Browsing by Author "Rivera-Letelier, Juan"

Now showing 1 - 5 of 5
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    A characterization of hyperbolic potentials of rational maps
    (2012) Inoquio-Renteria, Irene; Rivera-Letelier, Juan
    Consider a rational map f of degree at least 2 acting on its Julia set J(f), a Holder continuous potential phi : J(f) -> R and the pressure P(f, phi). In the case where
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    Invariant measures of minimal post-critical sets of logistic maps
    (2010) Isabel Cortez, Maria; Rivera-Letelier, Juan
    We construct logistic maps whose restriction to the omega-limit set of its critical point is a minimal Cantor system having a prescribed number of distinct ergodic and invariant probability measures. In fact, we show that every metrizable Choquet simplex whose set of extreme points is compact and totally disconnected can be realized as the set of invariant probability measures of a minimal Cantor system corresponding to the restriction of a logistic map to the omega-limit set of its critical point. Furthermore, we show that such a logistic map f can be taken so that each such invariant measure has zero Lyapunov exponent and is an equilibrium state of f for the potential -ln |f'|.
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    Low-temperature phase transitions in the quadratic family
    (2013) Coronel, Daniel; Rivera-Letelier, Juan
    We give the first example of a quadratic map having a phase transition after the first zero of the geometric pressure function. This implies that several dimension spectra and large deviation rate functions associated to this map are not (expected to be) real analytic, in contrast to the uniformly hyperbolic case. The quadratic map we study has a non-recurrent critical point, so it is non-uniformly hyperbolic in a strong sense. (c) 2013 Elsevier Inc. All rights reserved.
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    Robust sensitive dependence of geometric Gibbs states for analytic families of quadratic maps
    (2023) Coronel, Daniel; Rivera-Letelier, Juan
    For quadratic-like maps, we show a phenomenon of sensitive dependence of geometric Gibbs states: There are analytic families of quadratic-like maps for which an arbitrarily small perturbation of the parameter can have a definite effect on the low-temperature geometric Gibbs states. Furthermore, this phenomenon is robust: There is an open set of analytic 2-parameter families of quadratic-like maps that exhibit sensitive dependence of geometric Gibbs states. We introduce a geometric version of the Peierls condition for contour models ensuring that the low-temperature geometric Gibbs states are concentrated near the critical orbit. (c) 2023 Elsevier Inc. All rights reserved.
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    There are at most finitely many singular moduli that are S-units
    (2024) Herrero, Sebastian; Menares, Ricardo; Rivera-Letelier, Juan
    We show that for every finite set of prime numbers $S$, there are at most finitely many singular moduli that are $S$-units. The key new ingredient is that for every prime number $p$, singular moduli are $p$-adically disperse. We prove analogous results for the Weber modular functions, the $\lambda$-invariants and the McKay-Thompson series associated with the elements of the monster group. Finally, we also obtain that a modular function that specializes to infinitely many algebraic units at quadratic imaginary numbers must be a weak modular unit.