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These studies are available for the following degree students:</p><ul><li>Bachelor's Degree Programme in Mathematics</li><li>Master's Degree Programme in Mathematics</li><li>Bachelor's Degree Programme in Mathematics (Subject Teacher)</li><li>Master's Degree Programme in Mathematics&nbsp;(Subject Teacher)</li><li>Bachelor's Degree Programme in Mathematics, Chemistry or Physics Subject Teacher Education and Primary Teacher Education (Specialication in Mathematics)</li><li>Master's Degree Programme in Mathematics, Chemistry or Physics Subject Teacher Education and Primary Teacher Education (Specialication in Mathematics)</li><li>Doctoral Programme in Mathematics and Statistics</li><li>Doctoral Programme in Mathematics and Science (Specialication in Mathematics)</li><br></ul>","fi":"<p>Tämä opintojakso on tarjolla Matematiikan syventävät opinnot -ristiinopiskeluverkostossa. Verkoston opinnot ovat tarjolla seuraaville opiskelijoille:</p><ul><li>Matematiikan kandidaattiohjelma</li><li>Matematiikan maisteriohjelma</li><li>Matematiikan aineenopettajien kandidaattiohjelma</li><li>Matematiikan aineenopettajien maisteriohjelma</li><li>Matematiikan, kemian tai fysiikan aineenopettajan ja luokanopettajan kandidaattiohjelma (matematiikan opintosuunta)</li><li>Matematiikan, kemian tai fysiikan aineenopettajan ja luokanopettajan maisteriiohjelma (matematiikan opintosuunta)</li><li>Matematiikan ja tilastotieteen tohtoriohjelma</li><li>Matemaattisten tieteiden ja luonnontieteiden tohtoriohjelma (matematiikan opintosuunta)</li><br></ul>"},"cooperationNetwork":{"abbreviation":"matematiikansyventavat","name":{"en":"Cross-institutional studies in advanced courses in mathematics and statistics","fi":"Matematiikan ja tilastotieteen syventävien kurssien ristiinopiskelu","sv":"Korsstudier i fördjupade kurser i matematik och statistik"}}}],"gradeScaleId":"sis-0-5","outcomes":{"en":"To learn the main concepts of topological dynamical systems, symbolic dynamics and shift spaces. To be able to use topology as a tool in the context of symbolic dynamics.","fi":"To learn the main concepts of topological dynamical systems, symbolic dynamics and shift spaces. To be able to use topology as a tool in the context of symbolic dynamics.","sv":"To learn the main concepts of topological dynamical systems, symbolic dynamics and shift spaces. To be able to use topology as a tool in the context of symbolic dynamics."},"tweetText":null,"content":{"en":"The course introduces fundamental concepts of topological and symbolic dynamical systems as group or monoid actions, such as homomorphisms and factor maps, orbit closures and subsystems, mixing properties, recurrence properties and sensitivity to initial conditions. In symbolic systems the course introduces the concepts of shift spaces, subshifts of finite type, sofic and effective subshifts and sliding block codes. The course also teaches basic facts of the compact metric spaces and the Cantor topology on shift spaces.","fi":"The course introduces fundamental concepts of topological and symbolic dynamical systems as group or monoid actions, such as homomorphisms and factor maps, orbit closures and subsystems, mixing properties, recurrence properties and sensitivity to initial conditions. In symbolic systems the course introduces the concepts of shift spaces, subshifts of finite type, sofic and effective subshifts and sliding block codes. The course also teaches basic facts of the compact metric spaces and the Cantor topology on shift spaces.","sv":"The course introduces fundamental concepts of topological and symbolic dynamical systems as group or monoid actions, such as homomorphisms and factor maps, orbit closures and subsystems, mixing properties, recurrence properties and sensitivity to initial conditions. In symbolic systems the course introduces the concepts of shift spaces, subshifts of finite type, sofic and effective subshifts and sliding block codes. The course also teaches basic facts of the compact metric spaces and the Cantor topology on shift spaces."},"additional":{"en":"Course will be given every other spring (in odd years).","fi":"Course will be given every other spring (in odd years)."},"prerequisites":null,"compulsoryFormalPrerequisites":[],"recommendedFormalPrerequisites":[],"literature":[],"learningMaterial":null,"completionMethods":[]}],"prerequisiteCourseUnit":[],"prerequisiteModule":[]},"prerequisiteCourseUnitPage":{"nodes":[]},"prerequisiteModulePage":{"nodes":[]},"parentModulePage":{"nodes":[]}},"pageContext":{"type":"courseUnit","locale":"en","title":"Symbolic Dynamics","id":"otm-b3df2f31-7cb8-3ed6-9235-246c55210ba7","code":"MATE5411","prerequisiteCourseUnitIds":[],"prerequisiteModuleIds":[],"parentModuleIds":[],"curriculumPeriodStartDate":"2026-08-01","curriculumPeriodEndDate":"2027-08-01","coordinatingOrgIds":[],"searchable":false,"searchTags":null,"organisationIds":["otm-69f8ea31-de0c-3050-ab2a-e571fbab1f95"],"organisations":[],"attainmentLanguages":[],"hasSummerStudies":false,"teachingPeriods":[],"cooperationNetworkDirection":"INBOUND","hasCooperationNetworkSettings":true,"hasAvoinTeaching":false}}}