scholarly journals Multifractal analysis of the Lyapunov exponent for the backward continued fraction map

2009 ◽  
Vol 30 (1) ◽  
pp. 211-232 ◽  
Author(s):  
GODOFREDO IOMMI
Keyword(s):  
Thermodynamic Formalism ◽  
Multifractal Spectrum ◽  
Equilibrium Measures ◽  
Interval Maps ◽  
Markov Shifts ◽  
Inflection Points ◽  
Real Analytic ◽  
First Time ◽  
Uniqueness Of Equilibrium ◽  
Do So

AbstractIn this paper we study the multifractal spectrum of Lyapunov exponents for interval maps with infinitely many branches and a parabolic fixed point. It turns out that, in strong contrast with the hyperbolic case, the domain of the spectrum is unbounded and points of non-differentiability might exist. Moreover, the spectrum is not concave. We establish conditions that ensure the existence of inflection points. To the best of our knowledge this is the first time that conditions of this type have been given. We also study the thermodynamic formalism for such maps. We prove that the pressure function is real analytic in a certain interval and then becomes equal to zero. We also discuss the existence and uniqueness of equilibrium measures. In order to do so, we introduce a family of countable Markov shifts that can be thought of as a generalization of the renewal shift.

Thermodynamic formalism for countable Markov shifts

1999 ◽  
Vol 19 (6) ◽  
pp. 1565-1593 ◽  
Author(s):  
OMRI M. SARIG
Keyword(s):  
Equilibrium Measure ◽  
Thermodynamic Formalism ◽  
Finite Measure ◽  
Complete Characterization ◽  
Topological Pressure ◽  
Recurrent Functions ◽  
Equilibrium Measures ◽  
Positive Recurrence ◽  
Markov Shifts ◽  
Definition Of

We establish a generalized thermodynamic formalism for topological Markov shifts with a countable number of states. We offer a definition of topological pressure and show that it satisfies a variational principle for the metric entropies. The pressure of $\phi =0$ is the Gurevic entropy. This pressure may be finite even if the topological entropy is infinite. Let $L_\phi$ denote the Ruelle operator for $\phi$. We offer a definition of positive recurrence for $\phi$ and show that it is a necessary and sufficient condition for a Ruelle–Perron–Frobenius theorem to hold: there exist a $\sigma$-finite measure $\nu $, a continuous function $h>0$ and $\lambda >0$ such that $L_\phi ^{*}\nu =\lambda \nu$, $L_\phi h=\lambda h $ and $\lambda ^{-n}L_\phi ^nf\rightarrow h\int f\,d\nu$ for suitable functions $f$. We show that under certain conditions this convergence is uniform and exponential. We prove a decomposition theorem for positive recurrent functions and construct conformal measures and equilibrium measures. We give complete characterization of the situation when the equilibrium measure is a Gibbs measure. We end by giving examples where positive recurrence can be verified. These include functions of the form $$ \phi =\log f\left( \cfrac{1}{x_0+ \cfrac{1}{x_1+\dotsb }}\right), $$ where $f$ is a suitable function on a suitable shift $X$.

scholarly journals Hidden Gibbs measures on shift spaces over countable alphabets

2019 ◽  
Vol 20 (04) ◽  
pp. 2050028
Author(s):  
Godofredo Iommi ◽  
Camilo Lacalle ◽  
Yuki Yayama
Keyword(s):  
Variational Principle ◽  
Existence And Uniqueness ◽  
Gibbs Measures ◽  
Thermodynamic Formalism ◽  
Topological Pressure ◽  
Markov Shifts ◽  
Sofic Shifts ◽  
Uniqueness Of Equilibrium ◽  
Countable Markov Shifts ◽  
Generalized Gibbs Measures

We study the thermodynamic formalism for particular types of sub-additive sequences on a class of subshifts over countable alphabets. The subshifts we consider include factors of irreducible countable Markov shifts under certain conditions, which we call irreducible countable sofic shifts. We show the variational principle for topological pressure for some sub-additive sequences with tempered variation on irreducible countable sofic shifts. We also study conditions for the existence and uniqueness of invariant ergodic Gibbs measures and the uniqueness of equilibrium states. Applications are given to some dimension problems and study of factors of (generalized) Gibbs measures on certain subshifts over countable alphabets.

Thermodynamic formalism for a family of non-uniformly hyperbolic horseshoes and the unstable Jacobian

2010 ◽  
Vol 31 (2) ◽  
pp. 423-447 ◽  
Author(s):  
RENAUD LEPLAIDEUR
Keyword(s):  
Lyapunov Exponent ◽  
Existence And Uniqueness ◽  
Thermodynamic Formalism ◽  
Fractal Dimensions ◽  
Equilibrium States ◽  
Real Analytic ◽  
Homoclinic Tangencies ◽  
Uniqueness Of Equilibrium

AbstractIn this article we prove the existence and uniqueness of equilibrium states for the potential $\phi _{t}= -t\logju \ (t\in \R )$ and the class of non-uniformly hyperbolic horseshoes which was introduced in Rios [Unfolding homoclinic tangencies inside horseshoes: hyperbolicity, fractal dimensions and persistent tangencies. Nonlinearity14 (2001), 431–462]. We show that the pressure t↦𝒫(t) for −tlog Ju is real-analytic on $\R $. We give the exact equations of the two asymptotes to the graph of 𝒫(t) at ±∞ and we prove that these non-uniformly hyperbolic horseshoes do not have measures which minimize the unstable Lyapunov exponent.

In the Labyrinth

2017 ◽  
Author(s):  
Andrew McNeillie
Keyword(s):  
The Right ◽  
First Time ◽  
Do So

It is now widely acknowledged, and far beyond Ireland, that Tim Robinson’s two volumes jointly known as Stones of Aran (‘Pilgrimage’ and ‘Labyrinth’) are modern classics, exemplary in every way of how to write about place and to do so with a formal, literary accomplishment that more than earns the right to nod at Ruskin’s own classic. In 2012, Robinson went back to Árainn, the largest of the three islands, for the first time in nearly ten years. He did so at the urging of Andrew McNeillie, with whom he spent two and a half days revisiting old haunts. This chapter makes account of the occasion and uses, in the process, a unique document provided by Robinson as an experiment in annotating his work. This prompts McNeillie to investigate some of his own annotations and footnotes to Aran.

Reconstituting the International, 1918–1923

2017 ◽  
Author(s):  
Talbot C. Imlay
Keyword(s):  
Seismic Events ◽  
Great War ◽  
Bolshevik Revolution ◽  
Meaning And Purpose ◽  
Post War ◽  
The Impact ◽  
First Time ◽  
The Great War ◽  
Do So ◽  
Lengthy Process

This chapter examines the post-war efforts of European socialists to reconstitute the Socialist International. Initial efforts to cooperate culminated in an international socialist conference in Berne in February 1919 at which socialists from the two wartime camps met for the first time. In the end, however, it would take four years to reconstitute the International with the creation of the Labour and Socialist International (LSI) in 1923. That it took so long to do so is a testimony to the impact of the Great War and to the Bolshevik revolution. Together, these two seismic events compelled socialists to reconsider the meaning and purpose of socialism. The search for answers sparked prolonged debates between and within the major parties, profoundly reconfiguring the pre-war world of European socialism. One prominent stake in this lengthy process, moreover, was the nature of socialist internationalism—both its content and its functioning.

scholarly journals Breast Cancer Cell Re-Dissemination from Lung Metastases—A Mechanism for Enhancing Metastatic Burden

2021 ◽  
Vol 10 (11) ◽  
pp. 2340
Author(s):  
Lucia Borriello ◽  
John Condeelis ◽  
David Entenberg ◽  
Maja H. Oktay
Keyword(s):  
Breast Cancer ◽  
Cancer Cells ◽  
Cancer Cell ◽  
Metastatic Disease ◽  
Breast Cancer Cells ◽  
Lung Metastases ◽  
Primary Tumors ◽  
Metastatic Burden ◽  
First Time ◽  
Do So

Although metastatic disease is the primary cause of mortality in cancer patients, the mechanisms leading to overwhelming metastatic burden are still incompletely understood. Metastases are the endpoint of a series of multi-step events involving cancer cell intravasation, dissemination to distant organs, and outgrowth to metastatic colonies. Here we show, for the first-time, that breast cancer cells do not solely disseminate to distant organs from primary tumors and metastatic nodules in the lymph nodes, but also do so from lung metastases. Thus, our findings indicate that metastatic dissemination could continue even after the removal of the primary tumor. Provided that the re-disseminated cancer cells initiate growth upon arrival to distant sites, cancer cell re-dissemination from metastatic foci could be one of the crucial mechanisms leading to overt metastases and patient demise. Therefore, the development of new therapeutic strategies to block cancer cell re-dissemination would be crucial to improving survival of patients with metastatic disease.

scholarly journals Thermodynamics of smooth models of pseudo–Anosov homeomorphisms

2021 ◽  
pp. 1-43
Author(s):  
DOMINIC VECONI
Keyword(s):  
Thermodynamic Formalism ◽  
Equilibrium States ◽  
Maximal Entropy ◽  
Srb Measure ◽  
The Central Limit Theorem ◽  
Exponential Decay Of Correlations ◽  
Surface Homeomorphisms ◽  
Young Towers ◽  
Uniqueness Of Equilibrium ◽  
Measure Of Maximal Entropy

Abstract We develop a thermodynamic formalism for a smooth realization of pseudo-Anosov surface homeomorphisms. In this realization, the singularities of the pseudo-Anosov map are assumed to be fixed, and the trajectories are slowed down so the differential is the identity at these points. Using Young towers, we prove existence and uniqueness of equilibrium states for geometric t-potentials. This family of equilibrium states includes a unique SRB measure and a measure of maximal entropy, the latter of which has exponential decay of correlations and the central limit theorem.

ALMOST ADDITIVE THERMODYNAMIC FORMALISM: SOME RECENT DEVELOPMENTS

2010 ◽  
Vol 22 (10) ◽  
pp. 1147-1179 ◽  
Author(s):  
LUIS BARREIRA
Keyword(s):  
Variational Principle ◽  
Existence And Uniqueness ◽  
Gibbs Measures ◽  
Thermodynamic Formalism ◽  
Topological Pressure ◽  
Present Stage ◽  
The Past ◽  
Recent Developments ◽  
General Sequences ◽  
Uniqueness Of Equilibrium

This is a survey on recent developments concerning a thermodynamic formalism for almost additive sequences of functions. While the nonadditive thermodynamic formalism applies to much more general sequences, at the present stage of the theory there are no general results concerning, for example, a variational principle for the topological pressure or the existence of equilibrium or Gibbs measures (at least without further restrictive assumptions). On the other hand, in the case of almost additive sequences, it is possible to establish a variational principle and to discuss the existence and uniqueness of equilibrium and Gibbs measures, among several other results. After presenting in a self-contained manner the foundations of the theory, the survey includes the description of three applications of the almost additive thermodynamic formalism: a multifractal analysis of Lyapunov exponents for a class of nonconformal repellers; a conditional variational principle for limits of almost additive sequences; and the study of dimension spectra that consider simultaneously limits into the future and into the past.

Some Palaeoliths from South Arabia

1954 ◽  
Vol 19 (2) ◽  
pp. 189-218 ◽  
Author(s):  
G. Caton-Thompson
Keyword(s):  
Geographical Society ◽  
South Arabia ◽  
First Time ◽  
Do So

The material here described was found in the Hadhramaut by Elinor Gardner and myself between November 1937 and March 1938. My personal investigation of the Palaeolithic Age was limited by pre-Islamic excavations, and I am therefore indebted to her for the gathering of most of the specimens in situ in terrace gravels, and to her detailed study of their positions.The collection consists mainly of groups from four fairly widely separated localities; the physiography of these has already been outlined in a comprehensive paper published in the Geographical Journal. Whenever appropriate to the purpose of this account, which is to place for the first time on illustrated record all we observed about the palaeoliths, I have reused in this different context illustrations of Quaternary environment which appeared in that Journal. With thanks I acknowledge the permission of the Royal Geographical Society to do so.

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