Makoto Mori

Fredholm determinant for piecewise linear transformations

We call the number ξ the lower Lyapunov number. We will study Spec^) , the spectrum of P \BV> the restriction of P to the subspace BV of functions with bounded variation. The generating function of P is determined by the orbits of the division points of the partition, and the orbits are characterized by a finite dimensional matrix Φ(z) which is defined by a renewal equation (§ 3). Hence, we can show that D(z)=det(I— Φ(#))> which we call a Fredholm determinant, is the determinant of /— #P=ΣίΓ-o zP in the following sense: Theorem A. Let λ G C and assume that \\\>e~. Then λ belongs to Sρec(F) if and only if z—\~ is a zero of D(z):

Low discrepancy sequences generated by piecewise linear Maps

We construct low discrepancy sequences using piecewise linear Markov maps from the view point of a dynamical system. Our main tool is the spectrum of the Perron-Frobenius operator associated with the map. We give also a concrete way to get the spectrumof the -operator.

Discrepancy of van der Corput sequences generated by piecewise linear transformations

Discrepancy of van der Corput sequences constructed by piecewise linear transformations with same slope β > 1 is discussed. If there is no eigenvalues of the Perron-Frobenius operator associated with the dynamical system in |z| > 1/β except 1, the discrepancy equals (logN) k+1/N, where k is the number of the non-Markov endpoints.

Discrepancy of sequences generated by piecewise monotone maps

In [3], we constructed low discrepancy sequences using piecewise linear Markov maps from the view point of dynamical system. In this paper, we will determine discrepancies of sequences generated by general piecewise monotone mappings in terms of the Fredholm determinant. The main tool is the signed symbolic dynamics introduced in [1] and [2].

Construction of three-dimensional low discrepancy sequences

Three-dimensional low discrepancy sequence is constructed via dynamical system. Different from one-dimensional cases, the low discrepancy sequence cannot be constructed only by expanding transformation. The transformation used in this paper is not only expanding but also shuffling.