[Show abstract][Hide abstract] ABSTRACT: In recent work Cortez and Petite defined odometer actions of discrete,
finitely generated and residually finite groups G. In this paper we focus on
the case where G is the discrete Heisenberg group. We prove a structure theorem
for finite index subgroups of the Heisenberg group based on their geometry when
they are considered as subsets of Z^3. We provide a complete classification of
Heisenberg odometers based on the structure of their defining subgroups and we
provide examples of each class. Mackey has shown that all such actions have
discrete spectrum, i.e. that the unitary operator associated to the dynamical
system admits a decomposition into finite dimensional, irreducible
representations of the group G. Here we provide an explicit proof of this fact
for general G odometers. Our proof allows us to define explicitly those
representations of the Heisenberg group which appear in the spectral
decomposition of a Heisenberg odometer, as a function of the defining
subgroups. Along the way we also provide necessary and sufficient conditions
for a Z^d odometer to be a product odometer as defined by Cortez.
Proceedings of the American Mathematical Society 09/2011; 142(7). DOI:10.1090/S0002-9939-2014-11963-0 · 0.68 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We describe a general method of arithmetic coding of geodesics on the modular
surface based on a two parameter family of continued fraction transformations
studied previously by the authors. The finite rectangular structure of the
attractors of the natural extension maps and the corresponding "reduction
theory" play an essential role. In special cases, when an (a,b)-expansion
admits a so-called "dual", the coding sequences are obtained by juxtaposition
of the boundary expansions of the fixed points, and the set of coding sequences
is a countable sofic shift. We also prove that the natural extension maps are
Bernoulli shifts and compute the density of the absolutely continuous invariant
measure and the measure-theoretic entropy of the one-dimensional map.
Ergodic Theory and Dynamical Systems 05/2011; 17. DOI:10.3934/era.2010.17.20 · 0.78 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We study a two-parameter family of one-dimensional maps and related (a,b)-continued fractions suggested for consideration by Don Zagier. We prove that the associated natural extension maps have attractors with finite rectangular structure for the entire parameter set except for a Cantor-like set of one-dimensional Lebesgue zero measure that we completely describe. We show that the structure of these attractors can be "computed" from the data (a,b), and that for a dense open set of parameters the Reduction theory conjecture holds, i.e. every point is mapped to the attractor after finitely many iterations. We also show how this theory can be applied to the study of invariant measures and ergodic properties of the associated Gauss-like maps. Comment: 51 pages, 8 figures
Journal of Modern Dynamics 04/2010; 4(4). DOI:10.3934/jmd.2010.4.637 · 0.84 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Computer simulations have shown that several classes of population models, including the May host-parasitoid model and the Ginzburg-Taneyhill "maternal-quality" single species popula-tion model, exhibit extremely complicated orbit structures. These structures include islands-around-islands, ad infinitum, with the smaller islands containing stable periodic points of higher period. We identify the mechanism that generates this complexity and we discuss some biological implications.
[Show abstract][Hide abstract] ABSTRACT: Following ecologists discoveries, mathematicians have begun studying extensions of the ubiquitous age structured Leslie population model to allow some survival probabilities and/or fer-tility rates depend on population densities. These nonlinear ex-tensions commonly exhibit very complicated dynamics: through computer studies, some authors have discovered robust Hénon-like strange attractors in several families. Population researchers frequently wish to average a function over many generations and conclude that the average is indepen-dent of the initial population distribution. This type of "ergodic-ity" seems to be a fundamental tenet in population biology. In this manuscript we develop the first rigorous ergodic theoretic frame-work for density dependent Leslie population models. We study two generation models with Ricker and Hassell (recruitment type) fertility terms. We prove that for some parameter regions these models admit a chaotic (ergodic) attractor which supports a unique physical probability measure. This physical measure, having full Lebesgue measure basin, satisfies in the strongest possible sense the population biologist's requirement for ergodicity in their pop-ulation models. We use the celebrated work of Wang and Young , and our results are the first applications of their method to the biological sciences.
[Show abstract][Hide abstract] ABSTRACT: In this expository article we describe the two main methods of representing geodesics on surfaces of constant negative curvature by symbolic sequences and their development. A geometric method stems from a 1898 work of J. Hadamard and was developed by M. Morse in the 1920s. It consists of recording the successive sides of a given fundamental region cut by the geodesic and may be applied to all finitely generated Fuchsian groups. Another method, of arithmetic nature, uses continued fraction expansions of the end points of the geodesic at infinity and is even older—it comes from the Gauss reduction theory. Introduced to dynamics by E. Artin in a 1924 paper, this method was used to exhibit dense geodesics on the modular surface. For 80 years these classical works have provided inspiration for mathematicians and a testing ground for new methods in dynamics, geometry and combinatorial group theory. We present some of the ideas, results (old and recent), and interpretations that illustrate the multiple facets of the subject.
Bulletin of the American Mathematical Society 10/2006; 44(01):87-133. DOI:10.1090/S0273-0979-06-01115-3 · 2.11 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We prove that if a diffeomorphism on a compact manifold pre- serves a nonatomic hyperbolic Borel probability measure, then there exists a hyperbolic periodic point such that the closure of its unstable manifold has positive measure. Moreover, the support of the measure is contained in the closure of all such hyperbolic periodic points. We also show that if an ergodic hyperbolic probability measure does not locally maximize entropy in the space of invariant ergodic hyperbolic measures, then there exist hyperbolic periodic points that satisfy a multiplicative asymptotic growth and are uniformly dis- tributed with respect to this measure.
Discrete and Continuous Dynamical Systems 10/2006; 16(2):505-512. DOI:10.3934/dcds.2006.16.505 · 0.83 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: In this article we present three arithmetic methods for coding oriented geodesics on the modular surface using various continued fraction expansions and show that the space of admissible coding sequences for each coding is a one-step topological Markov chain with countable alphabet. We also present conditions under which these arithmetic codes coincide with the geometric code obtained by recording oriented excursions into the cusp of the modular surface.
[Show abstract][Hide abstract] ABSTRACT: Dedicated to Yu. S. Ilyashenko on the occasion of his sixtieth birthday Abstract. The Morse method of coding geodesics on a surface of constant negative curvature consists of recording the sides of a given fundamental region cut by the geodesic. For the modular surface with the standard fundamental region each geodesic (which does not go to the cusp in either direction) is represented by a bi-infinite sequence of non-zero integers called its geometric code. In this paper we show that the set of all geometric codes is not a finite-step Markov chain, and identify a maximal 1-step topological Markov chain of ad- missible geometric codes which we call, as well as the corresponding geodesics, geometrically Markov. We also show that the set of geometrically Markov codes is the maximal symmetric 1-step topological Markov chain of admissi- ble geometric codes, and obtain an estimate from below for the topological entropy of the geodesic flow restricted to this set.
[Show abstract][Hide abstract] ABSTRACT: We study the dynamics of an overcompensatory Leslie population model where the fertility rates decay exponentially with population size. We find a plethora of complicated dynamical behaviour, some of which has not been previously observed in population models and which may give rise to new paradigms in population biology and demography. We study the two-and three-dimensional models and find a large variety of complicated behaviour: all codimension 1 local bifurcations, period doubling cascades, attracting closed curves that bifurcate into strange attractors, multiple coexisting strange attractors with large basins (which cause an intrinsic lack of 'ergodicity'), crises that can cause a discontinuous large population swing, merging of attractors, phase locking and transient chaos. We find (and explain) two different bifurcation cascades transforming an attracting invariant closed curve into a strange attractor. We also find one-parameter families that exhibit most of these phenomena. We show that some of the more exotic phenomena arise from homoclinic tangencies.