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Статті в журналах з теми "Partially Hyperbolic System"

Автор: Grafiati

Опубліковано: 11 березня 2023

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1

RIOS, I., and J. SIQUEIRA. "On equilibrium states for partially hyperbolic horseshoes." Ergodic Theory and Dynamical Systems 38, no. 1 (2016): 301–35. http://dx.doi.org/10.1017/etds.2016.21.

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We prove the existence and uniqueness of equilibrium states for a family of partially hyperbolic systems, with respect to Hölder continuous potentials with small variation. The family comes from the projection, on the center-unstable direction, of a family of partially hyperbolic horseshoes introduced by Díaz et al [Destroying horseshoes via heterodimensional cycles: generating bifurcations inside homoclinic classes. Ergod. Th. & Dynam. Sys.29 (2009), 433–474]. For the original three-dimensional system we consider potentials with small variation, constant on local stable manifolds, obtaini

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2

RODRIGUEZ HERTZ, F., M. A. RODRIGUEZ HERTZ, A. TAHZIBI, and R. URES. "Maximizing measures for partially hyperbolic systems with compact center leaves." Ergodic Theory and Dynamical Systems 32, no. 2 (2011): 825–39. http://dx.doi.org/10.1017/s0143385711000757.

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AbstractWe obtain the following dichotomy for accessible partially hyperbolic diffeomorphisms of three-dimensional manifolds having compact center leaves: either there is a unique entropy-maximizing measure, this measure has the Bernoulli property and its center Lyapunov exponent is 0, or there are a finite number of entropy-maximizing measures, all of them with non-zero center Lyapunov exponents (at least one with a negative exponent and one with a positive exponent), that are finite extensions of a Bernoulli system. In the first case of the dichotomy, we obtain that the system is topological

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3

ANDERSSON, MARTIN, and CARLOS H. VÁSQUEZ. "On mostly expanding diffeomorphisms." Ergodic Theory and Dynamical Systems 38, no. 8 (2017): 2838–59. http://dx.doi.org/10.1017/etds.2017.17.

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In this work, we study the class of mostly expanding partially hyperbolic diffeomorphisms. We prove that such a class is$C^{r}$-open,$r>1$, among the partially hyperbolic diffeomorphisms and we prove that the mostly expanding condition guarantees the existence of physical measures and provides more information about the statistics of the system. Mañé’s classical derived-from-Anosov diffeomorphism on$\mathbb{T}^{3}$belongs to this set.

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4

Xu, Jiang, and Shuichi Kawashima. "Global Classical Solutions for Partially Dissipative Hyperbolic System of Balance Laws." Archive for Rational Mechanics and Analysis 211, no. 2 (2013): 513–53. http://dx.doi.org/10.1007/s00205-013-0679-8.

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5

BUZZI, J., T. FISHER, M. SAMBARINO, and C. VÁSQUEZ. "Maximal entropy measures for certain partially hyperbolic, derived from Anosov systems." Ergodic Theory and Dynamical Systems 32, no. 1 (2011): 63–79. http://dx.doi.org/10.1017/s0143385710000854.

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AbstractWe show that a class of robustly transitive diffeomorphisms originally described by Mañé are intrinsically ergodic. More precisely, we obtain an open set of diffeomorphisms which fail to be uniformly hyperbolic and structurally stable, but nevertheless have the following stability with respect to their entropy. Their topological entropy is constant and they each have a unique measure of maximal entropy with respect to which periodic orbits are equidistributed. Moreover, equipped with their respective measure of maximal entropy, these diffeomorphisms are pairwise isomorphic. We show tha

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6

Ondich, Jeffrey. "The reducibility of partially invariant solutions of systems of partial differential equations." European Journal of Applied Mathematics 6, no. 4 (1995): 329–54. http://dx.doi.org/10.1017/s0956792500001881.

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Ovsiannikov's partially invariant solutions of differential equations generalize Lie's group invariant solutions. A partially invariant solution is only interesting if it cannot be discovered more readily as an invariant solution. Roughly, a partially invariant solution that can be discovered more directly by Lie's method is said to be reducible. In this paper, I develop conditions under which a partially invariant solution or a class of such solutions must be reducible, and use these conditions both to obtain non-reducible solutions to a system of hyperbolic conservation laws, and to demonstr

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7

Vergara-Hermosilla, G., G. Leugering, and Y. Wang. "Boundary controllability of a system modelling a partially immersed obstacle." ESAIM: Control, Optimisation and Calculus of Variations 27 (2021): 80. http://dx.doi.org/10.1051/cocv/2021076.

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In this paper, we address the problem of boundary controllability for the one-dimensional nonlinear shallow water system, describing the free surface flow of water as well as the flow under a fixed gate structure. The system of differential equations considered can be interpreted as a simplified model of a particular type of wave energy device converter called oscillating water column. The physical requirements naturally lead to the problem of exact controllability in a prescribed region. In particular, we use the concept of nodal profile controllability in which at a given point (the node) ti

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8

Klar, A., and O. Tse. "An entropy functional and explicit decay rates for a nonlinear partially dissipative hyperbolic system." ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik 95, no. 5 (2014): 469–75. http://dx.doi.org/10.1002/zamm.201300275.

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9

GUPTA, CHINMAYA. "Extreme-value distributions for some classes of non-uniformly partially hyperbolic dynamical systems." Ergodic Theory and Dynamical Systems 30, no. 3 (2009): 757–71. http://dx.doi.org/10.1017/s0143385709000406.

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AbstractIn this note, we obtain verifiable sufficient conditions for the extreme-value distribution for a certain class of skew-product extensions of non-uniformly hyperbolic base maps. We show that these conditions, formulated in terms of the decay of correlations on the product system and the measure of rapidly returning points on the base, lead to a distribution for the maximum of Φ(p)=−log(d(p,p0)) that is of the first type. In particular, we establish the type I distribution for S1 extensions of piecewise C2 uniformly expanding maps of the interval, non-uniformly expanding maps of the int

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10

Rempel, M., and D. Przybylski. "Efficient Numerical Treatment of Ambipolar and Hall Drift as Hyperbolic System." Astrophysical Journal 923, no. 1 (2021): 79. http://dx.doi.org/10.3847/1538-4357/ac2c6d.

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Abstract Partially ionized plasmas, such as the solar chromosphere, require a generalized Ohm’s law including the effects of ambipolar and Hall drift. While both describe transport processes that arise from the multifluid equations and are therefore of hyperbolic nature, they are often incorporated in models as a diffusive, i.e., parabolic process. While the formulation as such is easy to include in standard MHD models, the resulting diffusive time-step constraints do require often a computationally more expensive implicit treatment or super-time-stepping approaches. In this paper we discuss a

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11

AFRAIMOVICH, V. S., SHUI-NEE CHOW та WENXIAN SHEN. "HYPERBOLIC HOMOCLINIC POINTS OF ℤd-ACTIONS IN LATTICE DYNAMICAL SYSTEMS". International Journal of Bifurcation and Chaos 06, № 06 (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 w

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12

CHOQUET, CATHERINE. "PARABOLIC AND DEGENERATE PARABOLIC MODELS FOR PRESSURE-DRIVEN TRANSPORT PROBLEMS." Mathematical Models and Methods in Applied Sciences 20, no. 04 (2010): 543–66. http://dx.doi.org/10.1142/s0218202510004337.

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We consider two models of flow and transport in porous media, the first one for consolidational flow in compressible sedimentary basins, the second one for flow in partially saturated media. Despite the differences in these physical settings, they lead to quite similar mathematical models with a strong pressure coupling. The first model is a coupled system of pde's of parabolic type. The second one involves a coupled system of pdes of degenerate parabolic–hyperbolic type. We state an existence result of weak solutions for both models.

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13

Nazir, Umar, Muhammad Sohail, Muhammad Bilal Hafeez, and Marek Krawczuk. "Significant Production of Thermal Energy in Partially Ionized Hyperbolic Tangent Material Based on Ternary Hybrid Nanomaterials." Energies 14, no. 21 (2021): 6911. http://dx.doi.org/10.3390/en14216911.

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Nanoparticles are frequently used to enhance the thermal performance of numerous materials. This study has many practical applications for activities that have to minimize losses of energy due to several impacts. This study investigates the inclusion of ternary hybrid nanoparticles in a partially ionized hyperbolic tangent liquid passed over a stretched melting surface. The fluid motion equation is presented by considering the rotation effect. The thermal energy expression is derived by the contribution of Joule heat and viscous dissipation. Flow equations were modeled by using the concept of

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14

ALVES, JOSÉ F., and HELDER VILARINHO. "Strong stochastic stability for non-uniformly expanding maps." Ergodic Theory and Dynamical Systems 33, no. 3 (2012): 647–92. http://dx.doi.org/10.1017/s0143385712000077.

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AbstractWe consider random perturbations of discrete-time dynamical systems. We give sufficient conditions for the stochastic stability of certain classes of maps, in a strong sense. This improves the main result in Alves and Araújo [Random perturbations of non-uniformly expanding maps. Astérisque 286 (2003), 25–62], where the stochastic stability in the $\mathrm {weak}^*$ topology was proved. Here, under slightly weaker assumptions on the random perturbations, we obtain a stronger version of stochastic stability: convergence of the density of the stationary measure to the density of the Sinai

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15

Medina, Virtudes, Antonio Lorenzo, and Mario Dı́az. "Electrogenic Na+-dependentl-alanine transport in the lizard duodenum. Involvement of systems A and ASC." American Journal of Physiology-Regulatory, Integrative and Comparative Physiology 280, no. 3 (2001): R612—R622. http://dx.doi.org/10.1152/ajpregu.2001.280.3.r612.

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l-Alanine transport across the isolated duodenal mucosa of the lizard Gallotia galloti has been studied in Ussing chambers under short-circuit conditions. Net l-alanine fluxes, transepithelial potential difference (PD), and short-circuit current ( Isc) showed concentration-dependent relationships. Na+-dependent l-alanine transport was substantially inhibited by the analog α-methyl aminoisobutyric acid (MeAIB). Likewise, MeAIB fluxes were completely inhibited byl-alanine, indicating the presence of system A for neutral amino acid transport. System A transport activity was electrogenic and exhib

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16

Harrison, David K., Mario Fasching, Mona Fontana-Ayoub, and Erich Gnaiger. "Cytochrome redox states and respiratory control in mouse and beef heart mitochondria at steady-state levels of hypoxia." Journal of Applied Physiology 119, no. 10 (2015): 1210–18. http://dx.doi.org/10.1152/japplphysiol.00146.2015.

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Mitochondrial control of cellular redox states is a fundamental component of cell signaling in the coordination of core energy metabolism and homeostasis during normoxia and hypoxia. We investigated the relationship between cytochrome redox states and mitochondrial oxygen consumption at steady-state levels of hypoxia in mitochondria isolated from beef and mouse heart (BHImt, MHImt), comparing two species with different cardiac dynamics and local oxygen demands. A low-noise, rapid spectrophotometric system using visible light for the measurement of cytochrome redox states was combined with high

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17

Clark Butler and Disheng Xu. "Uniformly quasiconformal partially hyperbolic systems." Annales scientifiques de l'École normale supérieure 51, no. 5 (2018): 1085–127. http://dx.doi.org/10.24033/asens.2372.

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18

Zhang, Pengfei. "Partially hyperbolic sets with positive measure and $ACIP$ for partially hyperbolic systems." Discrete & Continuous Dynamical Systems - A 32, no. 4 (2012): 1435–47. http://dx.doi.org/10.3934/dcds.2012.32.1435.

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19

Hammerlindl, Andy. "Ergodic components of partially hyperbolic systems." Commentarii Mathematici Helvetici 92, no. 1 (2017): 131–84. http://dx.doi.org/10.4171/cmh/409.

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20

Dolgopyat, Dmitry. "Limit theorems for partially hyperbolic systems." Transactions of the American Mathematical Society 356, no. 4 (2003): 1637–89. http://dx.doi.org/10.1090/s0002-9947-03-03335-x.

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21

Katok, A., and A. Kononenko. "Cocycles' stability for partially hyperbolic systems." Mathematical Research Letters 3, no. 2 (1996): 191–210. http://dx.doi.org/10.4310/mrl.1996.v3.n2.a6.

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22

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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23

XU, LAN, and BEIMEI CHEN. "TWO NOTES ABOUT THE ERGODICITY OF PARTIALLY HYPERBOLIC SYSTEMS." International Journal of Bifurcation and Chaos 23, no. 07 (2013): 1350123. http://dx.doi.org/10.1142/s021812741350123x.

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In this paper, two notes about the ergodicity of partially hyperbolic systems are given. First one is the ergodicity for a C2 volume preserving partially hyperbolic diffeomorphism of a smooth compact Riemannian manifold which is essentially accessible and weak central exponentially bunched. Second one is that for a C2 partially hyperbolic diffeomorphism, if both forward and backward center bunched are a full probability set, then it is center bunched in the sense of [Burns & Wilkinson, 2010].

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24

HU, HUYI, YUNHUA ZHOU, and YUJUN ZHU. "Quasi-shadowing for partially hyperbolic diffeomorphisms." Ergodic Theory and Dynamical Systems 35, no. 2 (2014): 412–30. http://dx.doi.org/10.1017/etds.2014.126.

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AbstractA partially hyperbolic diffeomorphism $f$ has the quasi-shadowing property if for any pseudo orbit $\{x_{k}\}_{k\in \mathbb{Z}}$, there is a sequence of points $\{y_{k}\}_{k\in \mathbb{Z}}$ tracing it in which $y_{k+1}$ is obtained from $f(y_{k})$ by a motion ${\it\tau}$ along the center direction. We show that any partially hyperbolic diffeomorphism has the quasi-shadowing property, and if $f$ has a $C^{1}$ center foliation then we can require ${\it\tau}$ to move the points along the center foliation. As applications, we show that any partially hyperbolic diffeomorphism is topological

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25

Climenhaga, Vaughn, Yakov Pesin, and Agnieszka Zelerowicz. "Equilibrium measures for some partially hyperbolic systems." Journal of Modern Dynamics 16 (2020): 155–205. http://dx.doi.org/10.3934/jmd.2020006.

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26

Burns, Keith, and Amie Wilkinson. "On the ergodicity of partially hyperbolic systems." Annals of Mathematics 171, no. 1 (2010): 451–89. http://dx.doi.org/10.4007/annals.2010.171.451.

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27

Wirth, Jens. "Diffusion phenomena for partially dissipative hyperbolic systems." Journal of Mathematical Analysis and Applications 414, no. 2 (2014): 666–77. http://dx.doi.org/10.1016/j.jmaa.2014.01.034.

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28

Tian, Xueting, and Weisheng Wu. "Unstable entropies and dimension theory of partially hyperbolic systems." Nonlinearity 35, no. 1 (2021): 658–80. http://dx.doi.org/10.1088/1361-6544/ac3dcb.

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Abstract In this paper we define unstable topological entropy for any subsets (not necessarily compact or invariant) in partially hyperbolic systems as a Carathéodory–Pesin dimension characteristic, motivated by the work of Bowen and Pesin etc. We then establish some basic results in dimension theory for Bowen unstable topological entropy, including an entropy distribution principle and a variational principle in general setting. As applications of this new concept, we study unstable topological entropy of saturated sets and extend some results in Bowen (1973 Trans. Am. Math. Soc. 184 125–36);

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29

De Simoi, Jacopo, and Carlangelo Liverani. "Limit theorems for fast–slow partially hyperbolic systems." Inventiones mathematicae 213, no. 3 (2018): 811–1016. http://dx.doi.org/10.1007/s00222-018-0798-9.

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30

Kalinin, Boris, and Victoria Sadovskaya. "Cocycles with one exponent over partially hyperbolic systems." Geometriae Dedicata 167, no. 1 (2012): 167–88. http://dx.doi.org/10.1007/s10711-012-9808-z.

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31

Beauchard, Karine, and Enrique Zuazua. "Large Time Asymptotics for Partially Dissipative Hyperbolic Systems." Archive for Rational Mechanics and Analysis 199, no. 1 (2010): 177–227. http://dx.doi.org/10.1007/s00205-010-0321-y.

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32

Suprun, E. N. "Prediction of uncontrolled course of bronchial asthma in children based on polymorphisms of genes of signaling molecules of the immune system and detoxification genes." Bulletin Physiology and Pathology of Respiration 1, no. 86 (2022): 56–61. http://dx.doi.org/10.36604/1998-5029-2022-86-56-61.

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Aim. To study the possibility of predicting the asthma control at various stages of the development of the disease, possibly on the basis of taking into account the genetic polymorphisms of Toll-like receptors, cytokines and detoxification system genes using the statistical method of learning neural networks.Materials and methods. We examined 167 children with bronchial asthma. The degree of asthma control was determined, the following mutations were detected: TLR2-Arg753Glu, TLR4-Asp299Gly, TLR4-Ghr399Ile, TLR9-T1237C, TLR9-A2848G; IL4-C589T, IL6- C174G, IL10-G1082A, IL10-C592A, IL10-C819T,

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33

HUA, YONGXIA, RADU SAGHIN, and ZHIHONG XIA. "Topological entropy and partially hyperbolic diffeomorphisms." Ergodic Theory and Dynamical Systems 28, no. 3 (2008): 843–62. http://dx.doi.org/10.1017/s0143385707000405.

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AbstractWe consider partially hyperbolic diffeomorphisms on compact manifolds. We define the notion of the unstable and stable foliations stably carrying some unique non-trivial homologies. Under this topological assumption, we prove the following two results: if the center foliation is one-dimensional, then the topological entropy is locally a constant; and if the center foliation is two-dimensional, then the topological entropy is continuous on the set of all $C^{\infty }$ diffeomorphisms. The proof uses a topological invariant we introduced, Yomdin’s theorem on upper semi-continuity, Katok’

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34

de Simoi, Jacopo, Carlangelo Liverani, Christophe Poquet, and Denis Volk. "Fast–Slow Partially Hyperbolic Systems Versus Freidlin–Wentzell Random Systems." Journal of Statistical Physics 166, no. 3-4 (2016): 650–79. http://dx.doi.org/10.1007/s10955-016-1628-3.

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35

Turo, Jan. "On some class of quasilinear hyperbolic systems of partial differential-functional equations of the first order." Czechoslovak Mathematical Journal 36, no. 2 (1986): 185–97. http://dx.doi.org/10.21136/cmj.1986.102083.

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36

Castorrini, Roberto, and Carlangelo Liverani. "Quantitative statistical properties of two-dimensional partially hyperbolic systems." Advances in Mathematics 409 (November 2022): 108625. http://dx.doi.org/10.1016/j.aim.2022.108625.

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37

Liu, Cunming, and Peng Qu. "Global classical solution to partially dissipative quasilinear hyperbolic systems." Journal de Mathématiques Pures et Appliquées 97, no. 3 (2012): 262–81. http://dx.doi.org/10.1016/j.matpur.2011.06.001.

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38

Dolgopyat, Dmitry. "On differentiability of SRB states for partially hyperbolic systems." Inventiones Mathematicae 155, no. 2 (2004): 389–449. http://dx.doi.org/10.1007/s00222-003-0324-5.

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39

Zhou, Yi. "Global classical solutions to partially dissipative quasilinear hyperbolic systems." Chinese Annals of Mathematics, Series B 32, no. 5 (2011): 771–80. http://dx.doi.org/10.1007/s11401-011-0666-z.

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40

WU, WEISHENG. "Local unstable entropies of partially hyperbolic diffeomorphisms." Ergodic Theory and Dynamical Systems 40, no. 8 (2019): 2274–304. http://dx.doi.org/10.1017/etds.2019.3.

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Consider a $C^{1}$-partially hyperbolic diffeomorphism $f:M\rightarrow M$. Following the ideas in establishing the local variational principle for topological dynamical systems, we introduce the notions of local unstable metric entropies (and local unstable topological entropy) relative to a Borel cover ${\mathcal{U}}$ of $M$. It is shown that they coincide with the unstable metric entropy (and unstable topological entropy, respectively), when ${\mathcal{U}}$ is an open cover with small diameter. We also define the unstable tail entropy in the sense of Bowen and the unstable topological condit

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41

CASTRO, ARMANDO, and TEÓFILO NASCIMENTO. "Statistical properties of the maximal entropy measure for partially hyperbolic attractors." Ergodic Theory and Dynamical Systems 37, no. 4 (2016): 1060–101. http://dx.doi.org/10.1017/etds.2015.86.

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We show the existence and uniqueness of the maximal entropy probability measure for partially hyperbolic diffeomorphisms which are semiconjugate to non-uniformly expanding maps. Using the theory of projective metrics on cones, we then prove exponential decay of correlations for Hölder continuous observables and the central limit theorem for the maximal entropy probability measure. Moreover, for systems derived from a solenoid, we also prove the statistical stability for the maximal entropy probability measure. Finally, we use such techniques to obtain similar results in a context containing pa

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42

De Simoi, Jacopo, and Carlangelo Liverani. "Statistical properties of mostly contracting fast-slow partially hyperbolic systems." Inventiones mathematicae 206, no. 1 (2016): 147–227. http://dx.doi.org/10.1007/s00222-016-0651-y.

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43

Zhou, Yunhua. "Non-zero Lyapunov exponents for some conservative partially hyperbolic systems." Proceedings of the American Mathematical Society 143, no. 7 (2015): 3147–53. http://dx.doi.org/10.1090/s0002-9939-2015-12498-7.

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44

Mi, Zeya, Yongluo Cao, and Dawei Yang. "A note on partially hyperbolic systems with mostly expanding centers." Proceedings of the American Mathematical Society 145, no. 12 (2017): 5299–313. http://dx.doi.org/10.1090/proc/13701.

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45

Kowalski, Julia, and Jim N. McElwaine. "Shallow two-component gravity-driven flows with vertical variation." Journal of Fluid Mechanics 714 (January 2, 2013): 434–62. http://dx.doi.org/10.1017/jfm.2012.489.

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AbstractGravity-driven geophysical mass flows often consist of a heterogeneous fluid–solid mixture. The complex interplay between the components leads to phenomena such as lateral levee formation in avalanches, or a granular front and an excess fluid pore pressure in debris flows. These effects are very important for predicting runout and the forces on structures, yet they are only partially represented in simplified shallow flow theories, since rearrangement of the mixture composition perpendicular to the main flow direction is neglected. In realistic flows, however, rheological properties an

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Koellermeier, Julian, and Manuel Torrilhon. "Numerical Study of Partially Conservative Moment Equations in Kinetic Theory." Communications in Computational Physics 21, no. 4 (2017): 981–1011. http://dx.doi.org/10.4208/cicp.oa-2016-0053.

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AbstractMoment models are often used for the solution of kinetic equations such as the Boltzmann equation. Unfortunately, standard models like Grad's equations are not hyperbolic and can lead to nonphysical solutions. Newly derived moment models like the Hyperbolic Moment Equations and the Quadrature-Based Moment Equations yield globally hyperbolic equations but are given in partially conservative form that cannot be written as a conservative system.In this paper we investigate the applicability of different dedicated numerical schemes to solve the partially conservative model equations. Cause

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Zhao, Liang, and Shuai Xi. "Convergence rate from systems of balance laws to isotropic parabolic systems, a periodic case." Asymptotic Analysis 124, no. 1-2 (2021): 163–98. http://dx.doi.org/10.3233/asy-211687.

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It is proved that partially dissipative hyperbolic systems converge globally-in-time to parabolic systems in a slow time scaling, when initial data are smooth and sufficiently close to constant equilibrium states. Based on this result, we establish the global-in-time error estimates between the smooth solutions to the partially dissipative hyperbolic systems and those to the isotropic parabolic limiting systems in a three dimensional torus, rather than in the one dimensional whole space (Appl. Anal. 100(5) (2021) 1079–1095). This avoids the condition raised for the strong connection between th

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Huang, Ping, Chenwei Wang, and Ercai Chen. "Unstable topological entropy in mean u-metrics for partially hyperbolic systems." Dynamical Systems 36, no. 3 (2021): 387–403. http://dx.doi.org/10.1080/14689367.2021.1923659.

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Burguet, David, and Todd Fisher. "Symbolic extensionsfor partially hyperbolic dynamical systems with 2-dimensional center bundle." Discrete & Continuous Dynamical Systems - A 33, no. 6 (2013): 2253–70. http://dx.doi.org/10.3934/dcds.2013.33.2253.

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Li, Xiaolong, and Katsutoshi Shinohara. "On super-exponential divergence of periodic points for partially hyperbolic systems." Discrete & Continuous Dynamical Systems 42, no. 4 (2022): 1707. http://dx.doi.org/10.3934/dcds.2021169.

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<p style='text-indent:20px;'>We say that a diffeomorphism <inline-formula><tex-math id="M1">\begin{document}$ f $\end{document}</tex-math></inline-formula> is super-exponentially divergent if for every <inline-formula><tex-math id="M2">\begin{document}$ b&gt;1 $\end{document}</tex-math></inline-formula> the lower limit of <inline-formula><tex-math id="M3">\begin{document}$ \#\mbox{Per}_n(f)/b^n $\end{document}</tex-math></inline-formula> diverges to infinity, where <inline-formula><tex-math id="M4">\b

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