Fat baker's transformations

Abstract

We investigate a variant of the baker's transformation in which the mapping is onto but is not one-to-one. The Bowen-Ruelle measure for this map is investigated.

On the discount sum of Bernoulli random variables

“…There are different formulae for the entropy dimension (Garsia, 1963;Alexander and Yorke, 1984;Alexander and Zagier, 1991;Ledrappier and Porzio). Using the above formula we can calculate that dim1(10 ~ 0.99571 which is compatible with the previous result.…”

“…Recently interest has been re-kindled, however, due to the development of the geometry of fractals, dynamical systems and iterated function systems (cf. Alexander and Yorke (1984), Bovier, Falconer (1990), Hu (to appear), Lalley, Lau and Ngai (to appear) and the references there). Let ~(x) = pjx + b~, xE~ d…”

The nature of the steady state in models of optimal growth under uncertainty

“…The authors, indeed, state that for all p and all > 1, if f is absolutely continuous with density in 1) .…”

“…and B r (x) denotes the ball of center x and radius r, holds for any interval (a; b) (0; 1) such that 1) and the transversality condition holds. As for the upper extremum of interval (a; b), they use the same estimation as in [56]: b = 0:649 < , which clearly works since, for 1=3 p 2=3 and 2, Necessary smoothness condition (iii ) in Theorem 2 is derived from a theorem of HardyLittlewood [31].…”

“…Using the speci…c structure of the example, it is possible, through an appropriate log-transformation, to convert this into an iterated function system (IFS) involving a¢ ne functions (or similarities). A straightforward application of the theory of iterated function systems leads to the functional equation governing the invariant distribution 1 of the (transformed) random dynamical system. We note that since our random dynamical system is generated by a …nite number of a¢ ne maps which are contractions, the existence, uniqueness and global stability of the invariant distribution (starting from an arbitrary initial distribution), which the literature is principally concerned with, are guaranteed by a straightforward application of the standard theory of iterated function systems, developed by Karlin [38], Norman [48], Hutchinson [35], Barnsley and Demko [4], and others.…”