2,034 research outputs found

    Topological pressure of simultaneous level sets

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    Multifractal analysis studies level sets of asymptotically defined quantities in a topological dynamical system. We consider the topological pressure function on such level sets, relating it both to the pressure on the entire phase space and to a conditional variational principle. We use this to recover information on the topological entropy and Hausdorff dimension of the level sets. Our approach is thermodynamic in nature, requiring only existence and uniqueness of equilibrium states for a dense subspace of potential functions. Using an idea of Hofbauer, we obtain results for all continuous potentials by approximating them with functions from this subspace. This technique allows us to extend a number of previous multifractal results from the C1+ϵC^{1+\epsilon} case to the C1C^1 case. We consider ergodic ratios Snϕ/SnψS_n \phi/S_n \psi where the function ψ\psi need not be uniformly positive, which lets us study dimension spectra for non-uniformly expanding maps. Our results also cover coarse spectra and level sets corresponding to more general limiting behaviour.Comment: 32 pages, minor changes based on referee's comment

    Geometric Foundation of Thermo-Statistics, Phase Transitions, Second Law of Thermodynamics, but without Thermodynamic Limit

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    A geometric foundation thermo-statistics is presented with the only axiomatic assumption of Boltzmann's principle S(E,N,V)=k\ln W. This relates the entropy to the geometric area e^{S(E,N,V)/k} of the manifold of constant energy in the finite-N-body phase space. From the principle, all thermodynamics and especially all phenomena of phase transitions and critical phenomena can unambiguously be identified for even small systems. The topology of the curvature matrix C(E,N) of S(E,N) determines regions of pure phases, regions of phase separation, and (multi-)critical points and lines. Within Boltzmann's principle, Statistical Mechanics becomes a geometric theory addressing the whole ensemble or the manifold of all points in phase space which are consistent with the few macroscopic conserved control parameters. This interpretation leads to a straight derivation of irreversibility and the Second Law of Thermodynamics out of the time-reversible, microscopic, mechanical dynamics. This is all possible without invoking the thermodynamic limit, extensivity, or concavity of S(E,N,V). The main obstacle against the Second Law, the conservation of the phase-space volume due to Liouville is overcome by realizing that a macroscopic theory like Thermodynamics cannot distinguish a fractal distribution in phase space from its closure.Comment: 26 pages, 6 figure

    The thermodynamic approach to multifractal analysis

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    Most results in multifractal analysis are obtained using either a thermodynamic approach based on existence and uniqueness of equilibrium states or a saturation approach based on some version of the specification property. A general framework incorporating the most important multifractal spectra was introduced by Barreira and Saussol, who used the thermodynamic approach to establish the multifractal formalism in the uniformly hyperbolic setting, unifying many existing results. We extend this framework to apply to a broad class of non-uniformly hyperbolic systems, including examples with phase transitions. In the process, we compare this thermodynamic approach with the saturation approach and give a survey of many of the multifractal results in the literature.Comment: 51 pages, minor corrections, added formal statements of new results to "applications" sectio

    Characterization for entropy of shifts of finite type on Cayley trees

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    The notion of tree-shifts constitutes an intermediate class in between one-sided shift spaces and multidimensional ones. This paper proposes an algorithm for computing of the entropy of a tree-shift of finite type. Meanwhile, the entropy of a tree-shift of finite type is 1plnλ\dfrac{1}{p} \ln \lambda for some pNp \in \mathbb{N}, where λ\lambda is a Perron number. This extends Lind's work on one-dimensional shifts of finite type. As an application, the entropy minimality problem is investigated, and we obtain the necessary and sufficient condition for a tree-shift of finite type being entropy minimal with some additional conditions
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