Ergodic transport theory and piecewise analytic subactions for analytic dynamics
Copyright policy )Ergodic transport theory and piecewise analytic subactions for analytic dynamics
We consider a piecewise analytic real expanding map $f: [0,1]\to [0,1]$ of degree $d$ which preserves orientation, and a real analytic positive potential $g: [0,1] \to \mathbb{R}$. We assume the map and the potential have a complex analytic extension to a neighborhood of the interval in the complex plane. We also assume $\log g$ is well defined for this extension. It is known in Complex Dynamics that under the above hypothesis, for the given potential $��\,\log g$, where $��$ is a real constant, there exists a real analytic eigenfunction $��_��$ defined on $[0,1]$ (with a complex analytic extension) for the Ruelle operator of $��\,\log g$. Under some assumptions we show that $\frac{1}��\, \log ��_��$ converges and is a piecewise analytic calibrated subaction. Our theory can be applied when $\log g(x)=-\log f'(x)$. In that case we relate the involution kernel to the so called scaling function.
6 figures
Orbit growth in dynamical systems, twist condition, Transportation, logistics and supply chain management, Gibbs state, large deviation, involution kernel, Dynamical aspects of measure-preserving transformations, scaling function, eigenmeasure, Dynamical Systems (math.DS), FOS: Mathematics, Thermodynamic formalism, variational principles, equilibrium states for dynamical systems, eigenfunction, Mathematics - Dynamical Systems, Complex Variables (math.CV), Ruelle operator, Mathematics - Optimization and Control, subaction, ergodic transport, Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc., Mathematics - Complex Variables, Expanding holomorphic maps; hyperbolicity; structural stability of holomorphic dynamical systems, turning point, Dynamical systems involving maps of the interval, 37C30, 37C35, 37A05, 37A45, 37F15, 90B06, Optimization and Control (math.OC), maximizing probability, Relations of ergodic theory with number theory and harmonic analysis, analytic dynamics
Orbit growth in dynamical systems, twist condition, Transportation, logistics and supply chain management, Gibbs state, large deviation, involution kernel, Dynamical aspects of measure-preserving transformations, scaling function, eigenmeasure, Dynamical Systems (math.DS), FOS: Mathematics, Thermodynamic formalism, variational principles, equilibrium states for dynamical systems, eigenfunction, Mathematics - Dynamical Systems, Complex Variables (math.CV), Ruelle operator, Mathematics - Optimization and Control, subaction, ergodic transport, Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc., Mathematics - Complex Variables, Expanding holomorphic maps; hyperbolicity; structural stability of holomorphic dynamical systems, turning point, Dynamical systems involving maps of the interval, 37C30, 37C35, 37A05, 37A45, 37F15, 90B06, Optimization and Control (math.OC), maximizing probability, Relations of ergodic theory with number theory and harmonic analysis, analytic dynamics
selected citations These citations are derived from selected sources.This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).9 popularity This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.Average influence This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).Average impulse This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.Average
Citations provided by BIP!Popularity provided by BIP!