Dynamics of semigroups generated by analytic functions of the Laplacian on homogeneous trees (Q2681056)

Let \(X\) be a homogeneous tree of degree \(q+1 \geq 3\) and denote by \(\mathcal L\) the canonical Laplacian on \(X\). For every \(p\in [1,\infty]\), \(\mathcal L\) defines a bounded linear operator on \(L^p(X)\). Let \(\Psi\) be a nonconstant holomorphic function on a domain that contains the \(L^p\)-spectrum of \(\mathcal L\). Then, the semigroup \(T(t):=\exp(t\Psi(\mathcal L)), t\geq 0\) consists of bounded linear operators. In this framework, the authors study the chaotic dynamics of the semigroup \(T(t)\). They show the following results: \begin{itemize} \item For \(2<p<\infty\) the following statements are equivalent: (1) \(T(t)\) is chaotic; (2) \(T(t)\) has a non-trivial periodic point; (3) The set of periodic points of \(T(t)\) is dense in \(L^p(X)\) \item For \(1\leq p \leq 2\) there holds: (1) \(T(t)\) has no non-trivial periodic point; (2) \(T(t)\) is not hypercyclic. Hence, \(T(t)\) is not chaotic. \item For a non-zero complex number \(a\), a real number \(b\) and \(2<p<\infty\), the following statements are equivalent: (1) \(\exp(t(a\Psi(\mathcal L)+b))\) is chaotic; (2) \(\exp(t(a\Psi(\mathcal L)+b))\) is hypercyclic; (3) \(\Re a\) and \(b\) satisfies certain inequality (given in the paper). \end{itemize}