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En este artículo revisamos un famoso teorema, descubierto por H. Steinhaus en 1936, en el que se da una condición suficiente que permite obtener las funciones coordenadas de una curva que llena el cuadrado unidad. Ponemos de manifiesto que el recíproco de este teorema no se cumple para la curva de Lebesgue. Aquí proponemos un teorema de caracterización de curvas que llenan el espacio, basado en una condición de llenado. Asimismo, damos una caracterización constructiva de esta condición de llenado...
This note contains a proof of the existence of a one-to-one function of onto itself with the following properties: is a rational-linear automorphism of , and the graph of is a non-measurable subset of the plane.
Let be a self-similar set with similarities ratio and Hausdorff
dimension , let be a probability vector. The
Besicovitch-type subset of is defined aswhere is the indicator function of the set . Let and be a gauge function, then we prove in this paper:(i) If
, thenmoreover both of and
are finite positive;(ii) If is a positive probability
vector other than , then the gauge functions can be
partitioned as follows
We prove that density preserving homeomorphisms form a Π11-complete subset in the Polish space ℍ of all increasing autohomeomorphisms of unit interval.
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