A mathematical framework is proposed for the " big bang ". It starts with some Cantor set and assumes the existence of a transformation from that set to the continuum set used in conventional physical theories like general relativity and quantum mechanics. Let us assume that the big bang started from a Cantor set (see e.g. [1]) with the power of continuum and an arbitrary cardinality higher tha...

The notion of a structure system | sets of structured objects that are composed of atomic objects | is introduced by a collection of axioms. By a uniform change of the atomic objects, the relationship, induced by the atomic objects, between the structured objects is preserved, resulting in a notion of isomorphism of sets of structured objects. By de nition, a relation R on such structured sets ...

In this paper we investigate algorithmic randomness on more general spaces than the Cantor space, namely computable metric spaces. To do this, we first develop a unified framework allowing computations with probability measures. We show that any computable metric space with a computable probability measure is isomorphic to the Cantor space in a computable and measure-theoretic sense. We show th...

The geometry of Cantor systems associated with an asymptotically non-hyperbolic family (f )0≤ ≤ 0 was studied by Jiang (Geometry of Cantor systems. Trans. Amer. Math. Soc. 351 (1999), 1975–1987). By applying the geometry studied there, we prove that the Hausdorff dimension of the maximal invariant set of f behaves like 1−K 1/γ asymptotically, as was conjectured by Jiang (Generalized Ulam–von Ne...

We define the notion of rotations on infinite binary trees, and construct an irrational tree rotation with bounded distortion. This lifts naturally to a Lipschitz circle homeomorphism having the middle-thirds Cantor set as its minimal set. This degree of smoothness is best possible, since it is known that no C1 circle diffeomorphism can have a linearly self-similar Cantor set as its minimal set.