For any given positive integer $l$, we prove that every plane deformation of a circle which preserves the $1/2$ and $1/(2l+1)$-rational caustics is trivial i.e. the deformation consists only of similarities (rescalings plus isometries).
We consider integrable Hamiltonian systems in three degrees of freedom near an elliptic equilibrium in 1:1:-2 resonance. The integrability originates from averaging along the periodic motion of the quadratic part and an imposed rotational symmetry ab
out the vertical axis. Introducing a detuning parameter we find a rich bifurcation diagram, containing three parabolas of Hamiltonian Hopf bifurcations that join at the origin. We describe the monodromy of the resulting ramified 3-torus bundle as variation of the detuning parameter lets the system pass through 1:1:-2 resonance.
Metric entropies along a hierarchy of unstable foliations are investigated for $C^1$ diffeomorphisms with dominated splitting. The analogues of Ruelles inequality and Pesins formula, which relate the metric entropy and Lyapunov exponents in each hierarchy, are given.
We classify integrable third order equations in 2+1 dimensions which generalize the examples of Kadomtsev-Petviashvili, Veselov-Novikov and Harry Dym equations. Our approach is based on the observation that dispersionless limits of integrable systems
in 2+1 dimensions possess infinitely many multi-phase solutions coming from the so-called hydrodynamic reductions. %Conversely, the requirement of the existence of hydrodynamic reductions proves to be an efficient classification criterion. In this paper we adopt a novel perturbative approach to the classification problem. Based on the method of hydrodynamic reductions, we first classify integrable quasilinear systems which may (potentially) occur as dispersionless limits of soliton equations in 2+1 dimensions. To reconstruct dispersive deformations, we require that all hydrodynamic reductions of the dispersionless limit are inherited by the corresponding dispersive counterpart. This procedure leads to a complete list of integrable third order equations, some of which are apparently new.
The dynamical structure of the rational map $ax+1/x$ on the projective line $P$ over the field $mathbb{Q}_p$ of $p$-adic numbers is described for $pgeq 3$.
V. Kaloshin
,C. E. Koudjinan
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(2021)
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"Non co-preservation of the $1/2$ & $1/(2l+1)$-rational caustics along deformations of circles"
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Comlan Edmond Koudjinan
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