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14
Orbitcounting in nonhyperbolic dynamical systems
 J. Reine Angew. Math
"... Abstract. There are wellknown analogs of the prime number theorem and Mertens ’ theorem for dynamical systems with hyperbolic behaviour. Here we consider the same question for the simplest nonhyperbolic algebraic systems. The asymptotic behaviour of the orbitcounting function is governed by a rot ..."
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Cited by 15 (14 self)
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Abstract. There are wellknown analogs of the prime number theorem and Mertens ’ theorem for dynamical systems with hyperbolic behaviour. Here we consider the same question for the simplest nonhyperbolic algebraic systems. The asymptotic behaviour of the orbitcounting function is governed by a rotation on an associated compact group, and in simple examples we exhibit uncountably many different asymptotic growth rates for the orbitcounting function. Mertens ’ Theorem also holds in this setting, with an explicit rational leading coefficient obtained from arithmetic properties of the nonhyperbolic eigendirections. 1.
Zeta functions for elements of entropy rank one actions
, 2006
"... An algebraic Z daction of entropy rank one is one for which each element has finite entropy. Using the structure theory of these actions due to Einsiedler and Lind, this paper investigates dynamical zeta functions for elements of the action. An explicit periodic point formula is obtained leading t ..."
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Cited by 14 (9 self)
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An algebraic Z daction of entropy rank one is one for which each element has finite entropy. Using the structure theory of these actions due to Einsiedler and Lind, this paper investigates dynamical zeta functions for elements of the action. An explicit periodic point formula is obtained leading to a uniform parameterization of the zeta functions that arise in expansive components of an expansive action, together with necessary and sufficient conditions for rationality in a more general setting.
Dirichlet series for finite combinatorial rank dynamics
 Trans. Amer. Math. Soc
"... Abstract. We introduce a class of group endomorphisms – those of finite combinatorial rank – exhibiting slow orbit growth. An associated Dirichlet series is used to obtain an exact orbit counting formula, and in the connected case this series is shown to to be a rational function of exponential vari ..."
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Cited by 13 (8 self)
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Abstract. We introduce a class of group endomorphisms – those of finite combinatorial rank – exhibiting slow orbit growth. An associated Dirichlet series is used to obtain an exact orbit counting formula, and in the connected case this series is shown to to be a rational function of exponential variables. Analytic properties of the Dirichlet series are related to orbitgrowth asymptotics: depending on the location of the abscissa of convergence and the degree of the pole there, various orbitgrowth asymptotics are found, all
PERIODIC POINTS OF ENDOMORPHISMS ON SOLENOIDS AND RELATED GROUPS
"... This paper investigates the problem of finding the possible sequences of periodic point counts for endomorphisms of solenoids. For an ergodic epimorphism of a solenoid, a closed formula is given which expresses the number of points of any given period in terms of sets of places of finitely many alge ..."
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Cited by 13 (9 self)
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This paper investigates the problem of finding the possible sequences of periodic point counts for endomorphisms of solenoids. For an ergodic epimorphism of a solenoid, a closed formula is given which expresses the number of points of any given period in terms of sets of places of finitely many algebraic number fields and distinguished elements of those fields. The result extends to more general epimorphisms of compact abelian groups.
Periodic point data detects subdynamics in entropy rank one
, 2006
"... A framework for understanding the geometry of continuous actions of Z d was developed by Boyle and Lind using the notion of expansive behavior along lowerdimensional subspaces. For algebraic Z dactions of entropy rank one, the expansive subdynamics is readily described in terms of Lyapunov exponen ..."
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Cited by 6 (5 self)
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A framework for understanding the geometry of continuous actions of Z d was developed by Boyle and Lind using the notion of expansive behavior along lowerdimensional subspaces. For algebraic Z dactions of entropy rank one, the expansive subdynamics is readily described in terms of Lyapunov exponents. Here we show that periodic point counts for elements of an entropy rank one action determine the expansive subdynamics. Moreover, the finer structure of the nonexpansive set is visible in the topological and smooth structure of a set of functions associated to the periodic point data.
Towards a Pólya–Carlson dichotomy for algebraic dynamics
, 2013
"... We present results and background rationale in support of a Pólya–Carlson dichotomy between rationality and a natural boundary for the analytic behaviour of dynamical zeta functions of compact group automorphisms. ..."
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Cited by 3 (3 self)
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We present results and background rationale in support of a Pólya–Carlson dichotomy between rationality and a natural boundary for the analytic behaviour of dynamical zeta functions of compact group automorphisms.
AUTOMORPHISMS WITH EXOTIC ORBIT GROWTH
"... Abstract. The dynamical Mertens ’ theorem describes asymptotics for the growth in the number of closed orbits in a dynamical system. We construct families of ergodic automorphisms of fixed entropy on compact connected groups with a continuum of growth rates on two different growth scales. This shows ..."
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Cited by 3 (2 self)
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Abstract. The dynamical Mertens ’ theorem describes asymptotics for the growth in the number of closed orbits in a dynamical system. We construct families of ergodic automorphisms of fixed entropy on compact connected groups with a continuum of growth rates on two different growth scales. This shows in particular that the space of all ergodic algebraic dynamical systems modulo the equivalence of shared orbitgrowth asymptotics is not countable. In contrast, for the equivalence relation of measurable isomorphism or equal entropy it is not known if the quotient space is countable or uncountable (this problem is a manifestation of Lehmer’s problem). 1.