Atractores en funciones lineales crecientes por parte en la recta real
DOI:
https://doi.org/10.37135/ns.01.08.03Palabras clave:
Atractor topológico,, Atractor global, Funciones lineales por parte, Transitividad, medidas invariantes ergódicasResumen
Las funciones lineales por parte aparecen como modelos matemáticos para describir sistemas provenientes de la ingeniería eléctrica, ciencias físicas y economía, recientemente también aparece en modelos de la actividad neuronal. Se considera una familia a 4 parámetros de funciones lineales crecientes por parte sobre la recta real usando la teoría de funciones continua crecientes por parte sobre intervalos compactos para estudiar la existencia de conjuntos atractores y describir la dinámica del atractor, verificando la existencia de órbitas periódicas, transitividad y la existencia de medidas ergódicas invariantes. Se demuestra específicamente los diferentes valores del parámetro donde la familia exhibe un intervalo atractor. Se prueba las condiciones necesarias y suficientes para que el conjunto atractor sea globalmente atractor, de hecho, en este caso se prueba que la dinámica de dicho atractor se comporta como la dinámica de la rotación de Poincaré del circulo unitario. También, se describe bajo qué condiciones en los parámetros la familia exhibe un atractor topológico. Finalmente se prueba la existencia de medidas invariantes ergódicas absolutamente continua a la medida de Lebesgue asociado a la familia restricta al atractor, inclusive se prueba el caso en que la medida es equivalente a la medida de Lebesgue.
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