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## Milnor Attractors of Skew Products with the Fiber a Circle

Okunev A.

In press

For a generic skew product with the fiber a circle over an Anosov diffeomorphism, we prove that the Milnor attractor coincides with the statistical attractor, is Lyapunov stable, and either has zero Lebesgue measure or coincides with the whole phase space. As a consequence, we conclude that such skew product is either transitive or has non-wandering set of zero measure. The result is proved under the assumption that the fiber maps preserve the orientation of the circle, and the skew product is partially hyperbolic.

Publication based on the results of:

Volk D., Kleptsyn V., Moscow Mathematical Journal 2014 Vol. 14 No. 2 P. 339-365

A one-dimensional confined nonlinear random walk is a tuple of N diffeomorphisms of the unit interval driven by a probabilistic Markov chain. For generic such walks, we obtain a geometric characterization of their ergodic stationary measures and prove that all of them have negative Lyapunov exponents. These measures appear to be probabilistic manifestations of physical measures for ...

Added: December 30, 2015

Alexey Okunev, / Cornell University. Series math "arxiv.org". 2015. No. 1508.02132.

We prove that for a generic skew product with circle fiber over an Anosov diffeomorphism the Milnor attractor (also called the likely limit set) coincides with the statistical attractor, is Lyapunov stable, and either has zero Lebesgue measure or coincides with the whole phase space. As a consequence we conclude that such skew product is ...

Added: November 24, 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

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

Minkov S. S., Okunev A., Functional Analysis and Its Applications 2016 Vol. 50 No. 1 P. 48-53

We prove that, for any E u ⊕ E cs partially hyperbolic C 2 diffeomorphism, the ω-limit set of a generic (with respect to the Lebesgue measure) point is a union of unstable leaves. As a corollary, we prove a conjecture made by Ilyashenko in his 2011 paper that the Milnor attractor is a union of unstable leaves. In the paper mentioned above, Ilyashenko reduced the ...

Added: May 4, 2016

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

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

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

А. В. Окунев, И. С. Шилин, Труды Математического института им. В.А. Стеклова РАН 2017 Т. 297 С. 260-280

Исследуются статистические аттракторы и аттракторы Милнора ступенчатых косых произведений над сдвигом Бернулли. В случае, когда слой – окружность, доказывается, что для топологически типичного косого произведения статистический аттрактор и аттрактор Милнора совпадают и являются устойчивыми по Ляпунову. Для этого рассматриваются некоторые свойства проекции аттрактора на слой, которые могут быть интересны сами по себе. В случае, когда слой – отрезок, дается ...

Added: October 14, 2018

Ilyashenko Y., Shilin I., Труды Математического института им. В.А. Стеклова РАН 2012 Т. 277 С. 91-100

В теории динамических систем существуют разные, неэквивалентные между собой определения аттракторов. Наиболее распространенными являются два определения: максимальный аттрактор и аттрактор Милнора. Максимальный аттрактор по определению устойчив по Ляпунову, однако зачастую он является в некотором смысле избыточным. Определение аттрактора Милнора более реалистично с физической точки зрения. Однако аттрактор Милнора может быть неустойчив по Ляпунову. Один из центральных вопросов ...

Added: October 14, 2018

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

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

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

Grines V., Mints D., Regular and Chaotic Dynamics 2023 Vol. 28 No. 3 P. 295-308

—In P. D. McSwiggen’s article, it was proposed Derived from Anosov type construction which leads to a partially hyperbolic diffeomorphism of the 3-torus. The nonwandering set of this diffeomorphism contains a two-dimensional attractor which consists of one-dimensional unstable manifolds of its points. The constructed diffeomorphism admits an invariant onedimensional orientable foliation such that it contains ...

Added: August 2, 2023

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

Nersisyan A., Zanasi R., International Journal of Robust and Nonlinear Control 1993 Vol. 3 No. 3 P. 199-209

A modified VS feedback is suggested for robust stabilization of continuous-time dynamical systems in the presence of parametric and external time-varying disturbances satisfying the ‘matching conditions’. The main feature of the proposed algorithm is that it contains additional switching ‘integral’ terms which track the unknown disturbances and make it possible to achieve the typical VSS ...

Added: September 10, 2021

Stankevich N., Kazakov A., Gonchenko S., Chaos 2020 Vol. 30 Article 123129

The generalized four-dimensional Rössler system is studied. Main bifurcation scenarios leading to a hyperchaos are described phenomenologically and their implementation in the model is demonstrated. In particular, we show that the formation of hyperchaotic invariant sets is related mainly to cascades (finite or infinite) of nondegenerate bifurcations of two types: period-doubling bifurcations of saddle cycles with a ...

Added: January 18, 2021

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

The contributions in this volume have been written by eminent scientists from the international mathematical community and present significant advances in several theories, methods and problems of Mathematical Analysis, Discrete Mathematics, Geometry and their Applications. The chapters focus on both old and recent developments in Functional Analysis, Harmonic Analysis, Complex Analysis, Operator Theory, Combinatorics, Functional ...

Added: May 30, 2014

Blank M., Doklady Mathematics 2016 Vol. 94 No. 3 P. 688-691

A novel approach to the fair division problem is proposed, which is based on the concept of a priori estimates and ideas of dynamical systems theory. For several problems on the division of a resource with discrete components, this approach leads to explicit constructive solutions in cases for which even the existence of solutions has ...

Added: February 20, 2017

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

V.L. Chernyshev, Tolchennikov A. A., Russian Journal of Mathematical Physics 2017 Vol. 24 No. 3 P. 290-298

In the problem of determining the asymptotics for the number of points moving along a metric tree, a polynomial approximation that uses Barnes’ multiple Bernoulli polynomials is found. The connection between the second term of the asymptotic expansion and the graph structure is discussed. ...

Added: October 3, 2017

Kudryashova E., Leonov G. A., Kuznetsov N. V., IFAC-PapersOnLine 2015

In this paper an approach to modeling of the Tunisian social system in 2011–2014 is considered and the revolution, bifurcation, and controlled stabilization are discussed. Using statistical analysis of socio-economic indicators of Tunisia there are selected two bifurcation parameters, which have influenced on stability of socio-economic system of Tunisia. Based on this analysis the recommendations ...

Added: March 28, 2015

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

Smilga I., / Cornell University. Series arXiv "math". 2012. No. 1205.4442.

In this paper, we give a few results on the local behavior of harmonic functions on the Sierpinski triangle - more precisely, of their restriction to a side of the triangle. First we present a general formula that gives the Hölder exponent of such a function in a given point. From this formula, we deduce ...

Added: September 26, 2018