DYNAMICS OF TRANSLATIONS ON MAXIMAL COMPACT SUBGROUPS Mauro Patrão, Ricardo Sandoval Discrete and Continuous Dynamical Systems Series A, 2025 In this article, we study the dynamics of translations of an element of a semisimple Lie group $ G $ acting on its maximal compact subgroup $ K $. First, we extend to our context some classical results in the context of general flag manifolds, showing that when the element is hyperbolic its dynamics is gradient and its fixed points components are given by some suitable right cosets of the centralizer of the element in $ K $. Second, we consider the dynamics of a general element and characterizes its recurrent set, its minimal Morse components and their stable and unstable manifolds in terms of the Jordan decomposition of the element, and we show that each minimal Morse component is normally hyperbolic.
The topological entropy of powers on Lie groups MAURO PATRÃO Ergodic Theory and Dynamical Systems, 2023 In this paper, we address the problem of computing the topological entropy of a map $\\psi : G \\to G$ , where G is a Lie group, given by some power $\\psi (g) = g^k$ , with k a positive integer. When G is abelian, $\\psi $ is an endomorphism and its topological entropy is given by $h(\\psi ) = \\dim (T(G)) \\log (k)$ , where $T(G)$ is the maximal torus of G, as shown by Patrão [The topological entropy of endomorphisms of Lie groups. Israel J. Math.234 (2019), 55–80]. However, when G is not abelian, $\\psi $ is no longer an endomorphism and these previous results cannot be used. Still, $\\psi $ has some interesting symmetries, for example, it commutes with the conjugations of G. In this paper, the structure theory of Lie groups is used to show that $h(\\psi ) = \\dim (T)\\log (k)$ , where T is a maximal torus of G, generalizing the formula in the abelian case. In particular, the topological entropy of powers on compact Lie groups with discrete center is always positive, in contrast to what happens to endomorphisms of such groups, which always have null entropy.
Entropy and its variational principle for locally compact metrizable systems ANDRÉ CALDAS, MAURO PATRÃO Ergodic Theory and Dynamical Systems, 2018 For a given topological dynamical system $T:X\\rightarrow X$ over a compact set $X$ with a metric $d$, the variational principle states that $$\\begin{eqnarray}\\sup _{\\unicode[STIX]{x1D707}}h_{\\unicode[STIX]{x1D707}}(T)=h(T)=h_{d}(T),\\end{eqnarray}$$ where $h_{\\unicode[STIX]{x1D707}}(T)$ is the Kolmogorov–Sinai entropy with the supremum taken over every $T$-invariant probability measure, $h_{d}(T)$ is the Bowen entropy and $h(T)$ is the topological entropy as defined by Adler, Konheim and McAndrew. In Patrão [Entropy and its variational principle for non-compact metric spaces. Ergod. Th. & Dynam. Sys. 30 (2010), 1529–1542], the concept of topological entropy was adapted for the case where $T$ is a proper map and $X$ is locally compact separable and metrizable, and the variational principle was extended to $$\\begin{eqnarray}\\sup _{\\unicode[STIX]{x1D707}}h_{\\unicode[STIX]{x1D707}}(T)=h(T)=\\min _{d}h_{d}(T),\\end{eqnarray}$$ where the minimum is taken over every distance compatible with the topology of $X$. In the present work, we drop the properness assumption and extend the above result for any continuous map $T$. We also apply our results to extend some previous formulae for the topological entropy of continuous endomorphisms of connected Lie groups that were proved in Caldas and Patrão [Dynamics of endomorphisms of Lie groups. Discrete Contin. Dyn. Syst. 33 (2013). 1351–1363]. In particular, we prove that any linear transformation $T:V\\rightarrow V$ over a finite-dimensional vector space $V$ has null topological entropy.
Income and wealth distributions in a neoclassical growth model with σ ≥ 1 Mauro Patrão Research on Economic Inequality, 2018 Abstract The publication of Capital in the Twenty-First Century by Piketty (2014) propelled the debate about the prospects of the evolution of income and wealth inequalities in this century. One of the main controversies is about the effects on the income and wealth inequalities of a decrease in the growth rate g. In Piketty (2014), it is claimed that a decrease in g will cause an increase in the wealth inequality, through an increase in the difference r−g, where r is the rate of return on capital. This claim was criticized by many authors. In this chapter, the author presents a neoclassical growth model with heterogeneous agents and uses it to shed more light on this issue. The author’s model generalizes and improves previous models introduced in Piketty and Zucman (2015) and in Aoki and Nirei (2016). The author also presents a result, relating income, wealth, and wage inequalities.
