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The Absolute Continuity of a Family of Self-Similar Measures

Qi-Rong Deng

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Abstract

For a class of self-similar measures defined by iterated function system on the line with equal equal probability weights, the absolute continuity has been proved by using a tech- nique of polynomial. This extends a result of Garsia on infinite Bernulli convolution. Also, this result provides examples of absolutely continuous self-similar measure without weak separa- tion condition.

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What this paper is about

For a class of self-similar measures defined by iterated function system on the line with equal equal probability weights, the absolute continuity has been proved by using a tech- nique of polynomial. This extends a result of Garsia on infinite Bernulli convolution. Also, this result provides examples of absolutely continuous self-similar measure without weak separa- tion condition.

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Available abstract

For a class of self-similar measures defined by iterated function system on the line with equal equal probability weights, the absolute continuity has been proved by using a tech- nique of polynomial. This extends a result of Garsia on infinite Bernulli convolution. Also, this result provides examples of absolutely continuous self-similar measure without weak separa- tion condition.

Key concepts: Absolute continuity, Mathematics, Measure (data warehouse), Convolution (computer science), Real line, Probability measure, Polynomial, Iterated function

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