• density of fractional parts
• IFS theory
• Kronecker's theorem
• left invertibility
• Mahler measure
• quantized systems

This thesis addresses left invertibility for dynamical discrete--time control systems evolving in a continuous state--space, with inputs and outputs in discrete sets. More precisely, inputs are arbitrary sequences of symbols in a finite alphabet, each symbol being associated to a specific action on the system. Information available on the system is represented by
sequences of output values in a discrete set. Such outputs are obtained by quantization, i.e. are generated by the system evolution according to a given partition of the state-space. Left
invertibility has to do with the injectivity of I/O map, that is the possibility of recovering the unknown inputs that generate the observed output sequences.

Two class of systems are examined: nonlinear systems with a contraction property (called joint contraction), and linear I/O quantized systems with uniform output generating partition (with no
contraction hypothesis). For joint contractive systems left invertibility is addressed using the theory of Iterated Function Systems (IFS), a tool developed for the study of fractals: the IFS
naturally associated to a system and the geometric properties of its attractor are linked to the invertibility property of the system.
Our main result here is a necessary and sufficient condition for left invertibility. For linear I/O quantized systems left invertibility is reduced, under suitable conditions, to the notion of left D-invertibility: these conditions have to do with density of fractional parts in the unit cube, and are found using number
theoretic methods, involving a generalization of the classical theorem of Kronecker in terms of Mahler measure of minimal polynomials. While left invertibility takes into account membership
to sets of a given partition, left D-invertibility considers only membership to a single set, and is therefore much easier to
detect: one important result here is a method to compute left D-invertibility for all linear systems with no eigenvalue of modulus one. The main results for linear I/O quantized systems show the equivalence between left D-invertibility and left invertibility for a full measure set of matrices, and stronger results in this
direction are proved for unidimensional system, and for multi-input single-output (MISO) systems.

Moreover a cryptographic system based on left invertibility of I/O quantized systems is presented, which provides a secure
communication method. Finally algorithms are given, to check left invertibility of a I/O quantized linear systems, and to recover
inputs from the knowledge of the output sequences.