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Journal articles on the topic 'Hyperbolic dynamical systems'

Author: Grafiati

Published: 7 December 2024

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1

Bandtlow, Oscar F., Wolfram Just, and Julia Slipantschuk. "A numerical study of rigidity of hyperbolic splittings in simple two-dimensional maps." Nonlinearity 37, no. 4 (March 4, 2024): 045007. http://dx.doi.org/10.1088/1361-6544/ad2b58.

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Abstract Chaotic hyperbolic dynamical systems enjoy a surprising degree of rigidity, a fact which is well known in the mathematics community but perhaps less so in theoretical physics circles. Low-dimensional hyperbolic systems are either conjugate to linear automorphisms, that is, dynamically equivalent to the Arnold cat map and its variants, or their hyperbolic structure is not smooth. We illustrate this dichotomy using a family of analytic maps, for which we show by means of numerical simulations that the corresponding hyperbolic structure is not smooth, thereby providing an example for a global mechanism which produces non-smooth phase space structures in an otherwise smooth dynamical system.

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2

Barinova, Marina K., and Evgenia K. Shustova. "Dynamical properties of direct products of discrete dynamical systems." Zhurnal Srednevolzhskogo Matematicheskogo Obshchestva 24, no. 1 (March 31, 2022): 21–30. http://dx.doi.org/10.15507/2079-6900.24.202201.21-30.

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A natural way for creating new dynamical systems is to consider direct products of already known systems. The paper studies some dynamical properties of direct products of homeomorphisms and diffeomorphisms. In particular, authors prove that a chain-recurrent set of the direct product of homeomorphisms is a direct product of the chain-recurrent sets. Another result established in the paper is that the direct product of diffeomorphisms holds hyperbolic structure on the direct product of hyperbolic sets. It is known that if a diffeomorphism has a hyperbolic chain-recurrent set, then this mapping is Ω-stable. Therefore, it follows from the results of the paper that the direct product of Ω-stable diffeomorphisms is also Ω-stable. Another question which is raised in the article concerns the existence of an energy function for the direct product of diffeomorphisms which already have such functions (recall that energy function is a smooth Lyapunov function whose set of critical points coincides with the chain-recurrent set of the system). Authors show that in this case the function can be found as a weighted sum of energy functions of initial diffeomorphisms.

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3

Whittaker, Michael. "Spectral triples for hyperbolic dynamical systems." Journal of Noncommutative Geometry 7, no. 2 (2013): 563–82. http://dx.doi.org/10.4171/jncg/127.

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4

Gogolev, Andrey, Pedro Ontaneda, and Federico Rodriguez Hertz. "New partially hyperbolic dynamical systems I." Acta Mathematica 215, no. 2 (2015): 363–93. http://dx.doi.org/10.1007/s11511-016-0135-3.

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5

Lokutsievskii, L. V. "Fractal structure of hyperbolic Lipschitzian dynamical systems." Russian Journal of Mathematical Physics 19, no. 1 (March 2012): 27–43. http://dx.doi.org/10.1134/s1061920812010050.

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6

CARVALHO, ALEXANDRE N., JOSÉ A. LANGA, and JAMES C. ROBINSON. "Lower semicontinuity of attractors for non-autonomous dynamical systems." Ergodic Theory and Dynamical Systems 29, no. 6 (February 3, 2009): 1765–80. http://dx.doi.org/10.1017/s0143385708000850.

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AbstractThis paper is concerned with the lower semicontinuity of attractors for semilinear non-autonomous differential equations in Banach spaces. We require the unperturbed attractor to be given as the union of unstable manifolds of time-dependent hyperbolic solutions, generalizing previous results valid only for gradient-like systems in which the hyperbolic solutions are equilibria. The tools employed are a study of the continuity of the local unstable manifolds of the hyperbolic solutions and results on the continuity of the exponential dichotomy of the linearization around each of these solutions.

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7

Blankers, Vance, Tristan Rendfrey, Aaron Shukert, and Patrick Shipman. "Julia and Mandelbrot Sets for Dynamics over the Hyperbolic Numbers." Fractal and Fractional 3, no. 1 (February 20, 2019): 6. http://dx.doi.org/10.3390/fractalfract3010006.

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Julia and Mandelbrot sets, which characterize bounded orbits in dynamical systems over the complex numbers, are classic examples of fractal sets. We investigate the analogs of these sets for dynamical systems over the hyperbolic numbers. Hyperbolic numbers, which have the form x + τ y for x , y ∈ R , and τ 2 = 1 but τ ≠ ± 1 , are the natural number system in which to encode geometric properties of the Minkowski space R 1 , 1 . We show that the hyperbolic analog of the Mandelbrot set parameterizes the connectedness of hyperbolic Julia sets. We give a wall-and-chamber decomposition of the hyperbolic plane in terms of these Julia sets.

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8

Pesin, Ya B. "Dynamical systems with generalized hyperbolic attractors: hyperbolic, ergodic and topological properties." Ergodic Theory and Dynamical Systems 12, no. 1 (March 1992): 123–51. http://dx.doi.org/10.1017/s0143385700006635.

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AbstractWe introduce a class of dynamical systems on a Riemannian manifold with singularities having attractors with strong hyperbolic behavior of trajectories. This class includes a number of famous examples such as the Lorenz type attractor, the Lozi attractor and some others which have been of great interest in recent years. We prove the existence of a special invariant measure which is an analog of the Bowen-Ruelle-Sinai measure for classical hyperbolic attractors and study the ergodic properties of the system with respect to this measure. We also describe some topological properties of the system on the attractor. Our results can be considered a dissipative version of the theory of systems with singularities preserving the smooth measure.

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9

Cong, Nguyen Dinh. "Structural stability of linear random dynamical systems." Ergodic Theory and Dynamical Systems 16, no. 6 (December 1996): 1207–20. http://dx.doi.org/10.1017/s0143385700009998.

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AbstractIn this paper, structural stability of discrete-time linear random dynamical systems is studied. A random dynamical system is called structurally stable with respect to a random norm if it is topologically conjugate to any random dynamical system which is sufficiently close to it in this norm. We prove that a discrete-time linear random dynamical system is structurally stable with respect to its Lyapunov norms if and only if it is hyperbolic.

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10

REY-BELLET, LUC, and LAI-SANG YOUNG. "Large deviations in non-uniformly hyperbolic dynamical systems." Ergodic Theory and Dynamical Systems 28, no. 2 (April 2008): 587–612. http://dx.doi.org/10.1017/s0143385707000478.

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AbstractWe prove large deviation principles for ergodic averages of dynamical systems admitting Markov tower extensions with exponential return times. Our main technical result from which a number of limit theorems are derived is the analyticity of logarithmic moment generating functions. Among the classes of dynamical systems to which our results apply are piecewise hyperbolic diffeomorphisms, dispersing billiards including Lorentz gases, and strange attractors of rank one including Hénon-type attractors.

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11

Diamond, P., P. Kloeden, V. Kozyakin, and A. Pokrovskii. "Expansivity of semi-hyperbolic Lipschitz mappings." Bulletin of the Australian Mathematical Society 51, no. 2 (April 1995): 301–8. http://dx.doi.org/10.1017/s0004972700014131.

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Semi-hyperbolic dynamical systems generated by Lipschitz mappings are shown to be exponentially expansive, locally at least, and explicit rates of expansion are determined. The result is applicable to nonsmooth noninvertible systems such as those with hysteresis effects as well as to classical systems involving hyperbolic diffeomorphisms.

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12

Urbański, Mariusz. "Recurrence rates for loosely Markov dynamical systems." Journal of the Australian Mathematical Society 82, no. 1 (February 2007): 39–57. http://dx.doi.org/10.1017/s1446788700017468.

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AbstractThe concept of loosely Markov dynamical systems is introduced. We show that for these systems the recurrence rates and pointwise dimensions coincide. The systems generated by hyperbolic exponential maps, arbitrary rational functions of the Riemann sphere, and measurable dynamical systems generated by infinite conformal iterated function systems are all checked to be loosely Markov.

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13

Zizza, Frank. "Automorphisms of Hyperbolic Dynamical Systems and K 2." Transactions of the American Mathematical Society 307, no. 2 (June 1988): 773. http://dx.doi.org/10.2307/2001198.

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14

Young, Lai-Sang. "Geometric and Ergodic Theory of Hyperbolic Dynamical Systems." Current Developments in Mathematics 1998, no. 1 (1998): 237–78. http://dx.doi.org/10.4310/cdm.1998.v1998.n1.a6.

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15

Feres, Renato. "Hyperbolic dynamical systems, invariant geometric structures, and rigidity." Mathematical Research Letters 1, no. 1 (1994): 11–26. http://dx.doi.org/10.4310/mrl.1994.v1.n1.a2.

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16

Wu, Xinxing, Xu Zhang, and Xin Ma. "Various Shadowing in Linear Dynamical Systems." International Journal of Bifurcation and Chaos 29, no. 03 (March 2019): 1950042. http://dx.doi.org/10.1142/s0218127419500421.

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This paper proves that the linear transformation [Formula: see text] on [Formula: see text] has the (asymptotic) average shadowing property if and only if [Formula: see text] is hyperbolic, where [Formula: see text] is a nonsingular matrix, giving a positive answer to a question in [Lee, 2012a]. Besides, it is proved that [Formula: see text] does not have the [Formula: see text]-shadowing property, thus does not have the ergodic shadowing property for every nonsingular matrix [Formula: see text].

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17

PARISÉ, PIERRE-OLIVIER, and DOMINIC ROCHON. "TRICOMPLEX DYNAMICAL SYSTEMS GENERATED BY POLYNOMIALS OF ODD DEGREE." Fractals 25, no. 03 (May 2017): 1750026. http://dx.doi.org/10.1142/s0218348x17500268.

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In this paper, we give the exact interval of the cross section of the Multibrot sets generated by the polynomial [Formula: see text] where [Formula: see text] and [Formula: see text] are complex numbers and [Formula: see text] is an odd integer. Furthermore, we show that the same Multibrots defined on the hyperbolic numbers are always squares. Moreover, we give a generalized 3D version of the hyperbolic Multibrot set and prove that our generalization is an octahedron for a specific 3D slice of the dynamical system generated by the tricomplex polynomial [Formula: see text] where [Formula: see text] is an odd integer.

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18

Massoukou, R. Y. M’pika, and K. S. Govinder. "Symmetry analysis for hyperbolic equilibria using a TB/dengue fever model." International Journal of Modern Physics B 30, no. 28n29 (November 10, 2016): 1640022. http://dx.doi.org/10.1142/s0217979216400221.

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We investigate the interplay between Lie symmetry analysis and dynamical systems analysis. As an example, we take a toy model describing the spread of TB and dengue fever. We first undertake a comprehensive dynamical systems analysis including a discussion about local stability. For those regions in which such analyzes cannot be translated to global behavior, we undertake a Lie symmetry analysis. It is shown that the Lie analysis can be useful in providing information for systems where the (local) dynamical systems analysis breaks down.

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19

Kifer, Yuri. "Limit theorems in averaging for dynamical systems." Ergodic Theory and Dynamical Systems 15, no. 6 (December 1995): 1143–72. http://dx.doi.org/10.1017/s0143385700009834.

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AbstractThis paper yields diffusion and moderate deviation type asymptotics for solutions of differential equations of the form dZε(t)/dt = εB(Zε(t), fty) where ft is a suspension flow (in particular, a hyperbolic flow) over a sufficiently fast mixing transformation. Such problems emerge in the study of perturbed Hamiltonian systems. These exhibit a new class of limit theorems for dynamical systems and extend a number of previously known results.

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20

COLLET, P. "Statistics of closest return for some non-uniformly hyperbolic systems." Ergodic Theory and Dynamical Systems 21, no. 2 (March 30, 2001): 401–20. http://dx.doi.org/10.1017/s0143385701001201.

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For non-uniformly hyperbolic maps of the interval with exponential decay of correlations we prove that the law of closest return to a given point when suitably normalized is almost surely asymptotically exponential. A similar result holds when the reference point is the initial point of the trajectory. We use the framework for non-uniformly hyperbolic dynamical systems developed by L. S. Young.

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21

BANKS, S. P., and SONG YI. "ELLIPTIC AND AUTOMORPHIC DYNAMICAL SYSTEMS ON SURFACES." International Journal of Bifurcation and Chaos 16, no. 04 (April 2006): 911–23. http://dx.doi.org/10.1142/s0218127406015209.

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We derive explicit differential equations for dynamical systems defined on generic surfaces applying elliptic and automorphic function theory to make uniform the surfaces in the upper half of the complex plane with the hyperbolic metric. By modifying the definition of the standard theta series we will determine general meromorphic systems on a fundamental domain in the upper half plane the solution trajectories of which "roll up" onto an appropriate surface of any given genus.

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22

Su, Yaofeng, and Leonid A. Bunimovich. "Poisson Approximations and Convergence Rates for Hyperbolic Dynamical Systems." Communications in Mathematical Physics 390, no. 1 (January 19, 2022): 113–68. http://dx.doi.org/10.1007/s00220-022-04309-w.

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23

Zizza, Frank. "Automorphisms of hyperbolic dynamical systems and $K\sb 2$." Transactions of the American Mathematical Society 307, no. 2 (February 1, 1988): 773. http://dx.doi.org/10.1090/s0002-9947-1988-0940227-2.

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24

Fokkink, R. J., J. Keesling, and L. G. Oversteegen. "Extensions of zero-dimensional dynamical systems and hyperbolic attractors." Topology and its Applications 152, no. 1-2 (July 2005): 83–86. http://dx.doi.org/10.1016/j.topol.2004.08.016.

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25

de la Llave, R. "Invariants for smooth conjugacy of hyperbolic dynamical systems II." Communications in Mathematical Physics 109, no. 3 (September 1987): 369–78. http://dx.doi.org/10.1007/bf01206141.

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26

Marco, J. M., and R. Moriy�n. "Invariants for smooth conjugacy of hyperbolic dynamical systems. I." Communications in Mathematical Physics 109, no. 4 (December 1987): 681–89. http://dx.doi.org/10.1007/bf01208962.

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27

de la Llave, R., and R. Moriyón. "Invariants for smooth conjugacy of hyperbolic dynamical systems. IV." Communications in Mathematical Physics 116, no. 2 (June 1988): 185–92. http://dx.doi.org/10.1007/bf01225254.

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28

Palmer, Kenneth J. "Normally Hyperbolic Invariant Manifolds in Dynamical Systems (Stephen Wiggins)." SIAM Review 37, no. 3 (September 1995): 472. http://dx.doi.org/10.1137/1037110.

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29

Marco, Jos� Manuel, and Roberto Moriy�n. "Invariants for smooth conjugacy of hyperbolic dynamical systems, III." Communications in Mathematical Physics 112, no. 2 (June 1987): 317–33. http://dx.doi.org/10.1007/bf01217815.

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30

Putnam, Ian F., and Jack Spielberg. "The Structure ofC*-Algebras Associated with Hyperbolic Dynamical Systems." Journal of Functional Analysis 163, no. 2 (April 1999): 279–99. http://dx.doi.org/10.1006/jfan.1998.3379.

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31

Guillarmou, Colin, and Thibault de Poyferré. "A paradifferential approach for hyperbolic dynamical systems and applications." Tunisian Journal of Mathematics 4, no. 4 (December 31, 2022): 673–718. http://dx.doi.org/10.2140/tunis.2022.4.673.

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32

Kryzhevich, Sergey. "Chaotic Dynamics and Bifurcations in Impact Systems." International Journal of Energy Optimization and Engineering 1, no. 4 (October 2012): 15–37. http://dx.doi.org/10.4018/ijeoe.2012100102.

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Bifurcations of dynamical systems described byseveral second order differential equations and by an impact condition are studied. It is shown that the variation of parameters when the number of impacts of a periodic solution increases, leads to the occurrence of a hyperbolic chaotic invariant set.

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33

Hoang, Manh Tuan, Thi Kim Quy Ngo, and Ha Hai Truong. "A simple method for studying asymptotic stability of discrete dynamical systems and its applications." An International Journal of Optimization and Control: Theories & Applications (IJOCTA) 13, no. 1 (January 23, 2023): 10–25. http://dx.doi.org/10.11121/ijocta.2023.1243.

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In this work, we introduce a simple method for investigating the asymptotic stability of discrete dynamical systems, which can be considered as an extension of the classical Lyapunov's indirect method. This method is constructed based on the classical Lyapunov's indirect method and the idea proposed by Ghaffari and Lasemi in a recent work. The new method can be applicable even when equilibia of dynamical systems are non-hyperbolic. Hence, in many cases, the classical Lyapunov's indirect method fails but the new one can be used simply. In addition, by combining the new stability method with the Mickens' methodology, we formulate some nonstandard finite difference (NSFD) methods which are able to preserve the asymptotic stability of some classes of differential equation models even when they have non-hyperbolic equilibrium points. As an important consequence, some well-known results on stability-preserving NSFD schemes for autonomous dynamical systems are improved and extended. Finally, a set of numerical examples are performed to illustrate and support the theoretical findings.

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34

MAULDIN, R. D., and M. URBAŃSKI. "Parabolic iterated function systems." Ergodic Theory and Dynamical Systems 20, no. 5 (October 2000): 1423–47. http://dx.doi.org/10.1017/s0143385700000778.

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In this paper we introduce and explore conformal parabolic iterated function systems. We define and study topological pressure, Perron–Frobenius-type operators, semiconformal and conformal measures and the Hausdorff dimension of the limit set. With every parabolic system we associate an infinite hyperbolic conformal iterated function system and we employ it to study geometric and dynamical features (properly defined invariant measures for example) of the limit set.

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35

Kong, De-Xing, and Fa Wu. "A New Type of Distributed Parameter Control Systems: Two-Point Boundary Value Problems for Infinite-Dimensional Dynamical Systems." Journal of Applied Mathematics 2013 (2013): 1–3. http://dx.doi.org/10.1155/2013/454857.

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This survey note describes a new type of distributed parameter control systems—the two-point boundary value problems for infinite-dimensional dynamical systems, particularly, for hyperbolic systems of partial differential equations of second order, some of the discoveries that have been done about it and some unresolved questions.

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36

MELBOURNE, IAN, VIOREL NIŢICĂ, and ANDREI TÖRÖK. "Transitivity of Heisenberg group extensions of hyperbolic systems." Ergodic Theory and Dynamical Systems 32, no. 1 (April 5, 2011): 223–35. http://dx.doi.org/10.1017/s014338571000091x.

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AbstractWe show that amongCrextensions (r>0) of a uniformly hyperbolic dynamical system with fiber the standard real Heisenberg group ℋnof dimension 2n+1, those that avoid an obvious obstruction to topological transitivity are generically topologically transitive. Moreover, if one considers extensions with fiber a connected nilpotent Lie group with a compact commutator subgroup (for example ℋn/ℤ), among those that avoid the obvious obstruction, topological transitivity is open and dense.

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37

Peidong, Liu, Qian Minping, and Tang Fuchang. "Pseudo-orbit tracing property for random diffeomorphisms." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 126, no. 5 (1996): 1027–33. http://dx.doi.org/10.1017/s0308210500023234.

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In this paper we consider the pseudo-orbit tracing property for dynamical systems generated by iterations of random diffeomorphisms. We first define a type of hyperbolicity by means of a ‘random’ multiplicative ergodic theorem, and then prove our shadowing result by employing the graph transformation methods. That result applies to, for example, the case of small random diffeomorphisms type perturbations of hyperbolic sets of deterministic dynamical systems.

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38

BENATTI, F., V. CAPPELLINI, M. DE COCK, M. FANNES, and D. VANPETEGHEM. "CLASSICAL LIMIT OF QUANTUM DYNAMICAL ENTROPIES." Reviews in Mathematical Physics 15, no. 08 (October 2003): 847–75. http://dx.doi.org/10.1142/s0129055x03001837.

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Two non-commutative dynamical entropies are studied in connection with the classical limit. For systems with a strongly chaotic classical limit, the Kolmogorov–Sinai invariant is recovered on time scales that are logarithmic in the quantization parameter. The model of the quantized hyperbolic automorphisms of the 2-torus is examined in detail.

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39

Grines, V. Z., E. V. Zhuzhoma, and O. V. Pochinka. "Dynamical Systems and Topology of Magnetic Fields in Conducting Medium." Contemporary Mathematics. Fundamental Directions 63, no. 3 (December 15, 2017): 455–74. http://dx.doi.org/10.22363/2413-3639-2017-63-3-455-474.

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We discuss application of contemporary methods of the theory of dynamical systems with regular and chaotic hyperbolic dynamics to investigation of topological structure of magnetic ﬁelds in conducting media. For substantial classes of magnetic ﬁelds, we consider well-known physical models allowing us to reduce investigation of such ﬁelds to study of vector ﬁelds and Morse-Smale diﬀeomorphisms as well as diﬀeomorphisms with nontrivial basic sets satisfying the A axiom introduced by Smale. For the point-charge magnetic ﬁeld model, we consider the problem of separator playing an important role in the reconnection processes and investigate relations between its singularities. We consider the class of magnetic ﬁelds in the solar corona and solve the problem of topological equivalency of ﬁelds in this class. We develop a topological modiﬁcation of the Zeldovich funicular model of the nondissipative cinematic dynamo, constructing a hyperbolic diﬀeomorphism with chaotic dynamics that is conservative in the neighborhood of its transitive invariant set.

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40

AFRAIMOVICH, V. S., SHUI-NEE CHOW, and WENXIAN SHEN. "HYPERBOLIC HOMOCLINIC POINTS OF ℤd-ACTIONS IN LATTICE DYNAMICAL SYSTEMS." International Journal of Bifurcation and Chaos 06, no. 06 (June 1996): 1059–75. http://dx.doi.org/10.1142/s0218127496000576.

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We study ℤd action on a set of equilibrium solutions of a lattice dynamical system, i.e., a system with discrete spatial variables, and the stability and hyperbolicity of the equilibrium solutions. Complicated behavior of ℤd-action corresponds to the existence of an infinite number of equilibrium solutions which are randomly situated along spatial coordinates. We prove that the existence of a homoclinic point of a ℤd-action implies complicated behavior, provided the hyperbolicity of the homoclinic solution with respect to the lattice dynamical system (this is a generalization of the previous work of the first two authors). Similar result holds for hyperbolic partially homoclinic and heteroclinic points. We show the equivalence of stability for any equilibrium solutions and the equivalence of hyperbolicity for homoclinic points under various norms.

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41

Nusse, Helena E., and James A. Yorke. "Analysis of a procedure for finding numerical trajectories close to chaotic saddle hyperbolic sets." Ergodic Theory and Dynamical Systems 11, no. 1 (March 1991): 189–208. http://dx.doi.org/10.1017/s0143385700006076.

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AbstractIn dynamical systems examples are common in which there are regions containing chaotic sets that are not attractors, e.g. systems with horseshoes have such regions. In such dynamical systems one will observe chaotic transients. An important problem is the ‘Dynamical Restraint Problem’: given a region that contains a chaotic set but contains no attractor, find a chaotic trajectory numerically that remains in the region for an arbitrarily long period of time.We present two procedures (‘PIM triple procedures’) for finding trajectories which stay extremely close to such chaotic sets for arbitrarily long periods of time.

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42

SAUSSOL, BENOIT. "AN INTRODUCTION TO QUANTITATIVE POINCARÉ RECURRENCE IN DYNAMICAL SYSTEMS." Reviews in Mathematical Physics 21, no. 08 (September 2009): 949–79. http://dx.doi.org/10.1142/s0129055x09003785.

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We present some recurrence results in the context of ergodic theory and dynamical systems. The main focus will be on smooth dynamical systems, in particular, those with some chaotic/hyperbolic behavior. The aim is to compute recurrence rates, limiting distributions of return times, and short returns. We choose to give the full proofs of the results directly related to recurrence, avoiding as much as possible to hide the ideas behind technical details. This drove us to consider as our basic dynamical system a one-dimensional expanding map of the interval. We note, however, that most of the arguments still apply to higher dimensional or less uniform situations, so that most of the statements continue to hold. Some basic notions from the thermodynamic formalism and the dimension theory of dynamical systems will be recalled.

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43

Babalic, Elena, and Calin Lazaroiu. "Cosmological flows on hyperbolic surfaces." Facta universitatis - series: Physics, Chemistry and Technology 17, no. 1, spec.issue (2019): 1–9. http://dx.doi.org/10.2298/fupct1901001b.

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We outline the geometric formulation of cosmological flows for FLRW models with the scalar matter as well as certain aspects which arise in their study with methods originating from the geometric theory of dynamical systems. We briefly summarize certain results of numerical analysis which we carried out when the scalar manifold of the model is a hyperbolic surface of the finite or infinite area.

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44

WARD, T. B. "Almost all $S$-integer dynamical systems have many periodic points." Ergodic Theory and Dynamical Systems 18, no. 2 (April 1998): 471–86. http://dx.doi.org/10.1017/s0143385798113378.

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We show that for almost every ergodic $S$-integer dynamical system the radius of convergence of the dynamical zeta function is no larger than $\exp(-\frac{1}{2}h_{\rm top})<1$. In the arithmetic case almost every zeta function is irrational.We conjecture that for almost every ergodic $S$-integer dynamical system the radius of convergence of the zeta function is exactly $\exp(-h_{\rm top})<1$ and the zeta function is irrational.In an important geometric case (the $S$-integer systems corresponding to isometric extensions of the full $p$-shift or, more generally, linear algebraic cellular automata on the full $p$-shift) we show that the conjecture holds with the possible exception of at most two primes $p$.Finally, we explicitly describe the structure of $S$-integer dynamical systems as isometric extensions of (quasi-)hyperbolic dynamical systems.

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45

Mohtashamipour, Maliheh, and Alireza Zamani Bahabadi. "Non-stationary dynamical systems; Shadowing theorem and some applications." Filomat 37, no. 18 (2023): 6245–53. http://dx.doi.org/10.2298/fil2318245m.

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In the present paper, we mean a sequence of maps along a sequence of spaces by a non-stationary dynamical system. We use an Anosov family as a generalization of an Anosov map, which is a sequence of diffeomorphisms along a sequence of compact Riemannian manifolds, so that the tangent bundles split into expanding and contracting subspaces, with uniform bounds for the contraction and the expansion. Also, we introduce the shadowing property on non-stationary dynamical systems. Then, we prepare the necessary conditions for the existence of the shadowing property to prove the shadowing theorem in nonstationary dynamical systems. The shadowing theorem is a known result in dynamical systems, which states that any dynamical system with a hyperbolic structure has the shadowing property. Here, we prove that the shadowing theorem is established on any invariant Anosov family in a non-stationary dynamical system. Then, as in some applications of the shadowing theorem, we check the stability of Anosov families, and also we peruse the stability of isolated invariant Anosov families in non-stationary dynamical systems.

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46

Molaei, Mohammadreza. "Hyperbolic dynamics of discrete dynamical systems on pseudo-riemannian manifolds." Electronic Research Announcements 25 (2018): 8–15. http://dx.doi.org/10.3934/era.2018.25.002.

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Demers, Mark F., and Paul Wright. "Behaviour of the escape rate function in hyperbolic dynamical systems." Nonlinearity 25, no. 7 (June 19, 2012): 2133–50. http://dx.doi.org/10.1088/0951-7715/25/7/2133.

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48

Putnam, Ian F. "Functoriality of the C *-Algebras Associated with Hyperbolic Dynamical Systems." Journal of the London Mathematical Society 62, no. 3 (December 2000): 873–84. http://dx.doi.org/10.1112/s002461070000140x.

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Alves, José F., and Vilton Pinheiro. "Slow Rates of Mixing for Dynamical Systems with Hyperbolic Structures." Journal of Statistical Physics 131, no. 3 (January 31, 2008): 505–34. http://dx.doi.org/10.1007/s10955-008-9482-6.

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Holland, Mark, and Matthew Nicol. "Speed of convergence to an extreme value distribution for non-uniformly hyperbolic dynamical systems." Stochastics and Dynamics 15, no. 04 (October 14, 2015): 1550028. http://dx.doi.org/10.1142/s0219493715500288.

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Abstract:

Suppose (f, 𝒳, ν) is a dynamical system and ϕ : 𝒳 → ℝ is an observation with a unique maximum at a (generic) point in 𝒳. We consider the time series of successive maxima Mn(x) := max {ϕ(x),…,ϕ ◦ fn-1(x)}. Recent works have focused on the distributional convergence of such maxima (under suitable normalization) to an extreme value distribution. In this paper, for certain dynamical systems, we establish convergence rates to the limiting distribution. In contrast to the case of i.i.d. random variables, the convergence rates depend on the rate of mixing and the recurrence time statistics. For a range of applications, including uniformly expanding maps, quadratic maps, and intermittent maps, we establish corresponding convergence rates. We also establish convergence rates for certain hyperbolic systems such as Anosov systems, and discuss convergence rates for non-uniformly hyperbolic systems, such as Hénon maps.

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