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"Randomness and Recurrence in Dynamical Systems" was published by the Mathematical Association of America in 2010, as volume 31 in the Carus Monographs Series. The main intention in writing this work was to make more easily accessible some famous and/or interesting ideas and results of 20th century mathematics whose proofs usually or often depend upon measure theory, thus avoiding the need to study measure theory before learning about them. Among these results are: Weyl's Theorem which says that the fractional parts of the multiples of an irrational number are uniformly distributed in the unit interval, Borel's result that almost all numbers in the unit interval have an asymptotically equal number of 0s and 1s in their expansion to base 2, the uniqueness of a rotationally invariant finitely additive set function on the circle group, and the uniqueness of Benford's Law, according to which certain sequences have the property that their terms have a leading digit d with probability log(1+1/d). Related phenomena concerning recurrence of observations, and the associated recurrence times and standard deviations are discussed. As it says in the Introduction to the work:
"The approach also endeavours to connect the material with recent research that is accessible at this level, with the aim of giving a more "contemporary" feel about the material and contributing in a small way towards "bridging the gap" between undergraduate teaching and the world of current mathematical ideas and research." .....Read more about this book.
"Difference spaces and Invariant Linear Forms" was published by Springer-Verlag in 1994 as volume 1586 in the Lecture Notes in Mathematics series. The main theme is the difference spaces, which are Hilbert spaces of square-integrable functions determined by how the Fourier transform of the function behaves near the origin or near certain subsets of n-dimensional Euclidean space. A linear form (not necessarily continuous) that vanishes on one of these spaces is invariant. For example, on the real line a translation invariant form on the space of square-integrable functions is one that vanishes on the first order difference space. Then, the usual differentiation operator maps the first order Sobolev space, whose members are determined by how their Fourier transforms behave at infinity, isometrically onto the first order difference space, with corresponding results for higher order derivatives. There are also connections with the theory of wavelets. .......Read more about these ideas.
 Download slides from the seminar "Dynamical systems and the Carathéodory definition of a measurable set"", given at the Mathematical Institute, Slovak Academy of Sciences, Bratislava, 14th September 2011. This discusses how the Carathépodroy definition can be motivated by calculating the outer measure of invariant sets for two dynamical systems -- rotations on the circle group, and the system introduced by Kakutani to give an example of a weakly mixing but not strong mixing transformation.
 Download slides from the seminar "Some mathematical analysis without measure theory", given in Banská Bystrica, Slovakia, in September 2011. This talk discusses topics such as recurrence phenomena, normal numbers, Borel's Theorem, and the uniqueness of invariant finitely additive set functions, but indicates approaches to such results that avoid the traditional measure theory, thus making the matter more accessible at the undergraduate level.
 Download slides from a seminar on the Fourier transform, given at the University of Wollongong in 2005. This deals with characterising the functions whose Fourier transforms have a prescribed behaviour near the origin, on the real line. The Fourier transform of a function behaves like |x| near the origin if and only if it is the sum of a finite number of first order differences. In this case, it is the sum of three such differences. All of this occurs in the Hilbert space of square-integrable functions on the real line.
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