2,034 research outputs found
Topological pressure of simultaneous level sets
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 case to the case. We consider ergodic ratios
where the function 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
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
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
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 for some , where 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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