Poincaré Recurrence for Observations

Poincaré Types Solutions of Systems of Difference Equations

Statistics of Poincaré recurrences for a class of smooth circle maps

Statistics of Poincaré recurrence for a class of circle maps, including sub-critical, critical, and super-critical cases, are studied. It is shown how the topological differences in the various types of the dynamics are manifested in the statistics of the return times.

Spectra of Dimensions for Poincaré Recurrences for Special Flows

We prove the variational principle for dimensions for Poincaré recurrences, in the case of invariant sets of dynamical systems with continuous time. To achieve this goal we show that these dimensions can be expressed as roots of a non–homogeneous Bowen equation.

Parametric Poincaré-perron Theorem with Applications

We prove a parametric generalization of the classical PoincaréPerron theorem on stabilizing recurrence relations where we assume that the varying coefficients of a recurrence depend on auxiliary parameters and converge uniformly in these parameters to their limiting values. As an application we study convergence of the ratios of families of functions satisfying finite recurrence relations with ...

The Poincaré Recurrence Problem of Inviscid Incompressible Fluids

Nadirashvili presented a beautiful example showing that the Poincaré recurrence does not occur near a particular solution to the 2D Euler equation of inviscid incompressible fluids. Unfortunately, Nadirashvili’s setup of the phase space is not appropriate, and details of the proof are missing. This note fixes that.