| Summary: | Given a continuous map <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>X</mi></mrow></semantics></math></inline-formula> on a metric space, it induces the maps <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mover><mi>f</mi><mo>¯</mo></mover><mo>:</mo><mi mathvariant="script">K</mi><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow><mo>→</mo><mi mathvariant="script">K</mi><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, on the hyperspace of nonempty compact subspaces of <i>X</i>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mover accent="true"><mi>f</mi><mo>^</mo></mover><mo>:</mo><mi mathvariant="script">F</mi><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow><mo>→</mo><mi mathvariant="script">F</mi><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, on the space of normal fuzzy sets, consisting of the upper semicontinuous functions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>u</mi><mo>:</mo><mi>X</mi><mo>→</mo><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></semantics></math></inline-formula> with compact support. Each of these spaces can be endowed with a respective metric. In this work, we studied the relationships among the dynamical systems <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>X</mi><mo>,</mo><mi>f</mi><mo>)</mo></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi mathvariant="script">K</mi><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow><mo>,</mo><mover><mi>f</mi><mo>¯</mo></mover><mo>)</mo></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi mathvariant="script">F</mi><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow><mo>,</mo><mover accent="true"><mi>f</mi><mo>^</mo></mover><mo>)</mo></mrow></semantics></math></inline-formula>. In particular, we considered several dynamical properties related to chaos: Devaney chaos, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">A</mi></semantics></math></inline-formula>-transitivity, Li–Yorke chaos, and distributional chaos, extending some results in work by Jardón, Sánchez and Sanchis (Mathematics 2020, 8, 1862) and work by Bernardes, Peris and Rodenas (Integr. Equ. Oper. Theory 2017, 88, 451–463). Especial attention is given to the dynamics of (continuous and linear) operators on metrizable topological vector spaces (linear dynamics).
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