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:

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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 \(L^p(X)\).

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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.

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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) \(\operatorname{Re} a\) and \(b\) satisfy a certain inequality (given in the paper).