Find in Library

Assume that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>Y</mi><mo>,</mo><mi>ρ</mi><mo>)</mo></mrow></semantics></math></inline-formula> is a nontrivial complete metric space, and that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>Y</mi><mo>,</mo><msub><mi>g</mi><mrow><mn>1</mn><mo>,</mo><mo>∞</mo></mrow></msub><mo>)</mo></mrow></semantics></math></inline-formula> is a time-varying discrete dynamical system (T-VDDS), which is given by sequences <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mrow><mo>(</mo><msub><mi>g</mi><mi>l</mi></msub><mo>)</mo></mrow><mrow><mi>l</mi><mo>=</mo><mn>1</mn></mrow><mo>∞</mo></msubsup></semantics></math></inline-formula> of continuous selfmaps <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>g</mi><mi>l</mi></msub><mo>:</mo><mi>Y</mi><mo>→</mo><mi>Y</mi></mrow></semantics></math></inline-formula>. In this paper, for a given Furstenberg family <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">G</mi></semantics></math></inline-formula> and a given T-VDDS <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>Y</mi><mo>,</mo><msub><mi>g</mi><mrow><mn>1</mn><mo>,</mo><mo>∞</mo></mrow></msub><mo>)</mo></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">G</mi></semantics></math></inline-formula>-scrambled pairs of points of the system <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>Y</mi><mo>,</mo><msub><mi>g</mi><mrow><mn>1</mn><mo>,</mo><mo>∞</mo></mrow></msub><mo>)</mo></mrow></semantics></math></inline-formula> (which contains the well-known scrambled pairs) are provided. Some properties of the set of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">G</mi></semantics></math></inline-formula>-scrambled pairs of a given T-VDDS <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>Y</mi><mo>,</mo><msub><mi>g</mi><mrow><mn>1</mn><mo>,</mo><mo>∞</mo></mrow></msub><mo>)</mo></mrow></semantics></math></inline-formula> are studied. Moreover, the generically <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">G</mi></semantics></math></inline-formula>-chaotic T-VDDS and the generically strongly <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">G</mi></semantics></math></inline-formula>-chaotic T-VDDS are defined. A sufficient condition for a given T-VDDS to be generically strongly <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">G</mi></semantics></math></inline-formula>-chaotic is also presented.