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Thermodynamic formalism of interval maps for upper semi-continuous potentials: Makarov-Smirnov's formalism | | Yiwei Zhang
; | | Date: |
9 Mar 2015 | | Abstract: | In this paper, we study the thermodynamic formalism of interval maps $f$ with
sufficient regularity, for a sub class $mathcal{U}$ composed of upper
semi-continuous potentials which includes both H"{o}lder and geometric
potentials. We show that for a given $uin mathcal{U}$ and negative values of
$t$, the pressure function $P(f,-tu)$ can be calculated in terms of the
corresponding hidden pressure function $widetilde{P}(f,-tu)$. Determination of
the values $tin(-infty,0)$ at which $P(f,-tu)
eq widetilde{P}(f,-tu)$ is
also characterized explicitly. These results generalize the recent studies on
the absence of phase transitions for the H"{o}lder continuous potentials in
cite[Theo B]{LiRiv13b}, and develop a real version of Makarnov-Smirnov’s
formalism for geometric potentials, in parallel to the complex version shown in
cite[Theo A, B]{MakSmi00}. Moreover, our results also provide a new and
simpler proof (using cite[Coro6.3]{Rue92}) of the original Makarnov-Smirnov’s
formalism in the complex setting, under an additional assumption about
non-exceptionality, i.e., cite[Theo 3.1]{MakSmi00}. | | Source: | arXiv, 1503.2535 | | Services: | Forum | Review | PDF | Favorites | |
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