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

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 and =-log | < Emphasis Type='Italic' > T < /Emphasis > '|. For each s≥ 0 there is a unique < Emphasis Type='Italic' > t < /Emphasis > = ( < Emphasis Type='Italic' > s < /Emphasis > ) such that the pressure < Emphasis Type='Italic' > p < /Emphasis > ( < Emphasis Type='Italic' > T < /Emphasis > │A, < Emphasis Type='Italic' > sg < /Emphasis > + < Emphasis Type='Italic' > t < /Emphasis > ) equals zero. For the Rényi dimension R < Emphasis Type='Italic' > < Subscript > s < /Subscript > < /Emphasis > with < Emphasis Type='Italic' > s < /Emphasis > ∈ < Emphasis Type='Bold' > R < /Emphasis > < Superscript > + < /Superscript > {1} we show under certain assumptions that R < Emphasis Type='Italic' > < Subscript > s < /Subscript > < /Emphasis > ( < Emphasis Type='Italic' > m < /Emphasis > ) = ( < Emphasis Type='Italic' > s < /Emphasis > )/(1- < Emphasis Type='Italic' > s < /Emphasis > ).