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For a continuous map on a topological graph containing a loop $S$ it is possible to define the degree (with respect to the loop $S$) and, for a map of degree $1$, rotation numbers. We study the rotation set of these maps and the periods of…
We compactify and regularize the space of initial values of a planar map with a quartic invariant and use this construction to prove its integrability in the sense of algebraic entropy. The system turns out to have certain unusual…
Periodic Geometry studies isometry invariants of periodic point sets that are also continuous under perturbations. The motivations come from periodic crystals whose structures are determined in a rigid form but any minimal cells can…
Transitivity, the existence of periodic points and positive topological entropy can be used to characterize complexity in dynamical systems. It is known that for graphs that are not trees, for every $\varepsilon>0,$ there exist (complicate)…
Prime numbers appeared in contexts spanning statistical mechanics, quantum mechanics and dynamical systems. However, the mechanisms governing the irregularities observed in their sequence and linking them to physical systems remained…
Two discrete dynamical systems are discussed and analyzed whose trajectories encode significant explicit information about a number of problems in combinatorial probability, including graphical enumeration on Riemann surfaces and random…
Piecewise smooth maps are known to exhibit a wide range of dynamical features including numerous types of periodic orbits. Predicting regions in parameter space where such periodic orbits might occur and determining their stability is…
This article consists in two independent parts. In the first one, we investigate the geometric properties of almost periodicity of model sets (or cut-and-project sets, defined under the weakest hypotheses); in particular we show that they…
In the first part we associate a periodic sequence to a partition and study the connection the distribution of elements of uniform limit of the sequences. Then some facts of statistical independence of these limits are proved
This paper aims at presenting a few models of quantum dynamics whose description involves the analysis of random unitary matrices for which dynamical localization has been proven to hold. Some models come from physical approximations…
The phenomenon of turbulence is investigated in the context of globally coupled maps. The local dynamics is given by the Chat\'e-Manneville minimal map previously used in studies of spatiotemporal intermittency in locally coupled map…
By varying a parameter of a one-dimensional piecewise smooth map, stable periodic orbits are observed. In this paper, complete analytic characterization of these stable periodic orbits is obtained. An interesting relationship between the…
We exhibit a family of dynamical systems arising from rational points on elliptic curves in an attempt to mimic the familiar toral automorphisms. At the non-archimedean primes, a continuous map is constructed on the local elliptic curve…
The class of stochastic maps, that is, linear, trace-preserving, positive maps between the self-adjoint trace class operators of complex separable Hilbert spaces plays an important role in the representation of reversible dynamics and…
Coupled Tchebyscheff maps have recently been introduced to explain parameters in the standard model of particle physics, using the stochastic quantisation of Parisi and Wu. This paper studies dynamical properties of these maps, finding…
We present a unified framework for the quantization of a family of discrete dynamical systems of varying degrees of "chaoticity". The systems to be quantized are piecewise affine maps on the two-torus, viewed as phase space, and include the…
Spectral properties of Coupled Map Lattices are described. Conditions for the stability of spatially homogeneous chaotic solutions are derived using linear stability analysis. Global stability analysis results are also presented. The…
One of the most basic, longstanding open problems in the theory of dynamical systems is whether reachability is decidable for one-dimensional piecewise affine maps with two intervals. In this paper we prove that for injective maps, it is…
We characterize the sequences of fixed point indices $\{i(f^n, p)\}_{n\ge 1}$ of fixed points that are isolated as an invariant set and continuous maps in the plane. In particular, we prove that the sequence is periodic and $i(f^n, p) \le…
The dynamical systems of identical particles admitting quadratic integrals of motion are classified. The relevant integrals are explicitly constructed and their relation to separation of variables in H-J equation is clarified.