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## Smart criticality

Cornell University
,
2014.

A crucial fact established by Thurston in his 1985 preprint is that distinct \emph{minors} of quadratic laminations do not cross inside the unit disk; this led to his construction of a combinatorial model of the Mandelbrot set. Thurston's argument is based upon the fact that \emph{majors} of a quadratic lamination never enter the region between them, a result that fails in the cubic case. In this paper, devoted to laminations of any degree, we use an alternative approach in which the fate of sets of intersecting leaves of two distinct laminations is studied. It turns out that, under some natural assumptions, these sets of intersecting leaves behave like gaps of a lamination. Relying upon this, we rule out certain types of mutual location of critical sets of distinct laminations (this can be viewed as a partial generalization of the theorem that quadratic minors do not cross inside the unit disk). The main application is to the cubic case.

Stanislav Minkov, Ivan Shilin, / Cornell University. Series math "arxiv.org". 2020. No. arXiv:2011.04824.

For Milnor, statistical, and minimal attractors, we construct examples of smooth flows φ on S^2 for which the attractor of the Cartesian square of φ is smaller than the Cartesian square of the attractor of φ. In the example for the minimal attractors, the flow φ also has an SRB-measure such that its square does ...

Added: November 12, 2020

Skripchenko A., Troubetzkoy S., Annales de l'Institut Fourier 2015 Vol. 65 No. 5 P. 1881-1896

We study the billiard on a square billiard table with a one-sided vertical mirror.
We associate trajectories of these billiards with double rotations and study orbit behavior and questions of complexit ...

Added: March 2, 2016

Ekaterina Amerik, Misha Verbitsky, / Cornell University. Series math "arxiv.org". 2014.

Let $M$ be a simple holomorphically symplectic manifold, that is, a simply connected holomorphically symplectic manifold of Kahler type with $h^{2,0}=1$. We prove that the group of holomorphic automorphisms of $M$ acts on the set of faces of its Kahler cone with finitely many orbits. This is a version of the Morrison-Kawamata cone conjecture for ...

Added: September 5, 2014

Blokh A., Oversteegen L., Ptacek R. et al., / Cornell University. Series math "arxiv.org". 2014.

We study the closure of the cubic Principal Hyperbolic Domain and its
intersection $\mathcal{P}_\lambda$ with the slice $\mathcal{F}_\lambda$ of the
space of all cubic polynomials with fixed point $0$ defined by the multiplier
$\lambda$ at $0$. We show that any bounded domain $\mathcal{W}$ of
$\mathcal{F}_\lambda\setminus\mathcal{P}_\lambda$ consists of $J$-stable
polynomials $f$ with connected Julia sets $J(f)$ and is either of \emph{Siegel
capture} ...

Added: February 11, 2015

Kleptsyn V., Alvarez S., Malicet D. et al., / Cornell University. Series math "arxiv.org". 2015.

Added: June 22, 2016

Demina M.V., Kudryashov N. A., Regular and Chaotic Dynamics 2016 Vol. 21 No. 3 P. 351-366

Polynomial dynamical systems describing interacting particles in the plane are
studied. A method replacing integration of a polynomial multi-particle dynamical system
by finding polynomial solutions of partial differential equations is introduced. The method
enables one to integrate a wide class of polynomial multi-particle dynamical systems. The
general solutions of certain dynamical systems related to linear second-order partial differential
equations are ...

Added: October 5, 2018

Protasov V., Systems and Control Letters 2016 Vol. 90 P. 54-60

We prove the existence of positive linear switching systems (continuous time), whose trajectories grow to infinity, but slower than a given increasing function. This implies that, unlike the situation with linear ODE, the maximal growth of trajectories of linear systems may be arbitrarily slow. For systems generated by a finite set of matrices, this phenomenon ...

Added: February 22, 2016

Ivan Shilin, / Cornell University. Series math "arxiv.org". 2019. No. arXiv:1903.01933.

An orientation-preserving non-contractible separatrix loop of a hyperbolic saddle of a vector field on a two-dimensional surface may be accumulated by a separatrix of the same saddle. We study the unfolding of such loops in generic one-parameter families of vector fields as a semi-local bifurcation. As a byproduct, we construct a countable family of pairwise ...

Added: November 12, 2020

Blokh A., Oversteegen L., Ptacek R. et al., / Cornell University. Series math "arxiv.org". 2014.

To construct a model for a connectedness locus of polynomials of degree $d\ge
3$ (cf with Thurston's model of the Mandelbrot set), we define \emph{linked}
geolaminations $\mathcal{L}_1$ and $\mathcal{L}_2$. An \emph{accordion} is
defined as the union of a leaf $\ell$ of $\mathcal{L}_1$ and leaves of
$\mathcal{L}_2$ crossing $\ell$. We show that any accordion behaves like a gap
of one lamination ...

Added: February 11, 2015

Volk D., Ergodic Theory and Dynamical Systems 2014 Vol. 34 No. 2 P. 693-704

For a smooth manifold of any dimension greater than one, we present an open set of smooth endomorphisms such that any of them has a transitive attractor with a non-empty interior. These maps are m-fold non-branched coverings,m≥3. The construction applies to any manifold of the form S 1×M, where S 1 is the standard circle and Mis an arbitrary manifold. ...

Added: December 28, 2015

Stankevich N., Shchegoleva N. A., Sataev I. R. et al., Journal of Computational and Nonlinear Dynamics 2020 Vol. 15 No. 11 P. 111001

Using an example a system of two coupled generators of quasiperiodic oscillations, we study the occurrence of chaotic dynamics with one positive, two zero and several negative Lyapunov exponents. It is shown that such dynamic arises as a result of a sequence of bifurcations of two-frequency torus doubling and involve saddle tori occurring at their ...

Added: September 4, 2020

Shilin I., Доклады Академии наук 2016 Т. 469 № 3 С. 287-290

В работе показано, что неустойчивость аттракторов Милнора по Ляпунову является локально топологически типичным динамическим явлением, которое наблюдается в присутствии устойчивых гомоклинических касаний для 2-сжимающих периодических седел. ...

Added: October 14, 2018

NY : Springer, 2012

The volume is dedicated to Stephen Smale on the occasion of his 80th birthday. Besides his startling 1960 result of the proof of the Poincaré conjecture for all dimensions greater than or equal to five, Smale’s ground breaking contributions in various fields in Mathematics have marked the second part of the 20th century and beyond. ...

Added: December 19, 2012

Blank M., Nonlinearity 2012 Vol. 25 No. 12 P. 3389-3408

We study ergodic properties of a family of traffic maps acting in
the space of bi-infinite sequences of real numbers. The corresponding
dynamics mimics the motion of vehicles in a simple traffic flow, which
explains the name. Using connections to topological Markov chains we obtain
nontrivial invariant measures, prove their stochastic stability, and
calculate the topological entropy. Technically these results ...

Added: November 26, 2014

Romaskevich O. L., L'Enseignement Mathématique 2014

We consider 3 -periodic orbits in an elliptic billiard. Numerical experiments conducted by Dan Reznik have shown that the locus of the centers of inscribed circles of the corresponding triangles is an ellipse. We prove this fact by the complexification of the problem coupled with the complex law of reflection. ...

Added: December 25, 2014

Filimonov D., Клепцын В. А., Nonlinearity 2014 Vol. 27 No. 6 P. 1205-1223

We study possible one-end finitely presented subgroups of <img />, acting without finite orbits. Our main result, theorem 1, establishes that any such action possesses the so-called property (<img />), that allows one to make distortion-controlled expansion and is thus sufficient to conclude that the action is Lebesgue-ergodic. We also propose a path towards full ...

Added: October 23, 2014

Mikheev A. V., Теория. Практика. Инновации 2017 № 9 (21)

In this paper we consider the calculation of a dynamical system described by a second-order differential equation in which a fundamental system of solutions consisting of functions of exponential type is replaced by bounded functions of the Verhulst model. The time dependence of the forces acting on the dynamical system is analyzed, and the obtained ...

Added: September 6, 2017

Stanislav Minkov, Shilin I., Qualitative Theory of Dynamical Systems 2021 Vol. 20 No. 3 Article 77

For Milnor, statistical, and minimal attractors, we construct examples of smooth flows ϕ on S^2 for which the attractor of the Cartesian square of ϕ is smaller than the Cartesian square of the attractor of ϕ. In the example for the minimal attractors, the flow ϕ also has a global physical measure such that its ...

Added: September 16, 2021

Springer, 2009

Vladimir Arnold is one of the great mathematical scientists of our time. He is famous for both the breadth and the depth of his work. At the same time he is one of the most prolific and outstanding mathematical authors. This first volume of his Collected Works focuses on representations of functions, celestial mechanics, and ...

Added: February 20, 2013

Blokh A., Oversteegen L., Ptacek R. et al., / Cornell University. Series math "arxiv.org". 2015.

Thurston parameterized quadratic invariant laminations with a non-invariant lamination, the quotient of which yields a combinatorial model for the Mandelbrot set. As a step toward generalizing this construction to cubic polynomials, we consider slices of the family of cubic invariant laminations defined by a fixed critical leaf with non-periodic endpoints. We parameterize each slice by ...

Added: February 11, 2015

Зелик С. В., Chepyzhov V. V., Доклады Академии наук 2014 Т. 455 № 5 С. 512-517

We study regular global attractors of the dynamical systems corresponding to dissipative evolution equations and their nonautonomous perturbations. We prove that the nonautonomous dynamical system (process) resulting from a small nonautonomous perturbation of an autonomous dynamical system (semigroup) having a regular attractor has a regular nonautonomous attractor as well. Moreover, the symmetric Hausdorff deviation of ...

Added: August 26, 2014

Volk D., Discrete and Continuous Dynamical Systems 2014 Vol. 34 No. 5 P. 2307-2314

Interval translation maps (ITMs) are a non-invertible generalization of interval exchange transformations (IETs). The dynamics of finite type ITMs is similar to IETs, while infinite type ITMs are known to exhibit new interesting effects. In this paper, we prove the finiteness conjecture for the ITMs of three intervals. Namely, the subset of ITMs of finite ...

Added: December 30, 2015

Romanov A., Izvestiya. Mathematics 2011 Vol. 75 No. 6 P. 1165-1183

For a linear contraction U in a Banach space X we discuss conditions for the convergence of ergodic operator nets corresponding to the adjoint operator U* in the W*O-topology of the space End X*. The accumulation points of all possible nets of this kind form a compact convex set L = Ker G in End ...

Added: October 6, 2012

Pardalos P. M., Rassias T. undefined., Springer, 2014

This volume consists of chapters written by eminent scientists and engineers from the international community and presents significant advances in several theories, and applications of an interdisciplinary research. These contributions focus on both old and recent developments of Global Optimization Theory, Convex Analysis, Calculus of Variations, and Discrete Mathematics and Geometry, as well as several ...

Added: May 30, 2014