Kazuki Okamura

Singularity Results for Functional Equations Driven by Linear Fractional Transformations

We consider functional equations driven by linear fractional transformations, which are special cases of de Rham’s functional equations. We consider Hausdorff dimension of the measure whose distribution function is the solution. We give a necessary and sufficient condition for singularity. We also show that they have a relationship with stationary measures.

On regularity for de Rham’s functional equations

We consider regularity for solutions of a class of de Rham’s functional equations. Under some smoothness conditions of functions making up the equation, we improve some results in Hata (Japan J Appl Math 2:381–414, 1985). Our results are applicable to some cases when the functions making up the equation are non-linear functions on an interval, specifically, polynomials and linear fractional transformations. Our results imply the singularity of some well-known singular functions, in particular, Minkowski’s question-mark function, and, some small perturbed functions of the singular functions.

Long Time Behavior of the Volume of the Wiener Sausage on Dirichlet Spaces

In the present paper, we consider long time behaviors of the volume of the Wiener sausage on Dirichlet spaces. We focus on the volume of the Wiener sausage for diffusion processes on metric measure spaces other than the Euclid space equipped with the Lebesgue measure. We obtain the growth rate of the expectations and almost sure behaviors of the volumes of the Wiener sausages on metric measure Dirichlet spaces satisfying Ahlfors regularity and sub-Gaussian heat kernel estimates. We show that the growth rate of the expectations on a bounded modification of the Euclidian space is identical with the one on the Euclidian space equipped with the Lebesgue measure. We give an example of a metric measure Dirichlet space on which a scaled of the means fluctuates.

Enlargement of subgraphs of infinite graphs by Bernoulli percolation

We consider changes in properties of a subgraph of an infinite graph resulting from the addition of open edges of Bernoulli percolation on the infinite graph to the subgraph. We give the triplet of an infinite graph, one of its subgraphs, and a property of the subgraphs. Then, in a manner similar to the way Hammersleys critical probability is defined, we can define two values associated with the triplet. We regard the two values as certain critical probabilities, and compare them with Hammersleys critical probability. In this paper, we focus on the following cases of a graph property: being a transient subgraph, having finitely many cut points or no cut points, being a recurrent subset, or being connected. Our results depend heavily on the choice of the triplet. Most results of this paper are announced in \cite{O16} without proofs. This paper gives full details of them.

A new generalization of the Takagi function

Abstract We consider a one-parameter family of functions { F ( t , x ) } t on [ 0 , 1 ] and partial derivatives ∂ t k F ( t , x ) with respect to the parameter t . Each function of the class is defined by a certain pair of two square matrices of order two. The class includes the Lebesgue singular functions and other singular functions. Our approach to the Takagi function is similar to Hata and Yamaguti. The class of partial derivatives ∂ t k F ( t , x ) includes the original Takagi function and some generalizations. We consider real-analytic properties of ∂ t k F ( t , x ) as a function of x , specifically, differentiability, the Hausdorff dimension of the graph, the asymptotics around dyadic rationals, variation, a question of local monotonicity and a local modulus of continuity. Our results are extensions of some results for the original Takagi function and some generalizations.

Random Sequences with Respect to a Measure Defined by Two Linear Fractional Transformations

We define a probability measure on the Cantor space by using two linear fractional transformations consisting of computable real numbers. The measure can be a non-product measure on the Cantor space, on the other hand, it can also be the Bernoulli measure. We consider the constructive dimensions for the points which are random with respect to the measure. We examine limit frequencies of the outcome of 0 for such random points.