The Rényi dimension of a conformal measure for a piecewise monotonic map of the interval

Summary

We consider a continuous piecewise monotonic transformation <Emphasis Type=”Italic”>T</Emphasis> on the interval, which is expanding, and an <Emphasis Type=”Italic”>e</Emphasis><Superscript>-<Emphasis Type=”Italic”>g</Emphasis></Superscript>-conformal measure <Emphasis Type=”Italic”>m</Emphasis>. Set <Emphasis Type=”Italic”>A</Emphasis>=supp &mgr; and &phgr; =-log |<Emphasis Type=”Italic”>T</Emphasis>&apos;|. For each s&ge; 0 there is a unique <Emphasis Type=”Italic”>t</Emphasis>=&tgr;(<Emphasis Type=”Italic”>s</Emphasis>) such that the pressure <Emphasis Type=”Italic”>p</Emphasis> (<Emphasis Type=”Italic”>T</Emphasis>&#9474;A, <Emphasis Type=”Italic”>sg</Emphasis> + <Emphasis Type=”Italic”>t</Emphasis>&phgr;) equals zero. For the R&eacute;nyi dimension R<Emphasis Type=”Italic”><Subscript>s</Subscript></Emphasis> with <Emphasis Type=”Italic”>s</Emphasis>&isin; <Emphasis Type=”Bold”>R</Emphasis><Superscript>+</Superscript>&bsol;{1} we show under certain assumptions that R<Emphasis Type=”Italic”><Subscript>s</Subscript></Emphasis>(<Emphasis Type=”Italic”>m</Emphasis>) = &tgr;(<Emphasis Type=”Italic”>s</Emphasis>)/(1-<Emphasis Type=”Italic”>s</Emphasis>).