Cerbelli and Giona’s map is pseudo-Anosov (original) (raw)

Cerbelli and Giona's Map Is Pseudo-Anosov and Nine Consequences

Journal of Nonlinear Science, 2006

It is shown that a piecewise affine area-preserving homeomorphism of the 2-torus studied by Cerbelli and Giona is pseudo-Anosov. This enables one to prove various of their conjectures, quantify the multifractality of its "w-measures," calculate many other quantities for its dynamics, and construct an exact area-preserving tilt map of the cylinder with proved diffusive behaviour.

Invariant sets for discontinuous parabolic area-preserving torus maps

Nonlinearity, 2000

We analyze a class of piecewise linear parabolic maps on the torus, namely those obtained by considering a linear map with double eigenvalue one and taking modulo one in each component. We show that within this two parameter family of maps, the set of noninvertible maps is open and dense. For cases where the entries in the matrix are rational we show that the maximal invariant set has positive Lebesgue measure and we give bounds on the measure. For several examples we find expressions for the measure of the invariant set but we leave open the question as to whether there are parameters for which this measure is zero.

Topological entropy of pseudo-Anosov maps on punctured surfaces vs. homology of mapping tori

2022

We investigate the relation between the topological entropy of pseudo-Anosov maps on surfaces with punctures and the rank of the first homology of their mapping tori. On the surface SSS of genus ggg with nnn punctures, we show that the entropy of a pseudo-Anosov map is bounded from above by dfrac(k+1)log(k+3)∣chi(S)∣\dfrac{(k+1)\log(k+3)}{|\chi(S)|}dfrac(k+1)log(k+3)∣chi(S)∣ up to a constant multiple when the rank of the first homology of the mapping torus is k+1k+1k+1 and k,g,nk, g, nk,g,n satisfy a certain assumption. This is a partial generalization of precedent works of Tsai and Agol-Leininger-Margalit.

A construction of pseudo-Anosov homeomorphisms

Transactions of the American Mathematical Society, 1988

We describe a generalization of Thurston’s original construction of pseudo-Anosov maps on a surface F F of negative Euler characteristic. In fact, we construct whole semigroups of pseudo-Anosov maps by taking appropriate compositions of Dehn twists along certain families of curves; our arguments furthermore apply to give examples of pseudo-Anosov maps on nonorientable surfaces. For each self-map f : F → F f:F \to F arising from our recipe, we construct an invariant "bigon track" (a slight generalization of train track) whose incidence matrix is Perron-Frobenius. Standard arguments produce a projective measured foliation invariant by f f . To finally prove that f f is pseudo-Anosov, we directly produce a transverse invariant projective measured foliation using tangential measures on bigon tracks. As a consequence of our argument, we derive a simple criterion for a surface automorphism to be pseudo-Anosov.

A Note on Topological Entropy of Continuous Self-Maps

Journal of Mathematics and System Science, 2015

Topological entropy can be an indicator of complicated behavior in dynamical systems. It is first introduce by Adler, Konheim and McAndrew by using open covers in 1965. After that it is still an active research by many researchers to produce more properties and applications up to nowadays. The purpose of this paper is to review and explain most important concepts and results of topological entropies of continuous self-maps for dynamical systems on compact and non-compact topological and metric spaces. We give proofs for some of its elementary properties of the topological entropy. Slight modification on Adler's topological entropy is also presented.

Topological Entropy in a Parameter Range of the Standard Map

Progress of Theoretical Physics, 2009

We combine the trellis method and the braid method, and by estimating the lower bounds of the topological entropy of the standard map for a certain parameter range, we follow the change of the topological entropy. The trellis in a tangency situation of the stable and unstable manifolds is constructed. Applying the trellis method to this trellis, the lower bound of topological entropy is calculated. There exist systems in which the trellis method can not be applicable. For such systems, we look for non-Birkhoff periodic orbits existent in the trellises, form braids from those periodic orbits, and estimate the topological entropy from their braid types. We perform these tasks for a sequence of trellises and numerically visualize the change in the topological entropy. In addition, we take particular sequence of connecting orbits to obtain homoclinic or heteroclinic orbits. As a natural extension, we assign topological entropy to these homoclinic and heteroclinic orbits.

A lower bound for topological entropy of generic non-Anosov symplectic diffeomorphisms

Ergodic Theory and Dynamical Systems, 2013

We prove that a C1{C}^{1} C1 generic symplectic diffeomorphism is either Anosov or its topological entropy is bounded from below by the supremum over the smallest positive Lyapunov exponent of its periodic points. We also prove that C1{C}^{1} C1 generic symplectic diffeomorphisms outside the Anosov ones do not admit symbolic extension and, finally, we give examples of volume preserving surface diffeomorphisms which are not points of upper semicontinuity of the entropy function in the C1{C}^{1} C1 topology.