Entropy and growth rate of periodic points of algebraic ℤ^{𝕕}-actions

AbstractCyclic algebraic ${\mathbb {Z}^{d}}$-actions are defined by ideals of Laurent polynomials in $d$ commuting variables. Such an action is expansive precisely when the complex variety of the ideal is disjoint from the multiplicative $d$-torus. For such expansive actions it is known that the limit of the growth rate of periodic points exists and is equal to the entropy of the action. In an earlier paper the authors extended this result to ideals whose variety intersects the $d$-torus in a finite set. Here we further extend it to the case where the dimension of intersection of the variety with the $d$-torus is at most $d-2$. The main tool is the construction of homoclinic points which decay rapidly enough to be summable.

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Isomorphism Classes of Solenoidal Algebras I

Canadian Mathematical Bulletin ◽

10.4153/cmb-1993-056-2 ◽

1993 ◽

Vol 36 (4) ◽

pp. 414-418 ◽

Cited By ~ 1

Author(s):

Berndt Brenken

Keyword(s):

Dynamical System ◽

Growth Rate ◽

Unit Circle ◽

Compact Space ◽

Periodic Points ◽

Crossed Product ◽

Roots Of Unity ◽

Isomorphism Classes ◽

Crossed Product Algebra ◽

Product Algebra

AbstractEach g ∊ ℤ[x] defines a homeomorphism of a compact space We investigate the isomorphism classes of the C*-crossed product algebra Bg associated with the dynamical system An isomorphism invariant Eg of the algebra Bg is shown to determine the algebra Bg up to * or * anti-isomorphism if degree g ≤ 1 and 1 is not a root of g or if degree g = 2 and g is irreducible. It is also observed that the entropy of the dynamical system is equal to the growth rate of the periodic points if g has no roots of unity as zeros. This slightly extends the previously known equality of these two quantities under the assumption that g has no zeros on the unit circle.

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Bowen Measure From Heteroclinic Points

Canadian Journal of Mathematics ◽

10.4153/cjm-2011-083-0 ◽

2012 ◽

Vol 64 (6) ◽

pp. 1341-1358 ◽

Cited By ~ 1

Author(s):

D. B. Killough ◽

I. F. Putnam

Keyword(s):

Growth Rate ◽

Invariant Probability Measure ◽

Periodic Points ◽

Direct Computation ◽

Shift Of Finite Type ◽

New Construction ◽

Smale Space ◽

Factor Maps ◽

Smale Spaces ◽

Heteroclinic Points

Abstract We present a new construction of the entropy-maximizing, invariant probability measure on a Smale space (the Bowen measure). Our construction is based on points that are unstably equivalent to one given point, and stably equivalent to another, i.e., heteroclinic points. The spirit of the construction is similar to Bowen's construction from periodic points, though the techniques are very different. We also prove results about the growth rate of certain sets of heteroclinic points, and about the stable and unstable components of the Bowen measure. The approach we take is to prove results through direct computation for the case of a Shift of Finite type, and then use resolving factor maps to extend the results to more general Smale spaces.

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On super-exponential divergence of periodic points for partially hyperbolic systems

Discrete and Continuous Dynamical Systems ◽

10.3934/dcds.2021169 ◽

2021 ◽

Vol 0 (0) ◽

pp. 0

Author(s):

Xiaolong Li ◽

Katsutoshi Shinohara

Keyword(s):

Growth Rate ◽

Lower Limit ◽

Open Subset ◽

Exponential Growth ◽

Dense Subset ◽

Hyperbolic Systems ◽

Periodic Points ◽

Residual Subset ◽

Partially Hyperbolic ◽

Smooth Closed Manifold

<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>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 $\end{document}</tex-math></inline-formula> diverges to infinity, where <inline-formula><tex-math id="M4">\begin{document}$ \mbox{Per}_n(f) $\end{document}</tex-math></inline-formula> is the set of all periodic points of <inline-formula><tex-math id="M5">\begin{document}$ f $\end{document}</tex-math></inline-formula> with period <inline-formula><tex-math id="M6">\begin{document}$ n $\end{document}</tex-math></inline-formula>. This property is stronger than the usual super-exponential growth of the number of periodic points. We show that for any <inline-formula><tex-math id="M7">\begin{document}$ n $\end{document}</tex-math></inline-formula>-dimensional smooth closed manifold <inline-formula><tex-math id="M8">\begin{document}$ M $\end{document}</tex-math></inline-formula> where <inline-formula><tex-math id="M9">\begin{document}$ n\ge 3 $\end{document}</tex-math></inline-formula>, there exists a non-empty open subset <inline-formula><tex-math id="M10">\begin{document}$ \mathcal{O} $\end{document}</tex-math></inline-formula> of <inline-formula><tex-math id="M11">\begin{document}$ \mbox{Diff}^1(M) $\end{document}</tex-math></inline-formula> such that diffeomorphisms with super-exponentially divergent property form a dense subset of <inline-formula><tex-math id="M12">\begin{document}$ \mathcal{O} $\end{document}</tex-math></inline-formula> in the <inline-formula><tex-math id="M13">\begin{document}$ C^1 $\end{document}</tex-math></inline-formula>-topology. A relevant result about the growth rate of the lower limit of the number of periodic points for diffeomorphisms in a <inline-formula><tex-math id="M14">\begin{document}$ C^r $\end{document}</tex-math></inline-formula>-residual subset of <inline-formula><tex-math id="M15">\begin{document}$ \mbox{Diff}^r(M)\ (1\le r\le \infty) $\end{document}</tex-math></inline-formula> is also shown.</p>

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Growth rate of the number of periodic points

Normal Forms, Bifurcations and Finiteness Problems in Differential Equations ◽

10.1007/978-94-007-1025-2_10 ◽

2004 ◽

pp. 355-385 ◽

Cited By ~ 2

Author(s):

Vadim Yu. Kaloshin

Keyword(s):

Growth Rate ◽

Periodic Points

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The threshold energy for atom displacement in fcc metals

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100175788 ◽

1990 ◽

Vol 48 (4) ◽

pp. 530-531

Author(s):

Wilfried Sigle ◽

Matthias Hohenstein ◽

Alfred Seeger

Keyword(s):

Growth Rate ◽

Elevated Temperatures ◽

Threshold Energy ◽

Dislocation Loops ◽

Absolute Minimum ◽

Voltage Electron ◽

Atom Displacement ◽

Electron Microscopes ◽

Fcc Metal

Prolonged electron irradiation of metals at elevated temperatures usually leads to the formation of large interstitial-type dislocation loops. The growth rate of the loops is proportional to the total cross-section for atom displacement,which is implicitly connected with the threshold energy for atom displacement, Ed . Thus, by measuring the growth rate as a function of the electron energy and the orientation of the specimen with respect to the electron beam, the anisotropy of Ed can be determined rather precisely. We have performed such experiments in situ in high-voltage electron microscopes on Ag and Au at 473K as a function of the orientation and on Au as a function of temperature at several fixed orientations.Whereas in Ag minima of Ed are found close to <100>,<110>, and <210> (13-18eV), (Fig.1) atom displacement in Au requires least energy along <100>(15-19eV) (Fig.2). Au is thus the first fcc metal in which the absolute minimum of the threshold energy has been established not to lie in or close to the <110> direction.

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