The author presents an improved version of work on the fundamental theorem of Sinai and Chernov (1987). It can be used to study the ergodicity of dynamical systems with singularities if an increasing nondegenerate quadratic form is defined almost everywhere. The author proves the ergodicity of the billiard map in the cardioid. The methods used in this proof allow one to check which if the billiards with Pesin region of measure one are ergodic.
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Roberto Markarian (1993) studied this question.
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