Dynamical Directions in Numeration

International audienceWe survey definitions and properties of numeration from a dynamical point of view. That is we focuse on numeration systems, their associated compactifications, and the dynamical systems that can be naturally defined on them. The exposition is unified by the notion of fibred numeration system. A lot of examples are discussed. Various numerations on natural, integral, real or complex numbers are presented with a special attention payed to beta-numeration and its generalisations, abstract numeration systems and shift radix systems. A section of applications ends the paper

The law of series

We prove a general ergodic-theoretic result concerning the return time statistic, which, properly understood, sheds some new light on the common sense phenomenon known as {\it the law of series}. Let \proc be an ergodic process on finitely many states, with positive entropy. We show that the distribution function of the normalized waiting time for the first visit to a small cylinder set $B$ is, for majority of such cylinders and up to epsilon, dominated by the exponential distribution function $1-e^{-t}$. This fact has the following interpretation: The occurrences of such a "rare event" $B$ can deviate from purely random in only one direction -- so that for any length of an "observation period" of time, the first occurrence of $B$ "attracts" its further repetitions in this period

Language-Based Analysis Of Differential Privacy

Differential privacy (Dwork, 2006; Dwork et al., 2006a) has achieved prominence over the past decade as a rigorous formal foundation upon which diverse tools and mechanisms for performing private data analysis can be built. The guarantee of differential privacy is that it protects privacy at the individual level: if the result of a differentially private query or operation on a dataset is publicly released, any individual present in that dataset can claim plausible deniability. This means that any participating individual can deny the presence of their information in the dataset based on the query result, because differentially private queries introduce enough random noise/bias to make the result indistinguishable from that of the same query run on a dataset which actually does not contain the individual’s information. Additionally, differential privacy guarantees are resilient against any form of linking attack in the presence of auxiliary information about individuals. Both static and dynamic tools have been developed to help non-experts write differentially private programs: static analysis tools construct a proof without needing to run the program; dynamic analysis tools construct a proof while running the program, using a dynamic monitor executed by the unmodified runtime system. The resulting proof may apply only to that execution of the program. Many of the static tools take the form of statically-typed programming languages, where correct privacy analysis is built into the soundness of the type system. Meanwhile dynamic systems typically take either a prescriptive or descriptive approach to analysis when running the program. This dissertation proposes new techniques for language-based analysis of differential privacy of programs in a variety of contexts spanning static and dynamic analysis. Our approach towards differential privacy analysis makes use of ideas from linear type systems and static/dynamic taint analysis. While several prior approaches towards differential privacy analysis exist, this dissertation proposes techniques which are designed to, in several regards, be more flexible and usable than prior work

Genericity in Topological Dynamics

We study genericity of dynamical properties in the space of homeomorphisms of the Cantor set and in the space of subshifts of a suitably large shift space. These rather different settings are related by a Glasner-King type correspondence: genericity in one is equivalent to genericity in the other. By applying symbolic techniques in the shift-space model we derive new results about genericity of dynamical properties for transitive and totally transitive homeomorphisms of the Cantor set. We show that the isomorphism class of the universal odometer is generic in the space of transitive systems. On the other hand, the space of totally transitive systems displays much more varied dynamics. In particular, we show that in this space the isomorphism class of every Cantor system without periodic points is dense, and the following properties are generic: minimality, zero entropy, disjointness from a fixed totally transitive system, weak mixing, strong mixing, and minimal self joinings. The last two stand in striking contrast to the situation in the measure-preserving category. We also prove a correspondence between genericity of dynamical properties in the measure-preserving category and genericity of systems supporting an invariant measure with the same property.Comment: 48 pages, to appear in Ergodic Theory Dynamical Systems. v2: revised exposition, added proof that the universal odometer is generic among transitive Cantor homeomorphism