On bounded distortions of maps in the line

GARCIA, IGNACIO; MOREIRA, CARLOS GUSTAVO

DYNAMICAL SYSTEMS

TAYLOR & FRANCIS LTD

Lugar: Londres; Año: 2012 vol. 27 p. 501 - 506

1468-9367

Let $I_1$ and $I_2$ be disjoint closed subintervals of $[0,1]$ and let $F:I_1cup I_2 o [0,1]$ be a $mathcal C^1$ map such that $F|_{I_i}$ is a diffeomorphism onto $[0,1]$ and $F´>1$ on its domain. The repeller $K$ of $F$ is a Cantor set. If $F$ satisfies the bounded distortion property ({f BD}), then $K$ has Hausdorff dimension $t$ smaller than $1$ and its $t$-dimensional Hausdorff measure is finite and positive. Another related property is the strong bounded distortion ({f SBD}), which is needed to define the scaling function of repellers associated with $mathcal C^{1+}$ maps. In this note we give an example that shows that {f SBD} is indeed stronger that {f BD}.