Topological entropy for noncompact sets

Abstract

For <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f colon upper X right-arrow upper X"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>:</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false">→</mml:mo> <mml:mi>X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">f:X \to X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> continuous and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Y subset-of upper X"> <mml:semantics> <mml:mrow> <mml:mi>Y</mml:mi> <mml:mo>⊂</mml:mo> <mml:mi>X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">Y \subset X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> a topological entropy <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h left-parenthesis f comma upper Y right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>h</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>f</mml:mi> <mml:mo>,</mml:mo> <mml:mi>Y</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">h(f,Y)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is defined. For <italic>X</italic> compact one obtains results generalizing known theorems about entropy for compact <italic>Y</italic> and about Hausdorff dimension for certain <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Y subset-of upper X equals upper S Superscript 1"> <mml:semantics> <mml:mrow> <mml:mi>Y</mml:mi> <mml:mo>⊂</mml:mo> <mml:mi>X</mml:mi> <mml:mo>=</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">Y \subset X = {S^1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . A notion of entropy-conjugacy is proposed for homeomorphisms.