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Journal of Modern Dynamics

2010 , Volume 4 , Issue 1

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Axiom A diffeomorphisms derived from Anosov flows
Christian Bonatti and Nancy Guelman
2010, 4(1): 1-63 doi: 10.3934/jmd.2010.4.1 +[Abstract](85) +[PDF](750.2KB)
Abstract:
Let $M$ be a closed $3$-manifold, and let $X_t$ be a transitive Anosov flow. We construct a diffeomorphism of the form $f(p)=Y_{t(p)}(p)$, where $Y$ is an Anosov flow equivalent to $X$. The diffeomorphism $f$ is structurally stable (satisfies Axiom A and the strong transversality condition); the non-wandering set of $f$ is the union of a transitive attractor and a transitive repeller; and $f$ is also partially hyperbolic (the direction $\RR.Y$ is the central bundle).
The Ricci flow for nilmanifolds
Tracy L. Payne
2010, 4(1): 65-90 doi: 10.3934/jmd.2010.4.65 +[Abstract](76) +[PDF](296.6KB)
Abstract:
We consider the Ricci flow for simply connected nilmanifolds. This translates to a Ricci flow on the space of nilpotent metric Lie algebras. We consider the evolution of the inner product and the evolution of structure constants, as well as the evolution of these quantities modulo rescaling. We set up systems of O.D.E.'s for some of these flows and describe their qualitative properties. We also present some explicit solutions for the evolution of soliton metrics under the Ricci flow.
Banach spaces for piecewise cone-hyperbolic maps
Viviane Baladi and Sébastien Gouëzel
2010, 4(1): 91-137 doi: 10.3934/jmd.2010.4.91 +[Abstract](1189) +[PDF](552.7KB)
Abstract:
We consider piecewise cone-hyperbolic systems satisfying a bunching condition, and we obtain a bound on the essential spectral radius of the associated weighted transfer operators acting on anisotropic Sobolev spaces. The bunching condition is always satisfied in dimension two, and our results give a unifying treatment of the work of Demers-Liverani [9] and our previous work [2]. When the complexity is subexponential, our bound implies a spectral gap for the transfer operator corresponding to the physical measures in many cases (for example if $T$ preserves volume, or if the stable dimension is equal to $1$ and the unstable dimension is not zero).
Volume entropy of hyperbolic buildings
François Ledrappier and Seonhee Lim
2010, 4(1): 139-165 doi: 10.3934/jmd.2010.4.139 +[Abstract](142) +[PDF](318.6KB)
Abstract:
We characterize the volume entropy of a regular building as the topological pressure of the geodesic flow on an apartment. We show that the entropy maximizing measure is not Liouville measure for any regular hyperbolic building. As a consequence, we obtain a strict lower bound on the volume entropy in terms of the branching numbers and the volume of the boundary polyhedrons.
Schreier graphs of the Basilica group
Daniele D'angeli, Alfredo Donno, Michel Matter and Tatiana Nagnibeda
2010, 4(1): 167-205 doi: 10.3934/jmd.2010.4.167 +[Abstract](1180) +[PDF](827.3KB)
Abstract:
To any self-similar action of a finitely generated group $G$ of automorphisms of a regular rooted tree $T$ can be naturally associated an infinite sequence of finite graphs $\{\Gamma_n\}_{n\geq 1}$, where $\Gamma_n$ is the Schreier graph of the action of $G$ on the $n$-th level of $T$. Moreover, the action of $G$ on $\partial T$ gives rise to orbital Schreier graphs $\Gamma_{\xi}$, $\xi\in \partial T$. Denoting by $\xi_n$ the prefix of length $n$ of the infinite ray $\xi$, the rooted graph $(\Gamma_{\xi},\xi)$ is then the limit of the sequence of finite rooted graphs $\{(\Gamma_n,\xi_n)\}_{n\geq 1}$ in the sense of pointed Gromov-Hausdorff convergence. In this paper, we give a complete classification (up to isomorphism) of the limit graphs $(\Gamma_{\xi},\xi)$ associated with the Basilica group acting on the binary tree, in terms of the infinite binary sequence $\xi$.
Errata to "Measure rigidity beyond uniform hyperbolicity: Invariant measures for Cartan actions on tori" and "Uniqueness of large invariant measures for $\Zk$ actions with Cartan homotopy data"
Boris Kalinin, Anatole Katok and Federico Rodriguez Hertz
2010, 4(1): 207-209 doi: 10.3934/jmd.2010.4.207 +[Abstract](92) +[PDF](55.1KB)
Abstract:
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