An analytical construction of the SRB measures for Baker-type maps

Abstract The papers [O. M. Sarig. Symbolic dynamics for surface diffeomorphisms with positive entropy. J. Amer. Math. Soc.26(2) (2013), 341–426] and [S. Ben Ovadia. Symbolic dynamics for non-uniformly hyperbolic diffeomorphisms of compact smooth manifolds. J. Mod. Dyn.13 (2018), 43–113] constructed symbolic dynamics for the restriction of $C^r$ diffeomorphisms to a set $M'$ with full measure for all sufficiently hyperbolic ergodic invariant probability measures, but the set $M'$ was not identified there. We improve the construction in a way that enables $M'$ to be identified explicitly. One application is the coding of infinite conservative measures on the homoclinic classes of Rodriguez-Hertz et al. [Uniqueness of SRB measures for transitive diffeomorphisms on surfaces. Comm. Math. Phys.306(1) (2011), 35–49].

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SRB measures for attractors with continuous invariant splittings

Mathematische Zeitschrift ◽

10.1007/s00209-017-1883-2 ◽

2017 ◽

Vol 288 (1-2) ◽

pp. 135-165 ◽

Cited By ~ 1

Author(s):

Zeya Mi ◽

Yongluo Cao ◽

Dawei Yang

Keyword(s):

Srb Measures

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SRB measures for partially hyperbolic attractors of local diffeomorphisms

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2018.115 ◽

2018 ◽

Vol 40 (6) ◽

pp. 1545-1593

Author(s):

ANDERSON CRUZ ◽

PAULO VARANDAS

Keyword(s):

Thermodynamic Formalism ◽

Positive Lebesgue Measure ◽

Stable Bundle ◽

Local Diffeomorphism ◽

Local Diffeomorphisms ◽

Axiom A ◽

Srb Measures ◽

Cone Field ◽

Hyperbolic Attractors ◽

Partially Hyperbolic

We contribute to the thermodynamic formalism of partially hyperbolic attractors for local diffeomorphisms admitting an invariant stable bundle and a positively invariant cone field with non-uniform cone expansion at a positive Lebesgue measure set of points. These include the case of attractors for Axiom A endomorphisms and partially hyperbolic endomorphisms derived from Anosov. We prove these attractors have finitely many SRB measures, that these are hyperbolic, and that the SRB measure is unique provided the dynamics is transitive. Moreover, we show that the SRB measures are statistically stable (in the weak$^{\ast }$ topology) and that their entropy varies continuously with respect to the local diffeomorphism.

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Hyperbolic SRB Measures and the Leaf Condition

Communications in Mathematical Physics ◽

10.1007/s00220-021-04208-6 ◽

2021 ◽

Vol 387 (3) ◽

pp. 1353-1404 ◽

Cited By ~ 1

Author(s):

Snir Ben Ovadia

Keyword(s):

Srb Measures

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Sinai’s Work on Markov Partitions and SRB Measures

The Abel Prize - The Abel Prize 2013-2017 ◽

10.1007/978-3-319-99028-6_11 ◽

2019 ◽

pp. 257-285 ◽

Cited By ~ 3

Author(s):

Yakov Pesin

Keyword(s):

Markov Partitions ◽

Srb Measures

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Equivalence of physical and SRB measures in random dynamical systems

Nonlinearity ◽

10.1088/1361-6544/aafaa8 ◽

2019 ◽

Vol 32 (4) ◽

pp. 1494-1524 ◽

Cited By ~ 2

Author(s):

Alex Blumenthal ◽

Lai-Sang Young

Keyword(s):

Dynamical Systems ◽

Random Dynamical Systems ◽

Srb Measures

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Adaptation of the control synthesized by the ACAR method to the control based on the implementation of the maximum principle

MATEC Web of Conferences ◽

10.1051/matecconf/201822602012 ◽

2018 ◽

Vol 226 ◽

pp. 02012 ◽

Cited By ~ 1

Author(s):

Viktor P. Lapshin ◽

Ilya A. Turkin ◽

Alexey A. Zakalyuzhnyy ◽

Viktor F. Khlystunov ◽

Gennadiy A. Kuzin

Keyword(s):

Optimal Control ◽

Control System ◽

Maximum Principle ◽

Synthesis Method ◽

The Maximum Principle ◽

Analytical Construction ◽

Optimal Linear ◽

The Moment ◽

Moment Of Resistance ◽

Maximum Method

A special case of synthesizing the electromechanical control system by the maximum method and using the Analytical Construction method of Aggregate Regulators (ACAR) is considered in the article. For the basis the task of synthesizing the optimal for speed electromechanical positioning system was chosen, while the moment of resistance to movement linearly depended on the output coordinate of the system, that is, on the angle of the engine rotor rotation. Synthesis of the optimal system for speed makes it possible to increase the efficiency of the entire production process in many production tasks, and the synthesis of the optimal linear control system based on the maximum principle is a fairly well-formalized problem. Here it should be noted that the procedure for synergistic synthesis of the optimal control system has no such formalization. An approach that brings together the solutions obtained by these two methods, which makes it possible to increase the efficiency of the ACAR method by adding some features of the methodology for synthesizing optimal systems by introducing nonlinearity of the “saturation” type is proposed in the article. The results obtained made it possible to formulate the following basic scientific proposition: the synthesis of a control system based on the synergetic approach makes it possible to obtain a system close to optimal (quasi-optimal, but after the modification of the synergetic synthesis method itself.) Here we also formulate the hypothesis of a connection between the time constants, using the ACAR method, with the optimal control switching time determined in the maximum method.

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