iterations of the map is a Cantor-like union of unstable fibers. We establish the existence and uniqueness of a conditionally invariant measure on this set, whose conditional distributions on unstable fibers are smooth. This generalizes previous works by Pianigiani, Yorke, and others.

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