linear mappings and generalized upper spectrum for

21

Upload: lamdung

Post on 10-Feb-2017

218 views

Category:

Documents


0 download

TRANSCRIPT

Page 1: LINEAR MAPPINGS AND GENERALIZED UPPER SPECTRUM FOR

��������� ���� � �������������� ��������������������� � ��� �� ��� "!#$��� %'&$ ����(�����)�*&$���

+-,�.0/21.43654798;:�1.'1.<.48;=?><.�.�3)@BA</21.�365C798;:�1.D1.EGFIHKJML�NPOQHKRTS�UWVCX6SYJ;Z[ O]\YS_^`Z[_a_b L�c [ NQRPd$L�FeRDUWVgf [ RPhIL�d [ RPHji�O ak2lnm#lpoqU�r�sMt a9ugvxw-y{zM| sMtY}�X6SYJ;Z[ O]\YS_^`Z[_a_u HjFI^ [ FI~L�O [��Q� d [ RPh)l � S_�)l�� [ FI~ [ d �Q� d [ RPh)l � S_�)l��

�����x�W���W���I���#�q�����P���q�`�*���������������� ¢¡q�`���¤£*�*�6�¥���Q¦¨§*©«ª¬�¥­�®�©��¯­"���°�±�*�'©«ªq��²¯�¤���)�³ ���±�¥´������¯µ���µ�©�´¶©��¸·-ª¬�Q��®±£��±�g©�®)��¹x��­*®��±�¯­��°£�­*§*�]�º�¸·»��©¶¹¥��´6´�©¶­��Q����ª¤����®g¼½���º�¸·»��©¶¹¥��´����±�±�*��²�¦�¯­���´_�*�±��¾��¥¹]�±©��¯­�®`¿TÀ4Á��*©¶®2¹x��­*®��±�¯­��gªq�Q·Gµ6�¬®±ª¤��´¶´Â�¥�g�±����­��±�*��£*�*�6�]�g�Q¦¨§*©«ª¬�Q­*®±©Â��­��¯�p�`������`©Â²�©�­���´Iª¬�Q��®±£��±�¯À

ÃIÄÆÅ 8)A)3ºÇ4=4È;ɺA°@¨ÇC8<.�8;=?8;Ç4Ae._A)@¨Ç48Ê hIL�NPL [ NPL�OQL�JML�N [ ^)i�UWFIi�L�cIRPO'UWV4~IH½dËL�FIOQH½UWFÌDhIHji�h�h [ JMLGÍ°L�L�FÎO¥RQ�I~IHjL�~Í°U6RPhHjFd [ RPh yL�d [ RQHji�O [ FI~�HjF�c9heSYOQHji�O�l Ê hIL�OQLÏi�UWFIi�L¯cIRPO [ NPL�NPL�^ [ RQL�~ÎRPUÐL�H½RPhIL¯NÑOQL�RQOÑU6N-d$L [ OQ�_NPL�O�l u U6NL¯r [ d$c9^jL a HjF�RQhILÏRPhIL�U6NQSÎUWV2~_S_F [ dËH½i [ ^4O]S_O¥RPL�dËOÑH½R-HjOÑO¥UWdËL�RPH½dËL�O-d$U6NPL��IOQL�V¨�I^gRPUËO¥RP�I~_SRPhILÆ~IHjdËL�FIOQHjUWFÒUWV [ cINPU6Í [ Í9Hj^½H½RTS�dËL [ OQ�_NPL6l v FÒV [ i�R a4[ F [ RQR¥N [ i�RPU6N�UWV [ ~_SYF [ dËHji [ ^CO¥S_O yRPL�dÓd [ SÆi [ NQNQS [ F [ RP�_N [ ^pHjFMJ [ NPH [ FMR¬dËL [ OQ�_NPL#ÔÕOQUÏi [ ^½^jL�~�ÖYH½F [ H y`× �IL�^½^jL y o¤U�ÌqL�F y dËL [ OQ�_NQL�ØÌDhIHji�hÙi�UWFMR [ H½FIOÐdËU6NQLÚHjFIV¨U6NPd [ RQHjUWFÛRPh [ F¢H½RPO"OQ�_cIcºU6NQRÎÔÕÌ-hIH½i�hÙHjO#RPhIL [ RQRQN [ i�RPU6N�دl Ê hILJ [ ^j�IL�UWVCRPhIL�~IHjdËL�FIO¥HjUWFÎUWV [ OQL�RDU6N [ dËL [ OQ�_NQL�d [ SÚJ [ NQS�V¨U6NÑ~IHKÜ°L�NPL�FeR-~IL��9FIH½RPH½UWFIO�lÊ hIL�dËUWO¥R4i�UWdËd$UWF�~IL��9FIHKRPHjUWFÏUWVI~IH½dËL�FIOQH½UWFÏHjO�RPhIL�Ý [ �IO¥~IU6NQÜË~IH½dËL�FIOQH½UWF a ~IHjd$Þ#l u U6NOQL�RQOGH½RÑH½OG~IL��9FIL�~ÒHjF�RPL¯NPdËO-UWVßÝ [ �IOQ~IU6N¥ÜÛdËL [ OQ�_NQL�O�l u U6N [ oqU6NPL�^;cINPU6Í [ Í9H½^jH½RTS�dËL [ OQ�_NQLà UWFâá�ãäH½R�i [ FâÍ°Lå~IL��9FIL�~æL�H½RPhIL�N�ÍeSçdËL [ FIO�UWV�Ý [ �IOQ~IU6NQÜè~IHjd$L�FIOQHjUWFIO�UWV�OQL�RPO�U6N aL�éê�IH½J [ ^jL�FMRP^½S a H½F�RPhIL�V¨UW^j^½U*ÌDHjFIë"Ì [ S°ì

Ô]}Wí½}*Øî~IHjd$Þ à�ï OQ�_cÐð°ñ�ò |�ó ^jHjdçH½FIVô�õ�ö^jUWë à Ô±÷ÔÕøCùPú6ØQØ^jUWë�ú òâñ�V¨U6N à y{[ ^jdËUWO¥R [ ^½^)ø�û�á ã¬ü í

ÝGL�NPL�÷ÔÕø4ùPúWØ�H½O"RQhIL�i�^½UWOQL�~¢Í [ ^j^qUWV-i�L�FeR¥NPL�øèûýá�ã [ FI~¢N [ ~IH½�IOÐú�ÌDH½RPh |<þ ú þ ÿ l× L�i�L�FMRP^½S�RPhIL�NPLÏh [ O'Í°L�L�F�dÆ�Ii�h�H½FeRPL¯NPL�O¥RGH½FÎRPhIL�c [ i�\YHjFIëË~IH½dËL�FIOQH½UWF a ~IH½d � a ~IL��9FIL�~�ÍêSNPL�c9^ [ i�HjFIëÆRPhIL�^½U*̬L�N-^½HjdËHKRqÌ-HKRPh�RQhIL��_cIc°L¯NÑ^jH½dËH½R¤H½FÛÔ]}Wl½}*Ø a RPh [ RDHjO aÔ]}Wí��WØî~IHjd � à�ï OQ�_c ð ñ�ò |Úó ^½HjdçOQ�_cô�õ�ö

^jUWë à Ô±÷ÔÕøCùPú6ØQØ^jUWë�ú òâñ�V¨U6N à y{[ ^jdËUWO]R [ ^½^°øåûÎá ã ü íÊ hIL¤Í [ O¥Hji¤ëWL�UWdËL�R¥NPHji [ ^ecINPU6c°L¯NQRPHjL�O�UWVpÝ [ �IOQ~IU6NQÜ�~IH½dËL�FIOQH½UWF���RPhILqcINPU � L�i¯RPHjUWF [ ^_cINPU6c yL�NQRQHjL�O�UWV�OQL�RQOÚÔ�� f [ N� a �Âf [ R»}� a ��� [ � �¸Ø [ FI~ dËL [ OQ�_NPL�OËÔ��ÂÝ Ê �¸Ø a RPhIL$~IH½dËL�FIOQH½UWF [ ^ßcINQU6c°L�N yRPH½L�O¬UWV;HjFMRPL�NPO¥L�i�RPH½UWFIO'UWV)O¥L�RPO�Ô�� f [ N� a � f [ R»}� a �Âf [ R���� a ��� [ h �¸Ø aY[ FI~�OQL�i�RPH½UWFIOqUWV;dËL [ OQ�_NPL�O¡�����¡��������������������! #"%$�&�'��(���*),+��- ( .� /0�(���1��2�3�À54�6�7�¡498:4�6�7;6��<8%=�6�>°¡4�À

Áº·»�6�¥®��]��µ»·@? ACBM¦¨Á<D<E¡

Page 2: LINEAR MAPPINGS AND GENERALIZED UPPER SPECTRUM FOR

4 F������±©��4�¯­*§ D ®���G�H�������¥­*�IH��H�Ô��ÂXJ��� a �ÂXMfK�¨ØL� [ NPL�ÌqL�^j^ y \_FIU�Ì-FÛÔ�OQL�L [ ^½OQUK� u }� a � u ��� ap[ FI~M�Âf [ R�s��¸Ø�l v FÎc [ NQRPH½i��I^ [ N a H½V à HjO[ oqU6NPL�^;cINQU6Í [ Í9Hj^jHKRTS�dËL [ OQ�_NPL�UWFÎá ã a RPhIL�F�V¨U6NON ã P Q yT[ ^½dËUWO¥R [ ^j^SR ûUT ã P QÔ]}Wí¶sMØ ~IHjdÚÞ àWV¢ï dËHjFYX�~IH½dËÞ à ù[Z]\MíÝGL�NPL�^ [ FI~_Z [ NQLqHjFMRPL�ëWL�NPOgÌDH½RQh |"þ Z þ ^ a N ã P Q HjOgRPhILq�IFIH½éY�IL¬NPU6R [ RPHjUWF [ ^½^½S�HjFMJ [ NPH [ FeRoqU6NPL�^�cINPU6Í [ Í9Hj^½H½RTSÛdËL [ O¥�_NPLÚUWFÛRPhILa`�N [ OQO¥d [ FIF d [ FIHjV¨UW^½~bT ã P Q i�UWFIOQH½O¥RPH½FIëÒUWV [ ^j^LZ y~IHjd$L�FIOQHjUWF [ ^2^½HjFIL [ N�OQ�_Í9O]c [ i�L�OÏUWV�á�ã a�[ FI~ àWV ï ÔÕcINQU � V Ø-c à H½O�RPhILÆHjd [ ëWL#UWV à �IFI~IL�NRPhIL�U6NQRPhIUWëWUWF [ ^°cINPU � L�i�RQHjUWF�cINQU � V ì6á�ãKd R$lÊ hILDëWL�UWd$L�RQNQS"UWVºc [ iP\YHjFIë�~IH½dËL�FIOQH½UWF$H½O�^½L�OQO�NQL�ëW�I^ [ N�RPh [ F$RPh [ R�UWV°Ý [ �IO¥~IU6NQÜÎ~IH½dËL�F yOQH½UWFæÔ�� u Ý#}� a � u Ýe��� a � u X%� a � u fK� a �ÂXWfU� a �ÂX9}� a �ÂXf���¸Ø¯lÒEGFI^jHK\MLËRPhIL�Ý [ �IO¥~IU6NQÜç~IHjdËL�FIOQHjUWF aRPhILÎc [ iP\YHjFIëÛ~IHjd$L�FIOQHjUWF0H½OËFIU6R$cINQL�OQL�NQJML�~ �IFI~IL�NÚRTSêc9Hji [ ^¤cINPU � L�i�RPH½UWFIO�l w L¯JML�NQRPhIL�^½L�OQO aRPhIL�c [ iP\_HjFIëÐ~IHjdËL�FIOQHjUWFIO'UWV4RPhIL�RTSYc9Hji [ ^°cINPU � L�i�RPH½UWFIODUWV [ oqU6NPL�^°cINPU6Í [ Í9H½^jHKRxSdËL [ OQ�_NPL ài [ FIFIU6RÏͺL$RPUêU�OQd [ ^½^±ì u [ ^ji�UWFIL�N [ FI~Ûf [ RQRPH½^ [ � u fK��OQhIU�ÌqL�~ÛRPh [ R�V¨U6NgN ã P Q yT[ ^½dËUWO¥R [ ^j^RÓûhT ã P QÔ]}Wí z Ø ~IHjd � àWV ò ~IHjd � à Ô]}OiÛ~IHjd$Þ àWj ^�Ø}lkæÔ]} j Zmi0} j ^�Ø_~IH½d � à iÛ~IH½dËÞ àWj ZcINPU�J_H½~IL�~Ù~IH½dËÞ àon ZÒl Ê hIL�S [ ^½OQUÒë [ JML [ FÙL�r [ d$c9^½LËÌ-hIH½i�h<OQhIU�Ì-O"RPh [ R"RPhIHjO"^½U*ÌqL¯NÍ°UW�IFI~ýHjO�RPhIL�ͺL�O¥Rc°UWOQOQHKÍ9^jL�UWFIL6lmpqUWFMRPHjFe�IHjFIëÙRQhILÒÌqU6N¥\ UWV uº[ ^½i�UWFIL�N [ FI~æf [ RQRPH½^ [_au [ ^ji�UWFIL�N [ FI~�ÝGU�Ì'NPU*SY~q� u Ýe���gcINPU�JML�~�RPh [ R�V¨U6N [ FeSÒoqU6NPL�^gcINPU6Í [ Í9Hj^½H½RTSådËL [ OQ�_NQL à UWFá�ã0RPhIL$c [ iP\_H½FIëÒ~IH½dËL�FIOQH½UWFIOÏUWV¤H½RPO�cINPU � L�i�RPH½UWFIO#UWFeRPUCN ã P Q y{[ ^jdËUWO]R [ ^j^lR ûrT ã P Q [ NQLL�éê� [ ^±l u U6N¤RPhIH½Oqc9�_NQc°UWOQL�RPhIL�SËë [ JML [ FIL�Ì i�h [ N [ i�RPL¯NPH1s [ RQHjUWFUWV4RPhILGc [ iP\_H½FIëÐ~IH½dËL�FIOQH½UWFUWV à HjF�RQL�NPdËO'UWVCéê� [ FeRPHKRPHjL�ODi [ ^½^jL�~�~IHjdËL�FIOQHjUWFcINPU6�9^jL�O�lÊ hIHjOGc [ c°L�N�OQL�L�\_OGRPU�L�O¥R [ Í9^jHjO¥h [ F [ ^jUWëW�IL�O�UWVßRPhIL�O¥L"cINQU � L�i�RPHjUWFåNPL�O¥�I^½RPO�V¨U6N��_cIc°L�Nut y~IHjd$L�FIOQHjUWFIO�l u U6NutKv | ÝG�IFMR [ FI~w� [ ^jUWO¥hIHjFx� Ýy�e��cINPU6c°UWO¥L�~ RPhIL$V¨UW^½^jU�Ì-H½FIë�cºU6RPL�FMRPH [ ^RPhIL�U6NQL�RPHjiG~IL�V¨FIH½RPH½UWFV¨U6NqRQhIL�^½U*̬L�NOt y ~IHjdËL�FIO¥HjUWFÚUWV [ o¤U6NQL�^ cINPU6Í [ Í9H½^jHKRxS$dËL [ OQ�_NQL à ÌDH½RQhi�UWd$c [ i¯R-OQ�_cIcºU6NQR»ìÔ]}Wí«tWØ z {WÔ à Ø ï OQ�_c ð ñ�ò | ìJ| }5| ~ à Ô��IØó ø�i�� ó��J� { ~ à ÔÕø°Ø þâÿ ü í�ÛL�ëWH½JML [ F [ F [ ^jUWëW�IL�UWV4RQhIHjO-~IL¯�9FIH½RPH½UWFÎHjF�RPhIL�i [ OQL�UWVCRQhIL��_cIcºL�N�t y ~IHjd$L�FIOQHjUWF)ìÔ]}Wí��MØ z { Ô à Ø ï O¥�_cSX�ñ�ò | ìI� ã� P {WÔ à Ø þâÿ \MùÌDhIL�NPLÔ]}Wí��WØ � ã� P {6Ô à Ø ï ^jHjdçH½FIVô�õ�ö úJ� � { | }5| dËHjFYXM}WùPú ã ó ø�i�� ó � ã ~ à Ô��IØ�\ � { ~ à Ô¸ø)دíÊ hIHjOÆ~IL��9FIH½RPH½UWF [ ^½^jU�Ì-O"�IO"RPU [ cIc9^KSÛRPhIL�RPL�i�hIFIHjéê�IL�O$VBNPUWd � f [ R�}� a � u Ýe��� a � u fU� aß[ FI~�ÂÝy���DHjF0U6NP~IL�NÚRQU O¥RP�I~_S?RQhILÎÍ°L�h [ J_H½UW�_NÚUWVÑRPhIL��_cIcºL�N�t y ~IHjdËL�FIO¥HjUWF a z { a UWV [ oqU6NPL�^cINPU6Í [ Í9H½^jH½RTS�dËL [ OQ�_NQL$�IFI~IL�N"^½HjFIL [ N#d [ cIc9HjFIëWO�l Ê hILÚi�U6NQNPL�O¥c°UWFI~IH½FIë�éY�IL�O¥RQHjUWFIOÏV¨U6NÏRQhIL^jU�ÌqL�N�t y ~IHjdËL�FIOQHjUWF a z { a h [ JML�Í°L�L�F�O¥RP�I~IH½L�~ÎÍeS�Ö [ �IL¯N [ FI~a��U6NQ\ML$Ô��¶Öf���¨Ø [ FI~�ÍêSÝG�IFMR[ FI~�� [ ^½UWOQhIHjF�Ô�� Ýy�e�¨Ø¬Ô�V¨U6N�~IL��9FIH½RQHjUWFIOßOQL�LDÖYL�i�RQHjUWF��Wدl Ê hIL�OQL'éY� [ FMRPH½RQHjL�O a ÌDhIHji�h [ NPL [ ^jO¥Ui [ ^½^jL�~$RQhILÑëWL�FIL�N [ ^½H1s�L�~Ú^jU�ÌqL�N [ FI~Ú�_cIcºL�NqO¥cºL�i�RQNQ�Id V¨U6Nq~IH½dËL�FIOQH½UWFIO aY[ NQLÑUWFIL y c [ N [ dËL�RPL�NV [ dËH½^jHjL�OÏUWVD~IH½dËL�FIOQH½UWFIO#i�UWdËdËUWFI^KS��IO¥L�~ÙH½F RQhILO]RP�I~_S UWV¤~_S_F [ dËH½i [ ^�O¥SYO¥RPL�d$O�l Ê hIL�SÌqL�NQL�HjFMRQNPUê~I�Ii�L�~ýÍeS ÝGL�FeRQOQi�hIL�^ [ FI~ k�NPUêi [ i�i�H [ HjF��ÂÝGk�� [ O [ ëWL�FIL�N [ ^jH�s [ RPH½UWFýUWV�RQhIL

Page 3: LINEAR MAPPINGS AND GENERALIZED UPPER SPECTRUM FOR

� ©¶­��Q���gªq������©¶­�²¯®4��­�§G�±�*��£����6�]���P¦B§�©¶ª¬�Q­*®±©��¯­ �^jU�ÌqL�N [ FI~$�_cIcºL�Nqi�U6NQNQL�^ [ RPH½UWF$~IH½dËL�FIOQH½UWFIOÑÔÕÌDhIHji�h [ NQL�z � [ FI~ z � HjFÐUW�_N¬FIU6R [ RQHjUWFpØ [ FI~HjFI~IL¯c°L�FI~IL�FMRP^KSÒÍeS]`GN [ OQO¥ÍºL�NPëWL�N�HjF���`y�`l)o¬S�RPhIL"ëWL�FIL�N [ ^½H1s�L�~ÒO¥cºL�i�RQNP�Id V¨U6N�~IH½dËL�FIOQH½UWFIOÌqL ~IUÙFIU6R�dËL [ FýRQhIL�d"�I^½RPH½VBN [ i�R [ ^�O¥cºL�i�RQNQ�Id UWV [ dËL [ OQ�_NPLçÔ�O¥L�L V¨U6N�L¯r [ d$c9^jL��¶m��¸Ø[ ^½RQhIUW�IëWhÛHjFÛOQUWdËLÐi [ OQL�OÏRQhIL�OQLËi�UWFIi�L�cIRPO [ NPL$NPL�^ [ RPL�~ RQU�L [ i�h U6RPhIL�NÏÍeS�RPhIL��4L�ëWL�FI~_NPLRQN [ FIO¥V¨U6NPd [ RPH½UWF)l u U6NDV¨�_NQRPhIL�N-HjFIV¨U6NPd [ RQHjUWFUWF�RPhIL�O¥L�~IHjd$L�FIOQHjUWFIODOQL�L��ÂkL�±lÖ [ �IL�N [ FI~]��U6NQ\MLÏcINPU�JML�~�HjF��¶Öf���2RPh [ R [ c [ NQR�UWV�RQhILÐëWL�FIL¯N [ ^jH�s�L�~�^½U*̬L�N�O¥c°L�i¯RQNP�Id aF [ dËL�^KS a RPhILÎ^jU�ÌqL�NÚi�U6N¥NPL�^ [ RQHjUWF?~IH½dËL�FIOQH½UWF a z � a HjO$cINQL�OQL�NQJML�~â�IFI~IL¯N��¬ã%Q yT[ ^½dËUWO¥R [ ^j^^jH½FIL [ N�d [ c9OÑVBNQUWd á�ã RQUËáLQlÑÝGL�NQL���ã%Q HjOÑRPhIL�^�Z y ~IH½dËL�FIOQH½UWF [ ^;��L�Í°L�O¥ëW�ILÐdËL [ OQ�_NPL6lEGOQH½FIë�RPhIL�c°U6RPL�FeRPH [ ^ y RPhIL�U6NPL�RQHji~IL��9FIH½RPH½UWF?V¨U6NÆRPhILÎëWL�FIL¯N [ ^jH�s�L�~?^½U*ÌqL¯N$O¥cºL�i�RQNQ�Id Ô]}Wl«tWØ aÝG�IFMR [ FI~K� [ ^jUWOQhIH½F Ô��ÂÝy���¸ØqL¯rYRPL�FI~IL�~�RPhIH½ODNPL�OQ�I^½R-V¨U6N [ ^j^ |Ðþ t n }ÚÔ�HjFRPhIL�H½N-FIU6R [ RPH½UWFV¨U6N [ ^j^-} þ t n �WØ�l Ê hIL�SÛcINQU*JML�~ RPh [ R#H½V à H½O [ oqU6NPL�^�cINPU6Í [ Í9H½^jH½RTSÛd$L [ OQ�_NPLÚUWFÛá�ãÌDH½RPh�i�UWd$c [ i�RÑO¥�_cIc°U6NQR a RPhIL�F�V¨U6Ny��ã%Q yT[ ^jd$UWO¥R [ ^½^;^jHjFIL [ NGd [ cIc9HjFIëWO�� ìeá�ã]d á�Q RQhILV¨UW^j^½U*ÌDHjFIë [ F [ ^jUWëW�IL�UWVqÔ]}Wl¶sMجhIUW^j~IO�ìÔ]}Wí� MØ z {WÔ¡�Oc à Ø ï dËH½FYX9z {WÔ à Ø�ù[Z]\MíÝGL�NPL���c à HjO�RPhILËHjd [ ëWL$UWV à �IFI~IL�N_�Dl Ê hIL�S [ ^jOQU�ë [ JMLËL¯r [ d$c9^jL�O�OQhIU�Ì-H½FIë�RPh [ R�RQhIL^jU�ÌqL�N�t y ~IHjdËL�FIOQHjUWF�H½ODFIU6R'cINPL�O¥L�NQJML�~��IFI~IL�NDRTSYc9Hji [ ^�^½HjFIL [ NDd [ c9O'V¨U6N@t_v }Wlv F�RPhIHjOÑc [ cºL�N�ÌqL [ ~I~_NPL�OQO�RPhILÏcINPU6Í9^½L�d UWVß�9FI~IHjFIëUW�_R�hIU*Ì RPhIL"ëWL�FIL�N [ ^½H1s�L�~å�_cIcºL�NO¥cºL�i�RQNP�Id Í°L�h [ JML�OÎ�IFI~IL�N�^½HjFIL [ N�d [ cIc9HjFIëWO�Ô�U6N��IFI~IL�N�U6N¥RPhIUWëWUWF [ ^GcINPU � L�i�RPH½UWFIO�دl v R[ cIc°L [ NPO�RPh [ R a �IFI^½H½\ML"RPhILÆ^jU�ÌqL�Nut y ~IH½dËL�FIOQH½UWF a RQhIL$�_cIc°L¯NÏUWFILÆHjO�FIU6R�cINPL�OQL�NQJML�~��IFI~IL�NRTSYc9H½i [ ^ ^½HjFIL [ N�d [ cIc9HjFIëWOÑÔ�U6N�RxSêc9Hji [ ^9cINPU � L�i�RQHjUWFIO�Ø�V¨U6N |Æþ t n }Wl¢�ÛL-ÌDHj^½^IcINPU*JMLDRPh [ R�H½V^ [ FI~�Z [ NPLGH½FeRPL�ëWL�NPO¬ÌDH½RPh |"þ Z þ ^ a à HjO [ oqU6NPL�^pcINPU6Í [ Í9Hj^½H½RTSËdËL [ OQ�_NPLGUWFÚá�ãåÌDH½RQhi�UWd$c [ i¯RßOQ�_cIc°U6NQR aY[ FI~ |Ðþ t n } a RPhIL�FÆV¨U6N��¬ã%Q yT[ ^½dËUWO¥R [ ^j^Y^jHjFIL [ N�d [ c9O*�0ìMá�ãhd á�QÌqL�h [ JML

Ô]}Wí�£MØ z { Ô¡��c à Ø ï ~IHjd Q {ÐÔ à دùÌDhIL�NPL�~IH½d Q {ÆÔ à جHjO [ i�UWFIO¥R [ FMRÑU6ÍIR [ H½FIL�~�ÍeSÎi�UWFeJMUW^KJ_HjFIë"RPhIL�dËL [ OQ�_NQL à ÌDH½RQh [ i�L�NQR [ H½F\ML�NPFIL�^ [ FI~ H½R�i [ FÛÍ°LËO¥R¥NPHji�RQ^½S�^jL�OQO�RPh [ RÏRPhILË�_cIcºL�N_t y ~IHjdËL�FIOQHjUWFÛUWV à l Ê he�IO#V¨U6N |�þt n }�RPhIL��_cIc°L�Nyt y ~IH½dËL�FIOQH½UWF�Í°L�h [ JML�O-^jH½\ML�RPhIL�c [ i�\YHjFIëÐ~IHjdËL�FIO¥HjUWF��IFI~IL�N-U6NQRPhIUWëWUWF [ ^cINPU � L�i�RQHjUWFIO a ÌDhIHj^jL�RPhIL�^jU�ÌqL�NÑUWFIL�HjO¤cINPL�O¥L�NQJML�~�^½H½\ML�RPhIL�Ý [ �IOQ~IU6N¥Ü�~IH½dËL�FIOQH½UWF)loqL�i [ �IOQL'UWV ͺL�HjFIë�éY�IH½RQL'i�UWFeJML�FIH½L�FMR¬V¨U6N�RPhILDc9�_NQcºUWOQL�O�UWVºFe�IdËL�NPH½i [ ^9i [ ^ji��I^ [ RPHjUWFIO a RQhIL^jU�ÌqL�N [ FI~Ú�_cIc°L�N¤i�U6N¥NPL�^ [ RQHjUWFË~IH½dËL�FIOQH½UWFIO a z � [ FI~ z � a h [ JMLÑNQL�i�L�HKJML�~Úd"�Ii�h [ R¥RPL�FMRPHjUWF)lu U6NÑRPhILÏ�_cIcºL�N�i�U6NQNPL�^ [ RPH½UWF�~IHjdËL�FIO¥HjUWF�ÌqLÏO]RP�I~_S�hIU�Ì(OQd [ ^j^;RPhILÏi�UWFIO¥R [ FeR�~IH½d Q ��Ô à ØDH½FÔ]}Wl�£MØ-i [ F�ͺL6l¤�ÛL"OQhIU�Ì RPh [ R�RPhILÐV¨UW^½^jU�Ì-H½FIë [ F [ ^½UWëW�IL"UWVGÔ]}Wl z Ø-hIUW^j~IO�l v V�z �»Ô à Ø n Z aRPhIL�FÔ]}Wí½} | Ø z � Ô àWV جò z � Ô à Ø�Ô]}Oi�z �*Ô à Ø j ^�Ø}¥k Ô]} j Z¦i0} j ^�Ø z � Ô à Ø;i�z �*Ô à Ø j ZV¨U6N�N ãIP Q yT[ ^½dËUWO¥R [ ^½^�R û§T ã P Q [ FI~¨��ã%Q yT[ ^½dËUWO¥R [ ^½^Ñ^jH½FIL [ N�d [ c9O�� ì'á�ã§d á�Q�l© VBRPL�N�RPhIHjO�̬U6NQ\ÐÌ [ O¬i�UWd$c9^½L�RPL�~$̬LÑ^jL [ NQFeRqVBNQUWd [ c [ cºL�NqÍêS uº[ ^½i�UWFIL�N [ FI~m_ª w L�Hj^;� u my�NPL�^ [ RPHjFIëd"�I^½RPH½VBN [ i�R [ ^4cINPU6c°L�N¥RPHjL�O�UWV�dËL [ OQ�_NPL�O y HjFIi�^½�I~IHjFIëCt y ~IH½dËL�FIOQH½UWFIO y RPU�RPhIUWOQLÆUWVcINPU � L�i�RQHjUWFIO�l © cIc9^KS_H½FIëDRPhIL�HKNgdËL¯RPhIUY~IO;ÌqL�O¥hIU6NQRP^½S�H½FI~IHji [ RPL�hIU*ÌýÔ]}Wl½} | Ø°i [ F�Í°L�L¯r_RQL�FI~IL�~RPU [ ^j^ |$þ t n }Wl��ÛL�ÌDHj^½^ [ ^½OQUË~IH½OQi��IOQO [ FÎL¯r [ d$c9^½L�VBNQUWd«� u fU�;ÌDhIHji�h�OQhIU�Ì-ODRPh [ RDRQhIL^jU�ÌqL�N�ͺUW�IFI~ ëWHKJML�F H½F Ô]}Wlj} | Ø$H½OËRPhIL�Í°L�O¥R�cºUWOQOQH½Í9^½LåUWFIL6l v FýÖYL�i�RPH½UWF¬��ÌqLåëWH½JML�RQhIL~IL��9FIHKRPHjUWFÚUWVºcINQL�J [ ^½L�FIi�LÑH½FeR¥NPUY~I�Ii�L�~�ÍeSÐÝG�IFeR a Ö [ �IL�N aY[ FI~��¬U6NQ\ML'HjFK�ÂÝ�Öf��� [ FI~ÚL¯r_RQL�FI~UW�_N'NPL�OQ�I^KRPODRPUw­ [ ^jdËUWO]R¤L�JML�NQSI®$i�UWFMRPHjFe�IUWOQ^KSÎ~IHKÜ°L�NPL�FeRPH [ Í9^jL�V¨�IFIi�RPHjUWF)l

Page 4: LINEAR MAPPINGS AND GENERALIZED UPPER SPECTRUM FOR

¯ F������±©��4�¯­*§ D ®���G�H�������¥­*�IH��H�v F °9�IL�FIi�L�~?ÍeS RPhILÎd$L�RPhIUY~IO"VBNPUWd � u fU� a � u Ýe��� a�[ FI~±�ÂÝy��� a ̬LÎ�IOQLRPhIL�FIL¯Ìîi�h [ N y[ i�RPL¯NPH1s [ RQHjUWFçÔ�O¥L�LåÔ]}Wl��MØ [ FI~âÔ]}Wl��WØ¥Ø�UWV¬RPhILË�_cIc°L¯N_t y ~IH½dËL�FIOQH½UWF UWV [ oqU6NPL�^ßcINPU6Í [ Í9H½^jH½RTSdËL [ O¥�_NPL à V¨U6Nyt�v | l Ê hIH½OÑi�h [ N [ i�RPL�NQH1s [ RPH½UWF a ÌDhIHji�håHjOÑëWHKJML�F�ÍêSÎdËL [ FIOÑUWVßéY� [ FMRPHKRPHjL�O~IL��9FIL�~ÛÍeS�i�UWFeJMUW^KJ_HjFIë à Ì-HKRPh [ i�L�NQR [ H½F�\ML�NPFIL�^ a H½O [ d$UY~IH½�9i [ RPHjUWF�UWV�RPhILË~IH½dËL�FIOQH½UWFcINPU6�9^½L [ cIcINPU [ i�h�H½FeRQNQUY~I�Ii�L�~�ÍêS uº[ ^½i�UWFIL�N [ FI~�ÝGU�ÌDNPU�S_~�HjFw� u Ýe���`l�ÛLh [ JMLU6NPë [ FIH�s�L�~ÙRPhIH½OÆc [ cºL�NÐHjF<RQhILÎV¨UW^j^½U*ÌDHjFIëåÌ [ SMl v F?ÖYL�i�RPH½UWF²��ÌqLÎëWHKJML�RQhIL~IL��9FIHKRPHjUWF UWVqRPhILÚ�_cIc°L¯N_t y ~IH½dËL�FIOQH½UWF a i�UWFIOQH½~IL�N [ F¢L�r [ d$c9^½L$OQhIU�Ì-H½FIë�RPh [ RÏRPhILÚ^½U*ÌqL¯NÍ°UW�IFI~�HjF?Ô]}Wlj} | Ø-H½O�RQhILÆͺL�O¥R�c°UWOQOQHKÍ9^jLÆUWFIL a L�O¥R [ Í9^½HjOQhÒRPhILÆFIL�Ì i�h [ N [ i�RPL�NQH1s [ RPH½UWF0Ô]}Wl��MØ yÔ]}Wl��WØ-UWV�RPhILÆ�_cIc°L�Ngt y ~IHjdËL�FIO¥HjUWF a�[ FI~�cINPU�JMLÆRQhILÆNPL�OQ�I^½RËÔ]}Wl�£MØPl v F ÖYL�i¯RPHjUWFÛsÚÌqL"cINPU�JMLÔ]}Wl½} | Ø [ FI~Ù�9F [ ^j^KS HjF0ÖYL�i�RPHjUWF z ̬LcINPU�JML�RQL�i�hIFIHji [ ^qNPL�O¥�I^½RPOÆFIL�L�~IL�~?RQh_NPUW�IëWhIUW�_RÐRQhILc [ c°L¯N»l ³

É0´C8;Ç0µ�¶M79=S·g7,¸Ñ798;A°,Ê hIHjOßÌqU6NQ\"Ì [ O�~IUWFIL'Ì-hIH½^jL'RPhIL [ �_RPhIU6NQO�ÌqL�NQLDJ_H½OQH½RQHjFIë�RPhILypqL�FMRPL�N¬V¨U6N�f [ RPhIL�d [ RPH½i [ ^ÖYi�H½L�FIi�L�O × L�OQL [ NPi�h [ R�RPhIL × �_RPëWL¯NPO�EGFIHKJML�NPOQHKRTS [ FI~ÒRQhIL b L�c [ N¥RPdËL�FMR�UWV�f [ RPhIL�d [ RPHji�O[ R�RPhIL$EGFIHKJML�NPOQHKRxSÒUWVO`�L�FIL�J [ l Ê hILÆ�INPO¥R�F [ dËL�~ [ �_RPhIU6N [ iP\_FIU�Ì-^½L�~IëWL�O�RQhIL © i [ ~IL�d#SUWV u HjFI^ [ FI~âÔ�ë6N [ FMR#Fe�Id"ͺL�N"s: ¹�% : MØ [ FI~ RPhIL u UWFI~IO w�[ RPHjUWF [ ^�ÖY�IHjO¥OQLÚ~ILË^ [�× L�i�hIL�NPi�hILÖYi�H½L�FeRQH½�9éY�ILÚÔ�ë6N [ FMRÑFe�Id"ͺL�N�� |�y t | ª z £Ws_l�£¹�WØqV¨U6ND�9F [ FIi�H [ ^4OQ�_cIc°U6N¥R»l Ê hIL�OQL�i�UWFI~åF [ dËL�~[ �_RPhIU6N [ iP\_FIU�Ì-^½L�~IëWL�O�RQhILÐ�9F [ FIi�H [ ^�OQ�_cIcºU6NQR�UWV�RPhIL © i [ ~IL�d#SåUWV u HjFI^ [ FI~ [ FI~»º�H½^jhIU a��N � ZU [ FI~h� [ ^j^½LeºÐZ[ H½O�Z[ ^{Z["u �IFI~)l¼ºÄ�½u¾ 7�È;:9:97p3wt<¿�=�@À¸Ñ798�,�@¨Ç48<.�8;=?ÇC3eA ¾ Ç;·gÇ48;.S¶�:93 Ç;Á�7pÉ A)@¨ÇC8;,v FËRPhIHjO�OQL�i�RQHjUWFÚÌqLÑi�UWFIO¥Hj~IL�NqV¨U6N |Ðþ t n }ÑRPhIL-ͺL�h [ JYHjUW�_NqUWV°RQhILÑ�_cIc°L�N¥t y ~IH½dËL�FIOQH½UWFUWV [ i�UWd$c [ i¯RP^½S"OQ�_cIcºU6NQRPL�~Ëo¤U6NQL�^_cINPU6Í [ Í9Hj^½H½RTS"dËL [ O¥�_NPL à UWFÐá�ã��IFI~IL�N�cINPU � L�i¯RPHjUWFIO [ FI~ëWL�FIL�N [ ^�^½HjFIL [ NDd [ c9O�l��L�R à Í°L [ oqU6NPL�^�cINPU6Í [ Í9Hj^½H½RTS�dËL [ OQ�_NPL#UWF�á�ã ÌDH½RQhÒi�UWd$c [ i�RGOQ�_cIcºU6NQR»ly��L�R�ú�v | lu U6N-L�JML¯NQSÂtaÃï | ÌqL�~IL¯�9FIL<Ä { Ô à ùPúWØ ï | à Ô±÷ÔÕø4ùPúWØQØ { ~ à ÔÕø°Ø¯í

Ê hIL�^jU�ÌqL�N [ FI~��_cIcºL�N�t y ~IHjd$L�FIOQHjUWFIODUWV à [ NQL�~IL��9FIL�~�ÍêSz {WÔ à Ø ï ^½HjdçHjFIVô�õ�ö^½UWë

Ä { Ô à ùPúWØtg^jUWë�ú[ FI~ z { Ô à Ø ï ^jH½dçOQ�_côPõ�ö

^½UWëÄ { Ô à ùPúWØt2^jUWë�ú í

w U6RPL�RPh [ RÑH½FÎUW�_N-OQL�RQRPH½FIëËHjF�RPhIL�~IL��9FIHKRPHjUWFÎUWVÄ { Ô à ùPú6ØqRQhIL�HjFMRPL�ë6N [ FI~�HjO'N [ HjOQL�~ÎRPUÐRQhILc°U�ÌqL�N�tYl v F�d [ FeS�U6RPhIL�NDc [ c°L�NQO a HjFIi�^j�I~IHjFIëÅ� Ýy�e� a RPhIL�i�U6NQNPL�O]c°UWFI~IHjFIëÚcºU*ÌqL¯NGHjO�tui0}Wlu U6NLt ï | RPhILDi�U6NQNPL�O¥cºUWFI~IHjFIëÏ~IL¯�9FIH½RPH½UWFIO [ NPL'ëWH½JML�F$HjFh�ÂÝy� aWb L��9FIH½RPH½UWF z l����4Ô�i [ OQL�t ï }HjF<RPhIL�H½N$FIU6R [ RPH½UWFpدl Ê hIL�OQL�éê� [ FeRQH½RPH½L�O�z ö [ FI~ z ö [ NPLÎi [ ^j^jL�~ÙRPhIL�^½U*ÌqL¯N [ FI~?�_cIcºL�NHjFIV¨U6NQd [ RPHjUWF~IHjd$L�FIOQHjUWF)l v F�RPhIH½O'c [ cºL�N-̬L�ÌDHj^j^)i�UWFIOQHj~IL�N-UWFI^½S�RPhIL�i [ OQLet�v | lÊ hIL�V¨UW^j^jU�ÌDHjFIë L¯r [ dÐc9^jL�i�UWFIO]RQNP�Ii�RPL�~ H½FÆ� u f a¥Ç r [ d$c9^jLåtYl½}�ÑOQhIU�ÌDO�RQh [ RÚRPhIL�Í°L yh [ J_HjUW�_N$UWVÑRPhIL��_cIcºL�N�t y ~IHjdËL�FIOQHjUWF0�IFI~IL�NËU6NQRPhIUWëWUWF [ ^¤cINPU � L�i�RPH½UWFIO [ FI~0^jH½FIL [ NËd [ c9OHjOÚ~IHKÜ°L�NQL�FeRVBNPUWd RPh [ R�UWVGRPhILå^jU�ÌqL�N�t y ~IHjdËL�FIOQHjUWF a RPh [ RÚH½O a V¨U6N |âþ t n }åRQhILÒ�_c yc°L¯N@t y ~IHjd$L�FIOQHjUWF a �IFI^½H½\ML�RQhIL�^½U*ÌqL¯N-UWFILËÔ��ÂÝy� a Ê hIL�U6NPL�d�}Wlj}�¸Ø a HjO'FIU6RDcINQL�OQL�NQJML�~��IFI~IL�NcINPU � L�i�RQHjUWFIO-U6ND^jH½FIL [ NDd [ c9O�l

Page 5: LINEAR MAPPINGS AND GENERALIZED UPPER SPECTRUM FOR

� ©¶­��Q���gªq������©¶­�²¯®4��­�§G�±�*��£����6�]���P¦B§�©¶ª¬�Q­*®±©��¯­ =��ÈWÉËÊwÌ*ÍÏΨÐËÑCÒ;Ó�Ôl^ÆÕ�ÖY×�Z Ø�Ó�Ù�Ö,Ô.Ó.ÚfÓ�Û-Ü�Ý#ÙÏÔ¡Þ |åþ Z þ ^lßÂÒ5ÓàÔ |åþ ~ þ ~ þ ^�Õ�ÖY×~ þ Z�ß�áSÞ,Ó�Ö�Ô¡ÞâÓ�Û[ÓyÓ�ã:ÙÀÜ�Ô¡Ü�Õeäyå�Û[Óàæ9ç0Û[å:Ø-ÕfØàÙÏæ�ÙÏÔéèeê�Ó-Õ�Ü�ëIÛ[Ó à å�Ö�á ã Ý#Ù�Ô�Þ�ì-å�ê@ç,Õfì�Ô5Ü�ë9ç¹çËå�Û�ÔÜ�ëËì�ÞaÔ¡Þ Õ�ÔlÔ¡ÞâÓLíîå�æÀæ1å�Ý#ÙÏÖIÚeçYÛ[å[çïÓ�Û�ÔéÙðÓ�Ü�Þ,å�æ�×¹ñ

Ô]}*ػԡÞâÓ�Û[ÓgÙÀܤÕeçïå<Ü�ÙÏÔéÙÏò%Ó�ì-å�ÖâÜ�Ô.Õ�Ö,Ô;ó�Ü�ë,ì�ÞaÔ�Þ Õ%Ôó¯ú ô nâà Ô±÷ÔÕø4ùPúWØQØ n ú ôóíîå�Û�Õ�æÀæIø�û�O¥cIR à Õ�Ö0× |Æþ ú n }%õÔð�WػԡÞâÓ�Û[Ó�Ó�ã:ÙÀÜ�Ô*Ü9Ó[öàëËÓ�Ö0ì-Ó�Ü�Ô�ú9÷WؤÕ%ÖY×$Ô¡øe÷WØ�å!íLçËå�Ü�ÙÏÔéÙÏò%Ó�Û[Ó[Õ%æYÖËëJê�Ø�Ó�Û-Ü�Ô!Ó�ÖY×�Ù�ÖIÚ�Ô.å�ù<Ó�Û[åÜ�ëËì�ÞÂÔ¡Þ Õ�Ô

à Ô±÷ÔÕøCùàú ^°ú9÷WØ¥Ø ï ú ô÷ Õ�ÖY× à Ô±÷ÔÕø4ù[øu÷ j �WØQØ ï ø ô÷íîå�Û�Õ�æÀæIø�û�O¥cIR à õ�Õ�Ö0×Ô±sMØ~IHjd�ûGÔÕcINPU � V ÔÕO¥cIR à ØQØ n ~ Ô]}�i ~ j ^�Ø}lk Ô]} j Zmi?} j ^�Ø ~ i ~ j Zíîå�Û�Õ�æÀæ*R ûrT ãIP Q ß]ü¤ÓàÛ[ÓÏO]cIRCÙÀÜ�Ô¡ÞâÓ�Ü�ë9ç¹çËå�Û�Ôgå!í�ÕUê�Ó[Õ�Ü�ëJÛ-ÓaÕ%ÖY× ~IHjd�ûýÙÀÜ�Ô¡ÞâÓë9ç¹çËÓ�Û_Ø�åàã:þ[ì-å�ëIÖ,Ô¡Ù�ÖIÚ�×�ÙÏê�ÓàÖâÜ�Ùðå�Öw×fÓ�ÿ�Ö0Ó-×�íîå�Û�Õ�æÀæ¢Ø�å�ëJÖY×JÓ[×�ÜîÓ�Ô¡Ü����?á�ã¬Øàè

~IHjd�û�� ï ^jHjdçOQ�_c�Põ�ö ^jUWë��<Ô��Úù�Øi�^½UWë�Ý*ÞâÓàÛ[Ó��<Ô��Úù�ØaÙÀÜÂÔ¡ÞâÓCÜ�ê�Õ�æÀæ1Ó�Ü�Ô�Ö,ëJê�Ø�Ó�Û]å!í�ÜîÓ�Ô¡ÜKå!íK×�ÙéÕ�ê�ÓàÔ.Ó�ÛUÕ�Ôgê�å<Ü�Ô�²Ô�ÞâÕ�Ôì-å�ò:Ó�Û��Cß OÛ[å9å!íîßO� L ëWH½JMLÒRPhIL�Í [ OQH½iÛHj~IL [ O�UWV�RPhIL i�UWFIO¥R¥NP�Ii�RPH½UWFäH½F � u f a�Ç r [ dÐc9^jL�tYlj}�`lm� Li�UWFIi�L�FeRQN [ RPLËUWF�RQhILÐcINPU � L�i�RQHjUWF�cINQU6c°L�NQRTS?Ô±sMØ [ FI~�NPL�V¨L¯NÏRPU�� u f aSÇ r [ d$c9^½LÐtYlj}� [ O�V [ N[ O�Ô]}*Ø [ FI~ÛÔð�WØ [ NPL�i�UWFIi�L�NPFIL�~)lb L��9FILÐNQL�i��_NPOQHKJML�^½SÒOQL�éY�IL�FIi�L�OÚÔ�ú9÷WØ [ FI~çÔ�øu÷�Ø�UWV�c°UWOQHKRPH½JML#NPL [ ^2Fê�Id#Í°L¯NPO�RPL�FI~IHjFIë�RQUs�L�NQU [ FI~ [ OQL�éY�IL�FIi�LÒÔ�ZC÷WØGUWV¬c°UWO¥H½RPHKJMLÐH½FeRPL�ëWL�NPO�RQL�FI~IHjFIë�RPU�HjF_�9FIHKRxSMl��4L¯R�ø � ï ú � ïZ ö ï }Wl2Ý [ J_HjFIë�~IL��9FIL�~Úú9÷ [ FI~�øe÷ a i�hIUêUWOQL [ cºUWOQH½RQH½JML'HjFMRPL�ëWL�NLZC÷�HjF$O¥�Ii�h [ Ì [ S"RPh [ RZ ��� ã÷ HjO [ ^jOQU [ F�H½FeRPL�ëWL�NDÌDH½RQhÔð�Yí½}*Ø Z ��� ã÷ i?} n ú�� ô � ô�� � � ã � ô��÷ n Z ��� ã÷ íw L¯rYR-~IL��9FIL�ú9÷�� � [ FI~høe÷�� � OQ�Ii�h�RPh [ RÔð�Yí��WØ ZC÷6ú ô÷�� � ï ú ô÷ [ FI~ ZC÷�ø ô÷�� � ï ø ô÷ íÖYL�R@^W÷ ï Z ��� í»í»í � ZC÷êl�o¬S�Ôð�Yl��Wج̬L�h [ JMLÔð�Yí¶sMØ ú ô÷ ï ø ô÷ [ FI~ ^W÷�ø ô÷�� � ï ^W÷�ú ô÷�� � ï }Wíu �_NQRPhIL�N a Ôð�Yl��WØ a Ôð�Yl«sMØ a [ FI~ Ôð�Ylj}*جH½d$c9^½SÚRPh [ RÔð�Yí z Ø øe÷�� � ï øe÷�Z � ��� ô÷ ï ú ô � ô÷ Z � ��� ô÷ þ ú9÷�Z � ��� ã÷ í

Page 6: LINEAR MAPPINGS AND GENERALIZED UPPER SPECTRUM FOR

� F������±©��4�¯­*§ D ®���G�H�������¥­*�IH��H�b L��9FIL [ hIHjL�N [ NQi�hMSÏUWVYi��_Í°L�O��¤÷ P � a�� ï }Wù»í»í»í�ù[^W÷ � � UWV_O¥Hj~IL y ^jL�FIë6RPh#ú9÷¤HjF�RPhIL�V¨UW^½^jU�Ì-H½FIëÌ [ SMl u H½NPO]RËR [ \ML�� � P � ï � | ù»}� ã l b H½J_H½~IL�� � P � H½FeRQUMZ � OQ�_Í°i��_Í°L�O�Ì-HKRPhçOQHj~IL y ^jL�FIë6RPhú � Z � ��� ã� lgEGO¥HjFIëÐÔð�Yl z Ø)ÌqL¬OQL�L¤RQh [ RgL [ i�hÆUWV_RPhIL�OQLqú � Z � ��� ã� y i��_Í°L�O2i�UWFeR [ H½FIO [ i�UWFIi�L�FMRQNPH½ii��_ͺLÆUWVßOQH½~IL y ^½L�FIë6RPh]ø ��l w�[ dËL�RQhIL�OQL_ø � y i��_ͺL�O"!#� P $ a&% ï }Wù»í»í»í¯ù[Z � l w L¯rYRÑR [ \MLÏVBNPUWdHjFIO¥Hj~ILUWV-L [ i�h?i��_ͺL'!(� P $ [ i�UWFIi�L�FeR¥NPHji�i��_Í°L�UWVDOQH½~IL y ^½L�FIë6RPh?ú)� [ FI~?i [ ^½^¬RPhIL�OQLi��_ͺL�O�*� P � a+� ï }Wù»í»í»í�ù[^ � lLpqUWFeRPH½Fê�IL�ÍeSÚ~IH½JYHj~IHjFIë"L [ i�h�i��_Í°L,�*� P � HjFMRPU_Z-�GOQ�_ͺi��_Í°L�O'UWV�OQHj~IL y^jL�FIë6RPhÎú)�àZ � ��� ã� l © ë [ HjFÍêS�Ôð�Yl z جL [ i�h�UWVCRPhIL�O¥L�OQ�_ͺi��_Í°L�O-i�UWFeR [ HjFIO [ i�UWFIi�L�FMRQNPH½iÏi��_ͺLUWV�OQHj~IL y ^jL�FIë6RQhwø,.�l¤��L�R�RPhIL�OQL�ø,. y i��_ͺL�O�ͺL/!�. P $ a0% ï }Wù»í»í»í�ù[^1��l_pqhIUYUWOQL [ i�UWFIi�L�FMRQNPH½iú2. y i��_Í°L"�3. P � a4� ï }Wù»í»í»í¯ù[^1� a VBNPUWd H½FIOQHj~ILGUWV)L [ i�h�UWV)RPhILÑi��_ͺL�O�!(. P $ a5% ï }Wù»í»í»í¯ù[^1� aY[ FI~i�UWFMRPHjFe�IL�RPhIL�i�UWFIO¥R¥NP�Ii�RPH½UWF)l Ê hIL�dËL [ O¥�_NPL à HjO'~IL��9FIL�~�UWFÎRQhIL�^½HjdËHKRPHjFIë"OQL�R67÷�8 � ã+9�:<;=� 8 � �g÷ P �

ÍeS [ NPL�cºL [ RPL�~�O¥�_Í°~IH½JYHjOQH½UWF�OQ�Ii�h�RPh [ RÔð�Yí«tWØ à Ô�!¢÷ P � Ø ïæà Ô��¤÷ P � Ø ï } j ^W÷ � � ï ø ô÷ ï ú ô÷V¨U6N [ ^j^?> ï }Wù��Yù»í»í»í [ FI~ � ï }Wù»í»í»í�ù[^W÷ � � l Ê N [ FIOQ^ [ RQHjFIë�Ôð�YlntWØ�VBNPUWd i��_Í°L�O'RPU#Í [ ^½^jO¤ëWH½JML�OÔ]}*Ø [ FI~ÛÔð�WØ�Ô�V¨U6N'RPhIL�~IL¯R [ Hj^½ODOQL�L�� u f a,Ç r [ d$c9^jL�tYl½}�`l��L�R�R ûKT ãIP Q l u HKr [ NQL [ ^9Fê�Id#Í°L¯NA@�Ì-HKRPh | n @ n Z V¨U6N�RPhIL¤RPHjdËLqͺL�HjFIëIl v F$U6NP~IL¯NRPUÚcINPU�JML#RPh [ R ~IHjd�ûÑÔÕcINPU � V ÔÕO¥cIR à ØQØ n @Ù̬LÆFIL�L�~�RPUÚO¥hIU*Ì�RQh [ R"�ÙÔ¸cINPU � V Ô�O¥cIR à دù�Ø�CBHjOÑͺUW�IFI~IL�~ÒVBNPUWd [ Í°U�JML [ ODÆRPL�FI~IOÑRPU�s�L�NPUIl w L¯rYRÑÌqLÏÌDHj^j^;�9FI~ÒUW�_RÑÌ-h [ RGJ [ ^j�IL�OGUWV#@[ NPL�c°UWOQO¥H½Í9^jL�V¨U6N-RPhIL�J [ ^jHj~IHKRTS�UWVCRPhIHjO'i�UWFI~IHKRPHjUWF)l��L�R*wv | l»pqUWFIOQH½~IL�NE>�O¥�Ii�h¢RPh [ R"ú9÷�� � þ n ú9÷Yl u U6N [ ^j^ � ï }Wù»í»í»í�ù[^W÷ � � RQhILcINPU � L�i�RQHjUWF�cINPU � V ÔF�g÷ P � ØDi [ F�ͺLÏi�U�JML�NPL�~�ÌDH½RPhKó�Ô�ú9÷ j �Ø Q i��_ͺL�OGUWV2OQH½~IL y ^½L�FIë6RPh�"Ì-hIL�NQLó ï Ôð� ú ^�Ø Q l Ê hê�IOÑRPhILÏcINQU � L�i�RPHjUWF�cINQU � V Ô�O¥cIR à ØDi [ F�ͺLÏi�U�JML�NPL�~åÌ-HKRPhKó�^W÷ � � ÔÕú9÷ j �Ø Q y i��_ͺL�O-ÌDhIHji�h�ëWHKJML�O'ÍeS Ôð�Yl«sMØ�<Ô�O¥cIR à ù ú ^&�Ø� B n ó�^W÷ � � ú Q÷ B � Q ï ó�ú Q � ô÷ B � Q íÝGL�FIi�L �<Ô�O¥cIR à ù ú ^&�Ø�CB n óGHjVÔð�Yí��MØ #ò0ú � Q � ô � � � Q � B �÷ ímGF$RPhILÑU6RPhIL¯N¬h [ FI~ a V¨U6N [ ^j^ � ï }Wù»í»í»í¯ù[^W÷ � � RPhILDcINPU � L�i¯RPHjUWFËcINPU � V ÔÕO¥cIR àHG �¤÷ P � Øßi [ FÍ°L#i�U*JML�NQL�~ÒÌDH½RQh�RPhILÏcINPU � L�i�RPH½UWFIOGUWV2RPhIL�ZC÷Ëi��_Í°L�O,�¤÷�� � PJI ÌDH½RPh�O¥Hj~IL y ^jL�FIë6RPhåú9÷�� � þ i�UWFMR [ HjFIL�~�HjFK�¤÷ P � l�EGO¥HjFIë�Ôé�Yl«sMØ [ FI~ÛÔð�Yl��WØ�ÌqL�h [ JML�<Ô�O¥cIR à ùàú ^&6ØL B n ó�^W÷C B ï ó¯ú � ô÷�� � B ï ó¯ú � ô÷ Za÷M B í

v VN>ÚH½OD^ [ NPëWL�L�FIUW�IëWhÎRPhIL�FCZ ��� ã÷ n �IÔ¡Z ��� ã÷ i0}*جëWH½JYHjFIë�<Ô�O¥cIR à ù ú ^&�Ø� B n � ã ó¯ú ã � ô � ôO� � � ã � ôO� � ô÷ B

ÍeS Ôé�Ylj}*Ø�l Ê hIHjO-H½d$c9^jH½L�OqRPh [ R��ÙÔÕO¥cIR à ù ú ^P�Ø� B n � ã óGHjVÔð�Yí��WØ n ú ô � ã � ô � � � B � ã � ô �Q�÷ í

Page 7: LINEAR MAPPINGS AND GENERALIZED UPPER SPECTRUM FOR

� ©¶­��Q���gªq������©¶­�²¯®4��­�§G�±�*��£����6�]���P¦B§�©¶ª¬�Q­*®±©��¯­ REGOQH½FIë�Ôð�Yl��MØ [ FI~ Ôð�Yl��WØ�ÌqL�h [ JML�<Ô�O¥cIR à ù ú ^S6ØL B n � ã ó

V¨U6N [ ^½^)OQd [ ^½^&ÏÌ-hIL�F�ÌqL�d [ \ML�RQhIL�i�hIUWHji�LËÔ¡Zýi ~ Ø j Ô¡ZmiT@gØ ï Ô ~ Ô¡^�i ~ ØQØ j ÔF@qÔ�^Âi ~ ØQØëWH½JYHjFIë @ ï ~ Ô]}Oi ~ j ^�Ø}lkæÔ]} j Z¦i0} j ^�Ø ~ i ~ j Z íÊ hIHjO'i�UWd$c9^jL¯RPL�O¤RPhIL�cINPUêUWV{lVUW�Ó�ê�Õ�ÛYX9Üß�}*Ø��4L¯Rat ò i�}WlÆp¤^½L [ NP^½SÙÍêS Ô]}*Ø [ FI~(Ôð�WØ�z {WÔ à Ø ï ~ [ FI~ z { Ô à Ø ï ~ lu �_NQRPhIL�N a ÍeSÛÔ�sMج̬L�h [ JMLz { Ô àWV Ø n ~IH½d�ûÑÔÕO¥cIR àWV Ø n z { Ô à Ø�Ô]}Oi�z {WÔ à Ø j ^�Ø}lk Ô]} j Zmi?} j ^�Ø z { Ô à Ø;i�z {WÔ à Ø j Z þ z { Ô à ØV¨U6N [ ^½^5R ûUT ã P Q lqÝGL�NQL�ÌqL��IOQL�~�RPhIL�V [ i¯R-RPh [ R z { Ô à Ø n ~IH½d�ûGÔ�O]cIR à ØqV¨U6N [ ^½^StÆò i�}WlÊ hIHjO'HjO'L [ O¥S�RPUÆOQL�L��IO¥HjFIëÐRQhIL�U6NPH½ëWHjF [ ^)~IL��9FIHKRPHjUWF�V¨U6N-RPhIL��_cIc°L¯N�t y ~IHjdËL�FIOQHjUWF�ëWH½JML�F�ÍêS`�N [ OQO¥Í°L¯NPëWL�N�HjFa��`y�`l Ê hIHjO2~IL��9FIHKRPHjUWFÆëWH½JML�O [ FIUWFIH½FIi�NPL [ OQH½FIë�V¨�IFIi¯RPHjUWFÐUWVËtÆÔÕOQL�Lu��o a Ôx}*tWØ!�¸ØÌDhIHji�h�L�éê� [ ^jO ~IHjd�ûÑÔ�O¥cIR à ؤV¨U6N�t ï i�}Wlw U6RPLßRPh [ RgRPhIL�NPL�^ [ RPHjUWF�ͺL�RTÌqL�L�F z { [ FI~ ~IHjd�ûÒëWH½JML�O4RPh [ R z { Ô à Ø n ^$V¨U6N [ ^j^ft#ò i�}Wl�WØ Ê hILÆHjFIL�éY� [ ^jHKRTS¢Ô±sMØÑHjF Ç r [ dÐc9^jLÚ}$H½O [ ^jOQU�RQNP�ILÆHjV�ÌqLÆNQL�c9^ [ i�LÆcINPU � V ÍeS [ ^½HjFIL [ Nd [ c²� ì�á�ãÆd áLQ a OQH½FIi�LÎRPhIL�i�UWFIO¥RQNP�Ii�RQHjUWFçHjF � u f a�Ç r [ d$c9^jLÎtYlj}�D�IOQL�OÚUWFI^KS<RQhILcINPU6cºL�NQRPH½L�O$RPh [ RÆRPhILÎN [ FIëWL�UWV-RPhIL�d [ c0HjO�Z y ~IHjd$L�FIOQHjUWF [ ^ [ FI~?RPh [ RÐRPhILÎH½d [ ëWL�UWV [i��_ͺL�UWVgOQH½~IL�^½L�FIë6RPh�ú"H½ODH½FIOQHj~IL [ i��_ͺL�UWVgOQH½~IL�^½L�FIë6RPhKó�ú"V¨U6N-OQUWd$L�i�UWFIO]R [ FeR�ó�lsMØ v Fq� u fU�ßRPhIH½O�L¯r [ d$c9^½L#Ì [ O�i�UWFIO¥RQNP�Ii¯RPL�~ V¨U6N�OQhIU�Ì-H½FIë�RPh [ R�RPhILÆ^jU�ÌqL�N�Í°UW�IFI~�V¨U6Nc [ iP\_H½FIëÛ~IHjdËL�FIO¥HjUWFIOÚUWVGcINPU � L�i�RPH½UWFIOËU6ÍIR [ H½FIL�~0ÍêS u [ ^ji�UWFIL�N [ FI~çf [ RQRPHj^ [ HjOÐRPhIL�Í°L�O]Rc°UWO¥OQH½Í9^½L�UWFIL6l��ÛLÌ-H½^j^¬^ [ RPL�NÐ�IOQL�RPhIHjOÆL¯r [ dÐc9^jL�V¨U6NÐRQhIL�O [ d$L�c9�_N¥c°UWOQL�HjF<RQhIL��_cIcºL�Ni�U6NQNQL�^ [ RPH½UWFÎ~IH½dËL�FIOQH½UWFÎi [ OQL�lw U�ÌæÌqL�ëWH½JML [ �IO¥L�V¨�I^;i�h [ N [ i�RPL�NPH�s [ RPH½UWFÎUWV4RQhIL��_cIcºL�N�t y ~IH½dËL�FIOQH½UWF�UWV à H½F�RPL�NQdËOqUWVRPhIL�i�UWFeJMUW^½�_RPHjUWFIO'dËL�FMRPH½UWFIL�~ÎH½FÎRPhIL�HjFMRQNPUY~I�Ii¯RPHjUWF)l u U6N [ ^j^Z> ï }Wù��Yù»í»í»í�ÌqL�~IL��9FIL

~IHjd ÷{ Ô à Ø ï O¥�_cSX�ñ�ò |�ó � ÷� P { Ô à Ø þâÿ \ ï H½FIV�X�ñ�ò |Úó � ÷� P { Ô à Ø ï ÿ \MùÌDhIL�NPL�V¨U6N [ ^j^�ñ�ò |� ÷� P {6Ô à Ø ï ^jHjdçH½FIVô�õ�ö úJ� � { | } | dËHjFYXM}WùPú ÷ ó ø�i�� ó � ÷ \ ~ à Ô��IØ � { ~ à Ô¸ø)دíÊ hIL¬V¨UW^j^½U*ÌDHjFIë-cINQU6c°UWOQHKRPHjUWFÏH½O [ F [ F [ ^½UWëW�IL�UWVl� u Ýe� a pqU6NPUW^½^ [ NQS�s�� [ FI~Â�ÂÝy� a k�NPU6c°UWO¥H½RPH½UWF�Ylj}�;V¨U6NDRPhIL��_cIc°L�Nyt y ~IH½dËL�FIOQH½UWF)l [)\ËÌN\S]�^`_)^`\Sacb0ÑÂÒ;Ó�Ô à Ø�Ó_Õhì-å�ê@ç,ÕJì�Ô¡æ�è�Ü�ëîç¹çïå�Û�Ô!Ó[×�äyå�Û[Ó�æâçYÛ[å:Ø�Õ¹ØàÙÏæ�ÙÏÔéè�ê�Ó[Õ�Ü�ëJÛ-Ó�å�Öá�ãlßáSÞâÓàÖ_íîå�Û�Õ�æÀæât_v | z { Ô à Ø ï ~IHjd ã {;Ô à دí OÛ[å9å!íîß���L�RGñuved�v z { Ô à Ø�l Ê hIL�F Ä { Ô à ùPúWØlv0úMf { V¨U6N [ ^j^;OQd [ ^½^°ú_v | l Ê hê�IO� ã� P { Ô à Ø ï ^jH½dâH½FIVô�õ�ö ú � � { | }5| dËH½FYXM}WùPú ã ó ø�i�� ó � ã \ ~ à Ô¡�IØ � { ~ à ÔÕø)Øò0^jH½dâH½FIVô�õ�ö úf� � { Ä { Ô à ùPú6Ø�ò0^jH½dâH½FIVô�õ�ö ú { � f � � � ï ÿ ù

Page 8: LINEAR MAPPINGS AND GENERALIZED UPPER SPECTRUM FOR

6 F������±©��4�¯­*§ D ®���G�H�������¥­*�IH��H�ëWH½JYHjFIë z { Ô à Øqò0~IHjd ã {)Ô à دlmGF�RPhIL�U6RPhIL�NDh [ FI~ a ^½L�R z { Ô à Ø�ved¥v ñMl Ê hIL�FÎRPhIL¯NPL�H½O [ OQL�éê�IL�FIi�L�Ô�ú9÷6جRQL�FI~IHjFIë$RQUs�L�NQUO¥�Ii�håRPh [ R Ä { Ô à ùPú9÷6Ø n ú { f÷ V¨U6N [ ^j^#>pl��4L¯R |�þ þ } a°|�þhgÎþ } a)[ FI~ |þhiËþ }Í°L�OQ�Ii�hÎRPh [ R d¢i¢ñ�içÔ]}�i g Ø�^�Ô]}�kj�Ø�v i"[ FI~Kd¢i¢ñ�i»^&�v i í��L�RD>ÆÍ°LG^ [ NQëWLÑL�FIUW�IëWhOQ�Ii�hRPh [ R¤ú9÷ þlk � [ FI~�^½UWëpÔ]} j ú9÷WØ n ú �Pm÷ a ÌDhIL�NPL k � [ FI~on � [ NQL[ OÑH½FK�4L�d$d [ } z l��4L¯RG�IO-�INPO¥R [ OQOQ�IdËL�RQh [ R |Ëþ t n }Wl'EGOQHjFIë���L�dËd [ } zIa Ô z l��WØ aº[ FI~RPhIL�V [ i�RDRQh [ R#Ô g kep�Ø { n g { kep { V¨U6N [ ^j^ g ùqpDò |$[ FI~ |"þ t n } a ̬L�U6ÍIR [ H½F| }5| dËHjFYXM}WùPú ã÷ ó ø�iw� ó � ã \ ~ à Ô¡�_Ø � { ~ à ÔÕø°Ø n �rn � Ô�^jUWë �ô 9 Ø { |5s&t 9Nu ô 9 ÔÕø°Ø { ~ à ÔÕø°Øn �rn � n { ÔÕ^jUWë �ô 9 Ø { | v à Ô±÷ÔÕøCùPú9÷�ØQØ { ú { �xw � � � ã � � � � �÷ k à Ô�÷�Ô¸øCùPú9÷�ØQØ { ú � { ã �÷k<ú { ã÷Ty ~ à ÔÕø°Øn �rn � n { ÔÕ^jUWë �ô 9 Ø { v

Ä { Ô à ùPú9÷6Ø]ú { �zw � � � ã � � � � �÷ k Ä { Ô à ùPú9÷WØ]ú � { ã �÷ kÙú { ã÷{yn �rn � n { ÔÕú { � f � � � � w � ã � � � � � �Pm �÷ kÙú { � f � ã � �&m �÷ kÙú { � ã �&m �÷ Ø�íÊ hê�IO'RPhIL�i�hIUWHji�L�UWVZ a|g a9[ FI~ i H½d$c9^jH½L�O'RQh [ R� ã� P { Ô à Ø ï ^½HjdçHjFIVôPõ�ö ú � � { | };| dËH½FYXM}WùPú ã ó ø�i»� ó � ã \ ~ à Ô¡�IØ � { ~ à ÔÕø°Ø ï |ÌDhIHji�h�i�UWd$c9^½L�RPL�O¤RPhIL�cINPUYUWV4HjF�RQhIL�i [ OQL |Æþ t n }Wlw U�Ì ^jL¯R_twv }WlÒEGOQHjFIë�RPhILÚRQNPH [ FIëW^jLÚH½FIL�éY� [ ^½H½RTS V¨U6N"RPhIL�� { y FIU6NPd UWV¤RPhIL�V¨�IFIi�RPH½UWF

u ô 9 ó s&t 9 a ÌqL�U6ÍIR [ H½FÎ^½H½\ML [ Í°U�JML�RPh [ R^½HjdçHjFIVô�õ�ö ú � � }5| }5| d$HjF0XM}WùPú ã ó ø�i�� ó � ã \ ~ à Ô¡�_Ø � { ~ à ÔÕø°Ø � ;} ï | í

Ê hê�IO���ã� P {WÔ à Ø ï |Ð[ ^jOQUÐHjFRPhIHjODi [ OQL6l~UEGOQH½FIë$k�NQU6c°UWOQHKRPHjUWFC�"ÌqL�U6ÍIR [ HjF)ì

!*�;Î<\S[<Î Ê �WÑCÒ5Ó�Ô à Ø�ÓuÕ�ì�å�ê@ç,ÕfìàÔ¡æ�è�Ü�ëîç¹çïå�Û�Ô!Ó[×�äyå�Û[Óàæ%çYÛ-å:Ø-Õ¹ØàÙ�æ�ÙÏÔ¡è�ê�Ó[Õ�Ü�ëIÛ[Ógå�ÖÐá�ãlß�Ò5ÓàÔ|<þ t n }�Õ%ÖY×Åæ1Ó�Ô�Z Ø�ÓhÕ�Ö²Ù�Ö,Ô.Ó.ÚfÓ�Û�Ý#Ù�Ô�Þ |<þ Z þ ^lßoáSÞâÓàÖKíîå�ÛgN ã P Q þ�Õ�æ�ê�å<Ü�Ô_Õ�æÀæRÓûhT ã P Q Ý¥ÓeÞâÕ�ò%Ó z { Ô àWV Ø ï ~IH½d Q {ÆÔ à دí OÛ[å9å!íîß@pqUWFIOQH½~IL�N�R ûKT ã P Q l���L�R�ñ�ò | l�ÖYHjFIi�L� Q� P {WÔ à Ø ï ^jH½dâH½FIVô�õ�ö ú � � { | }5| dËHjFïXM}WùPú Q ó ø�i�� ó � Q \ ~ à Ô¡�IØ � { ~ à ÔÕø°Øn ^jH½dâH½FIVô�õ�ö úf� � { | }5| dËHjFïXM}WùPú Q ó ø�i�� ó � Q \ ~ àWV Ô¡�_Ø � { ~ àWV ÔÕø°Ø ï � Q� P {6Ô àWV دùÌqL�h [ JML�ÍêS�k�NQU6c°UWOQHKRPHjUWFC�"RPh [ R z { Ô àWV Ø ï ~IH½d Q {"Ô àWV Ø n ~IHjd Q {ÆÔ à دl

Page 9: LINEAR MAPPINGS AND GENERALIZED UPPER SPECTRUM FOR

� ©¶­��Q���gªq������©¶­�²¯®4��­�§G�±�*��£����6�]���P¦B§�©¶ª¬�Q­*®±©��¯­ ���L�RÚñ þ ~IHjd Q {ÐÔ à Ø [ FI~<ú�v | l<EGO¥HjFIë XWL�FIO¥L�F�ª¶OËH½FIL�éY� [ ^½H½RTSýÔ�OQL�LV¨U6NËL�r [ d$c9^½L»��ÖYH aÊ hIL�U6NPL�d v l¶s_l z �¨Ø#V¨U6NÐRPhILÎi�UWFIi [ JMLÎV¨�IFIi�RPHjUWF<ø��d ø { a¬u �_Í9HjFIHéª�OÐRPhIL�U6NPL�d a�[ FI~ �Âf [ R�s a��L�dËd [ s_lj}W}� a ÌqL�U6ÍIR [ HjF|L| àWV Ô�÷�Ô¸cINPU � V ÔÕø°Ø¯ùPúWØQØ { ~ N ãIP Q ÔðR#Ø ~ à ÔÕø°Øn | };| à Ô!X9�Úû�á ã óIó cINPU � V ÔÕø�i��IØ ó n úf\�Ø ~ N ã P Q ÔðR#Ø � { ~ à ÔÕø°Øï | };| N ã P Q Ô!X�R ûKT ãIP Q ó9ó cINPU � V Ô¸ø�i��IØ ó n úJ\�Ø ~ à Ô¡�IØ � { ~ à ÔÕø°Øn óW| } | dËH½FYXM}WùPú Q ó ø�iw� ó � Q \ ~ à Ô��IØ�� { ~ à ÔÕø)Ø

ÌDhIL�NPL¤óG~IUêL�O-FIU6RD~IL�c°L�FI~�UWF�úYl�o¬S u �_Í9H½FIH骶O¤RPhIL�U6NQL�d^jH½dâH½FIVô�õ�ö úf� � { | Ä { Ô àWV ùPúWØ ~ N ã P Q ÔéR#Ø þâÿ í

Ê hê�IO uº[ RPUW��ª�OD^½L�dËd [ H½d$c9^jH½L�OqRPh [ R-V¨U6N�N ãIP Q yT[ ^½dËUWO¥R [ ^j^SR ûKT ã P Q ÌqL�h [ JML^jH½dâH½FIVô�õ�ö úf� � { Ä { Ô àWV ùPúWØ þâÿ

ëWH½JYHjFIë z { Ô àWV ØqòâñMl Ê hIL�i�^ [ H½d V¨UW^½^jU�Ì-O�ÍêS�R [ \_H½FIë [ O¥L�éY�IL�FIi�L�Ô`ñ<÷�Ø;RPL�FI~IHjFIëÑRPUD~IHjd Q { Ô à ØVBNPUWd Í°L�^½U*Ì�l�UÊ hIL�NPL�OQ�I^KRßUWV Ê hIL�U6NPL�d s-i [ F"ͺL¬ëWL�FIL�N [ ^jH1s�L�~#RPUGhIUW^j~#V¨U6N [ ^½dËUWO¥R [ ^j^e^jHjFIL [ N2d [ cIc9HjFIëWOVBNPUWdèá�ãËRPU-á�Q�l Ê hIH½OgV¨UW^j^½U*ÌDOC~IHKNPL�i�RQ^½S�VBNPUWd Ê hIL�U6NPL�d�s-�IOQH½FIë-RPhIL¬V¨UW^½^jU�Ì-H½FIë [ NPëW�IdËL�FeRPOÌDhIHji�h0ÌqL�NPLÎcºUWHjFMRPL�~0UW�_RËRQU �IOËÍêS [ NPL�V¨L¯NPL�L6l v RËHjOËi�^½L [ NP^KSÙL�FIUW�IëWh?RQUÛO¥RP�I~_S?^½HjFIL [ Nd [ cIc9H½FIëWO¥�0ì6á�ãKd á�Q ÌDH½RPhÚV¨�I^½^ N [ F_\ºl�ÖY�Ii�h [ d [ c��<i [ FËÍ°L-�IFIHjéY�IL�^½SË~IL�i�UWd$c°UWO¥L�~[ OßRPhILDcINPU � L�i¯RPHjUWFÐUWFeRPU�RPhIL@Z y ~IH½dËL�FIOQH½UWF [ ^9U6NQRPhIUWëWUWF [ ^9i�UWd$c9^½L�dËL�FMR�UWV RPhIL'\ML�NPFIL�^9UWV0�[ FI~ [ ^jHjFIL [ N¤d [ cIc9HjFIë#VBNPUWd á Q RPU#á Q ÌDH½RPh�V¨�I^½^ºN [ F_\°l�ÖYH½FIi�L�^jHjFIL [ N¤HjOQUWdËU6N¥c9hIHjOQdËO�~IUFIU6R�i�h [ FIëWL'RPhIL'~IHjd$L�FIOQHjUWF a ÌqL¤U6ÍIR [ H½FÆÍeS u �_Í9HjFIHéª�O�RQhIL�U6NPL�d�RPh [ R z { Ô���c à Ø ï ~IHjd Q { Ô à ØV¨U6N¤��ã%Q yT[ ^jd$UWO¥R [ ^½^ß^jHjFIL [ NÏd [ c9Ou� ì;á�ãqd á�Ql��ÛLËëWHKJML [ ^jO¥U [ F [ ^½RQL�NPF [ RPHKJMLÆcINPUêUWVÔ�OQL�L Ê hIL�U6NPL�d tWØ�Ì-hIH½i�h�HjO'�IOQL�V¨�I^;Ì-hIL�FÎÌqL�L¯r_RPL�FI~ÎRPhIHjO¤NPL�O¥�_R-V¨�_NQRQhIL�N-RPUÐi�UWFMRPHjFe�IUW�IOQ^½S~IH½ÜºL�NPL�FMRPH [ Í9^jLÚV¨�IFIi�RPH½UWFIO�Ô�OQL�L Ê hIL�U6NPL�d«�Mدl u U6NÏRPhIH½OÏc9�_NQc°UWO¥LËÌqLÚFIL�L�~ÛRPhILÚV¨UW^½^jU�Ì-H½FIëRPL�i�hIFIH½i [ ^¬^jL�dËd [ l u U6N [ F [ F [ ^jUWëWUW�IO�NQL�OQ�I^½RÏHjFÛRPhILË^½U*ÌqL¯N�t y ~IHjdËL�FIO¥HjUWFÛi [ OQLÚO¥L�L]�¶Öf� a��L�dËd [ �Yl����±l

�lÎIÊ�ÊwÉE�5ÑCÒ5Ó�Ô&>CÕ%ÖY×OZ«Ø�ÓOÙÏÖ,Ô!Ó(ÚfÓ�Û�ÜlÝ�ÙÏÔ�Þ |Æþ Z n >�ߥÒ5Ó�Ô|pGûÎá�Q Õ�ÖY×��(ì6á ÷ d á�QØ�Ó�ÕÂæ�ÙÏÖ0Ó[Õ�Û¤ê�Õ�ç�Ü�ë,ì�ÞKÔ¡Þ Õ�Ô��ÆÔF�¤÷IÔx}*ØQØ��h� Q Ô i Ø¥íîå�Û¤Üîå�ê�Ó i v | Ý*ÞâÓàÛ[ÓH�g÷IÔ]}*Ø ï X)��ûá ÷ óÑó � $ ó n }�íîå�Û¤Õ�æÀæ % ï }Wù»í»í»í¯ùO>,\qÕ�ÖY×K� Q Ô i Ø ï X)�âûèá Q óÑó � $ ó n i í9å�Û¤Õ�æÀæ % ï}Wù»í»í»í�ù[Z]\ ß�áSÞâÓ�Ö�íîå�Û��v | Ô¡ÞâÓ�Û-Ó�Ó�ã:ÙÀÜ�Ô¡Ü�Õhì-å�ÖâÜ�Ô(Õ�ÖËÔ�� ×fÓ(çËÓ�Ö0×�ÙÏÖIÚÂå�Ö�KÜ�ëËì�ÞhÔ�Þ Õ%Ô0íîå�ÛÕ�æÀæ |"þ ú þ }|5� 9 � � � dËHjFYXM}WùPú Q ó �ÆÔ��MØSkep ó � Q \ ~ � ÷ ÔF�MØ n �(úf� � d$HjFYXM}WùPú Q i � Q \Mí OÛ[å9å!íîß v RDHjO'L�FIUW�IëWh�RPU"cINPU�JML�RPhIL�i [ OQL p ï | OQHjFIi�L| � 9 � � � dËHjFYXM}WùPú Q ó �ÆÔF�MØWk�p ó � Q \ ~ � ÷ ÔF�MØ n | � 9 � � � dËHjFYXM}WùPú Q ó �ÆÔF�MØ ó � Q \ ~ � ÷ ÔF�MØ

Page 10: LINEAR MAPPINGS AND GENERALIZED UPPER SPECTRUM FOR

¡]� F������±©��4�¯­*§ D ®���G�H�������¥­*�IH��H�V¨U6N [ ^j^1pÑûÎá�Q�l#�ÛL�d [ S [ OQOQ�IdËL�RPh [ R-ú þliea OQHjFIi�L�V¨U6N [ ^½^)ú#ò i ÌqL�h [ JML|�� 9 � � � dËHjF0XM}WùPú Q ó �ÆÔ��MØ ó � Q \ ~ � ÷ ÔF�MØ n |�� 9 � � � } ~ � ÷ Ô��MØ ï � ÷ n � ÷ úf� � d$HjFYXM}WùPú Q i � Q \Mí�ÛL��IOQL�RPhIL�FIU6R [ RPHjUWFIO"��Ô�>YØ [ FI~��ÚÔ¡ZÎØ'V¨U6N-RQhILÏU6NQRPhIUWëWUWF [ ^;ë6NPUW�_c9ODUWVgá ÷ [ FI~Îá Q l��L�RD� ï R*����Í°L�RPhIL�OQH½FIëW�I^ [ NDJ [ ^½�ILÏ~IL�i�UWd$cºUWOQH½RQHjUWFÎUWVZ� Ô�OQL�L��¶ÖêR a Ê hIL�U6NPL�d �_l��_lj}�¨Ø¯lÝGL�NPL�� û���ÔL>YØ a � ï ~IH [ ë Ô g � ù g ��ù»í»í»í�ù g Q ù | ù»í»í»í�ù | Ø-HjO [ ~IH [ ëWUWF [ ^N>'��> y d [ RQNQHKr�ÌDH½RQhg � ò g �Gò ����� ò g Q ò |_a_[ FI~aR ï ÔéR � ù�RS�»Ø�HjO [ F�Z���> y d [ RQNPH�rÚOQ�Ii�h�RPh [ R�R � û��ÚÔ¡Z�Ø[ FI~�RS�¤H½Oßi�hIUWO¥L�FËOQ�Ii�h$RPh [ RlR HjO��IFIH½R [ N¥S a RPh [ R�HjO a R��*R ï v ~,÷ [ FI~�R�R�� ï v ~ Q l�ÖYHjFIi�L�ÆÔF�¤÷YÔ]}*ØQØ��l� Q Ô i Ø a ÌqL�h [ JML�RPh [ R g $ ò i j ú >ËV¨U6N [ ^½^ % ï }Wù»í»í»í�ù[Z�l��L�R�X)� � ù»í»í»í¯ùq�9÷J\"Í°L [ FåU6NQRPhIUWFIU6NQd [ ^4Í [ OQH½O�UWVßá ÷ OQ�Ii�h�RPh [ RGRQhIL�Z y c9^ [ FILÆO¥c [ FIFIL�~ÍeSwX)� � ù»í»í»í¯ùq� Q \"HjOGd [ cIcºL�~�ÍêS�� UWFeRPUÚRPhIL_Z y c9^ [ FILÆO]c [ FIFIL�~�ÍeS�RPhILÏ�INPO]RuZ�JML�i�RPU6NQOUWV4RQhIL�O¥R [ FI~ [ NQ~�Í [ OQH½O-UWV4á ÷ l#�ÛL�~IH½J_H½~IL��g÷IÔ]}*Ø�ÔÕÌDhIHji�hÎH½OD~IL��9FIL�~��IOQHjFIë$RQhIL�O¥R [ FI~ [ NP~Í [ OQH½O�ؤH½FeRQUÐRTÌqUåÔ�FIU6R'FIL�i�L�OQO [ NPH½^½S�~IHjO � UWH½FeR�ؤc [ NQRPO�� � [ FI~��0�GÌ-hIL�NQL

� � ï ð �Æû��¤÷IÔx}*Ø ó � ï ÷� $ 8 � � $ � $ [ FI~ ó � $ ó n ú i V¨U6N [ ^½^ % ï }Wù»í»í»í�ù[Z ü[ FI~ �0� ï ð �Æû��¤÷IÔ]}*Ø ó � ï ÷� $ 8 � � $ � $ [ FI~ Q� $ 8 � � �$ ò }> } úg � � � ü íw U6RPL�RQh [ RGRPhIL�NPL [ NPL#FIUÚNPL�O¥RQNPH½i�RPHjUWFIOÆÔÕU6RPhIL�NGRPh [ F�RPhIUWOQL#H½d$c9^jH½L�~�ÍeS�RPhIL#V [ i�RGRPh [ RH��û�¤÷IÔx}*ØQجV¨U6N-RPhIL�i�UYU6NQ~IHjF [ RPL�O�� Q � � ù»í»í»í�ùq�9÷ÆH½F�L�HKRPhIL�NDUWVCRPhIL�OQL�OQL�RPO�l w U�Ì|5� 9 � � � dËH½FYXM}WùPú Q ó �ÆÔF�MØ ó � Q \ ~ � ÷ ÔF�MØ n | � ; } ~ � ÷ ÔF�MØ�ke| �M¡°ú Q ó �ÆÔ��MØ ó � Q ~ � ÷ ÔF�MØ ï£¢ � k ¢ ��íu �_NQRPhIL�N a ¢ � n Ôð��ú6Ø Q i � Q Ôð� ú >YØ ÷ � Q n Ôð� ú >YØ ÷ ú � � ú Q i � Q íÖYHjFIi�L ó �ÆÔ��MØ ó ò i ¤ ¥ Q$ 8 � � �$ j ú > a ÌqL�h [ JML

¢ � n �J� Q Ôð� ú >YØ ÷ @qÔ�Z�Ø]ú Q i � Q |l¦ ÷t§ 9©¨ ; ~«ªªï �J� Q Ôð� ú >YØ ÷ @qÔ�Z�Ø]ú Q i � Q Ôð�ß^½UWë ú >�k¢^jUWë g � iÛ^jUWë�úWØ n �­¬½úJ� � ú Q i � Q ùÌDhIL�NPL3@¬Ô¡Z�ØqHjO'RPhIL�JMUW^j�IdËL�UWVCRPhIL$Ô¡Zýi?}*Ø y ~IHjd$L�FIOQHjUWF [ ^;�IFIHKR-O¥c9hIL¯NPL6l U!*�;Î<\S[<Î Ê ®YÑaÒ;Ó�Ô |Ðþ t n }aÕ�ÖY×�æ1Ó�ÔW^ Õ�Ö0×eZ Ø�ÓeÙÏÖËÔ.Ó(ÚJÓ�Û-Ü�Ý#Ù�Ô�Þ |Æþ Z þ ^lßOÒ5ÓàÔ à Ø�ÓÕ�äyå�Û-Ó�æIçYÛ-å:Ø-Õ¹ØÙÏæ�ÙÏÔéè�ê�Ó-Õ�Ü�ëIÛ[Ó_å�Ö�á�ãxÝ#ÙÏÔ¡ÞKì-å�ê@ç,Õfì�ÔlÜ�ë9ç¹çËå�Û�Ô.ß�áSÞâÓ�Ö_íîå�ÛO�¬ã%Q�þÕ%æ�ê�å<Ü�Ô�Õ�æÀææ�ÙÏÖ0Ó-Õ�Ûeê�Õ�çïÜ��0ìWá�ãhd á�Q Ý¥ÓgÞ Õ�ò%Óz { Ô¡��c à Ø ï ~IHjd Q {ÐÔ à دí OÛ[å9å!íîß Ê hIL�V [ i�R;RPh [ R ó �GÔ¸ø)Ø ó n°¯ � ¯ ó ø ó H½d$c9^jH½L�O)RPh [ R)RPhIL�HjFIL�éê� [ ^jH½RTS z { Ô¡��c à Ø n ~IHjd Q {ÐÔ à ØhIUW^j~IODV¨U6N [ ^½^W�(ÔÕOQL�L�RPhIL�cINPUêUWVCUWV Ê hIL�U6NPL�d sMدl

Page 11: LINEAR MAPPINGS AND GENERALIZED UPPER SPECTRUM FOR

� ©¶­��Q���gªq������©¶­�²¯®4��­�§G�±�*��£����6�]���P¦B§�©¶ª¬�Q­*®±©��¯­ ¡¯¡ÖYHjFIi�L z { Ô���c à Ø ï z { ÔQÔF@5�qØ-c à ØÑV¨U6N [ ^j^(@ v | H½R�HjO�L�FIUW�IëWh�RQUcINPU�JML"RPhIL"i�^ [ H½d V¨U6N� ã%Q y{[ ^jdËUWO¥R [ ^½^W^jH½FIL [ N�d [ c9O)Ì-HKRPh ó � $±� ó n }�V¨U6N [ ^½^ % ï }Wù»í»í»í�ù[Z [ FI~ � ï }Wù»í»í»í�ù[^"Ì-hIL�NQL� $²� [ NPLÐRPhIL$L�^jL�d$L�FeRQO�UWV�RPhILÐd [ RQNPH�råNPL�cINPL�OQL�FMRPHjFIëU� Ô��IOQHjFIëÎOQUWdËLÐ�_r_L�~ U6NQRQhIUWFIU6NPd [ ^Í [ OQL�ODHjFá ã [ FI~Îá Q Ø�l��H½\MLÏH½F�RPhILÏcINPUYUWV2UWV¤�ÂÝy� a k�NPU6c°UWOQHKRPHjUWFås_l����4̬L�Ì-H½^j^ [ cIc9^½Sh��L�dËd [Ðz RPUËRPhIL#^½HjFIL [ Nd [ c��H³�ìeá�ã:Q¦d á�Qî~IL¯�9FIL�~ÒÍeS-�H³pÔ��qØ ï �GÔÕø°ØDV¨U6NGOQUWd$L�ø ûÒO¥cIR à l w U6RQL�RPh [ RÑRQhIL^ [ NQëWL�O¥RÑO¥HjFIëW�I^ [ NDJ [ ^½�IL�UWV(�H³Ë~IL¯c°L�FI~IOGUWF�RPhIL�i�hIUWHji�LÏUWVgø4lqÝGU�ÌqL�JML�N a OQH½FIi�L3� B ³)�&´rµ ï@Z�H³gk·¶1�Hµ�V¨U6N [ ^j^N@�ù¶äû0á [ FI~Ûø4ù[�<û0á�ã [ FI~ÛRPhILÚOQ�_cIcºU6NQR"UWV à HjOÏÍ°UW�IFI~IL�~ a ̬Lh [ JMLDRPh [ R�OQ�_c ³r¸+¹Qº�»�¼ ¯ �H³ ¯ þâÿýaê[ FI~ËOQU�RPhIL-^ [ NPëWL�O¥R�OQHjFIëW�I^ [ N�J [ ^j�ILDUWV&�H³ÏHjO��IFIHjV¨U6NPd$^½SÍ°UW�IFI~IL�~�V¨U6N�ø û O¥cIR à l Ê he�IO�RQhIL#i�UWFIO¥R [ FMR�� HjF]�4L�dËd [Ëz ÔÕÌDhIHji�h�~IL¯c°L�FI~IO�UWFåRQhIL^ [ NQëWL�O¥RÐOQHjFIëW�I^ [ N$J [ ^j�IL�Ø#i [ FÙͺLÎi�hIUWOQL�F?RPUÒÍ°L�HjFI~IL�cºL�FI~IL�FMR�UWV-øCl u �_N¥RPhIL�N a V¨U6N$L [ i�høèûýá�ãçRPhIL�NPLL¯rIH½O¥RPO�} n % n ^ O¥�Ii�h¢RPh [ R ó ø $ ó ò ó ø ó j ú ^ a Ì-hIH½i�h<Hjd$c9^½HjL�OÏRQh [ RÆRQhIL�H³ y H½d [ ëWL�UWVgRPhIUWOQLg� û�� ã%Q Ô]}*ؤV¨U6N-ÌDhIHji�hK� � ÷ ï | V¨U6N [ ^½^¬Ô � ùO>YجÌ-HKRPh½>]Ãï % i�UWFMR [ H½FIORPhIL�i��_Í°L�� Q Ô ó ø ó j ú ^�دl��L�RGñ þ d þ ~IHjd Q { Ô à Ø [ FI~�^jL�R¾ ï d5i ñWl�EGO¥HjFIë u [ RPUW��ª�O'^½L�dËd [_aYu �_Í9H½FIH骶OqRPhIL�U6NPL�d aXWL�FIOQL�F�ª�OÑHjFIL�éY� [ ^jHKRTS aI[ FI~h��L�dËd ["zIa ÌqL�U6ÍIR [ HjF| ��¿�À � � � � Q� P { Ô���c à Ø ~ � ã%Q Ô¡�¬Øn ^jHjdçH½FIVô�õ�ö úf� � { |ÂÁï| |5��¿�À � � � d$HjFYXM}WùPú Q ó �GÔ¸ø)Ø;iw�ÑÔ¡�IØ ó � Q \ ~ � ã�Q Ô¡�qØ ~ à Ô¡�_Ø � { ~ à Ô¸ø)Øn � { ^jH½dçHjFIVô�õ�ö úJ� � � � � � { |ÃÁï| dËHjFïXM}Wù*Ô ú ^ºúWØ Q ó ø�i�� ó � Q \ ~ à Ô¡�IØ©� { ~ à ÔÕø°ØÔð�Yí� MØï ó9� Qf(P { Ô à دí

Ê hê�IOËÍeSÙk�NPU6c°UWO¥H½RPH½UWFx�ÒÌqL�h [ JML z { Ô���c à Ø ï ~IH½d Q {ÆÔ¡��c à ØÎò ñ�V¨U6N��¬ã%Q yT[ ^½dËUWO¥R [ ^j^�(û{� ã�Q Ôx}*دl Ê [ \YHjFIë [ OQL�éY�IL�FIi�L�Ô`ñ9÷WØ-RPL�FI~IHjFIëRPU~IHjd Q {ÆÔ à ØÑVBNPUWd ͺL�^jU�Ì Hjd$c9^½HjL�OÑRQhILi�^ [ Hjd�lÄUv F$U6NP~IL¯N�RPU�cINPU�JMLDRPhIL [ F [ ^jUWëW�ILDUWV Ê hIL�U6NPL�d t�V¨U6N�i�UWFMRPHjFe�IUW�IOQ^½SÐ~IH½ÜºL�NPL�FMRPH [ Í9^jL-V¨�IFIi yRPH½UWFIO-ÌqLÏ�IOQLÏRQhIL#FIU6R [ RPHjUWFÎUWVgcINPL�J [ ^jL�FIi�L#VBNQUWd �ÂÝ�Öf���±l u H�r�RTÌqUËÍ [ OQL�OGHjF�á ã [ FI~�á Q aNPL�O]c°L�i�RQH½JML�^KSMl Ǭ[ i�h<^jHjFIL [ NÐd [ c�� ìgá ã d á Q i [ F¢Í°L�ëWH½JML�F¢ÍeS [ F�Z ��^ y d [ RQNPHKr°l��L�R�� � ÔÕá ã ùQá Q Ø#Í°L�RQhILÎOQL�R$UWVÑ^jH½FIL [ NÆRQN [ FIOQV¨U6NPd [ RPHjUWFIO"Ì-HKRPh<RQhIL [ Í9OQUW^½�_RPL�J [ ^j�IL�OÐUWVRPhILL�FMRQNPHjL�OÐFIU6RÆë6NPL [ RQL�NÐRQh [ F<UWFIL6l b L�FIU6RQLÍeSÆÅ � ÔÕáßãßùQá#Q$ØÏRQhILO¥L�R$UWVDi�UWFMRPHjFe�IUW�IOQ^½S~IH½ÜºL�NPL�FMRPH [ Í9^jL"V¨�IFIi�RPH½UWFIO�VBNQUWd á ã RQU�á Q l © OQL�R"!Â�ÇÅ � ÔÕá ã ùQá Q Ø-H½OGi [ ^j^½L�~hçYÛ-Ó�ò�Õ�æ1ÓàÖ,ÔHjV4V¨U6N [ ^½^1Èåû'Å � ÔÕá�ã2ùQá�Q$جÌqL�h [ JML3È�k��0ûK!ÓV¨U6N@�¬ã%Q yT[ ^½dËUWO¥R [ ^½^S�0ûC� � ÔÕáßãßùQá#QÐدl

!*�;Î<\S[<Î Ê ÉWÑlÊ5Ö0×fÓ�ÛeÔ¡ÞâÓ�Õ�Ü-Ü�ëIê@çYÔéÙðå�ÖâÜ_å!í�áSÞâÓ�å�Û[Ó�êÌË�Ý¥ÓgÞ Õ�ò%Óz { ÔÍȹc à Ø ï ~IHjd Q { Ô à Øíîå�Û�ÕeçYÛ-Ó�ò�Õ�æ1ÓàÖ,Ô�Ü9Ó�ÔOå�í�í�ëIÖ0ì�ÔéÙðå�ÖâܾÈÒû'Å � ÔÕáßãßùQá#Q$Øîß OÛ[å9å!íîß u U6N'�_rIL�~�È�û'Å � ÔÕá�ã2ùQá#QËجRPhIL�cINPUYUWVgUWV Ê hIL�U6NPL�d tÆëWH½JML�O'RPh [ Rz { ÔQÔÍÈ_k��qØ-c à Ø ï ~IHjd Q { Ô à ØV¨U6N��¬ã%Q yT[ dËUWO]R [ ^½^;^jHjFIL [ N-d [ c9OO�çì6á�ãKd á�Q�l v FI~IL�L�~ a RPhIL�H½FIL�éY� [ ^½H½RTS

ó �ÑÔÕø°Ø�keÈ2ÔÕø°Ø;i��GÔ¡�_Ø;iTÈ2Ô¡�IØ ó n Ô ¯ � ¯ k<OQ�_cÎ ó È ¬ ÔF�MØ ó Ø ó ø�i�� ó

Page 12: LINEAR MAPPINGS AND GENERALIZED UPPER SPECTRUM FOR

¡4 F������±©��4�¯­*§ D ®���G�H�������¥­*�IH��H�[ FI~ÚRPhILGV [ i�RPO¬RQh [ R à h [ Oqi�UWd$c [ i�RqO¥�_cIc°U6NQR [ FI~'ÈÒûoÅ � ÔÕáßãßùQá#QÐØ�Hjd$c9^KSÐRQh [ R z { ÔQÔÍÈ�k�qØ[c à Ø n ~IHjd Q {ÆÔ à Ø-V¨U6N [ ^j^;�-l u U6N�RPhIL"U6cIc°UWOQHKRPL"HjFIL�éê� [ ^jH½RTS�̬LÐ�IOQLÆHjFÙÔð�Yl� MØ@�4L�d$d [ÚzÌDH½RPhKp ï È2Ô¸ø)Ø;iTÈ2Ô��IدlÄUW�Ó�ê�Õ�ÛYX9Üß�}*Ø Ê hIL [ F [ ^jUWë6S UWV Ê hIL�U6NPL�d �ÒV¨U6NÐRPhIL^jU�ÌqL�N�t y ~IHjdËL�FIOQHjUWF<i [ FÙÍ°LV¨UW�IFI~VBNPUWd �ÂÝy� a Ê hIL�U6NPL�dîs_lj}�`l�WØ u U6NLt�vè}'̬L-~IU�FIU6R�\_FIU�Ì0ÌDhIL�RPhIL�N�RPhILD�_cIcºL�N¥t y ~IH½dËL�FIOQH½UWFIO�UWV cINQU � L�i�RPHjUWFIO�UWFeRQU[ ^jd$UWO¥R [ ^j^Ic9^ [ FIL�O [ NPL-L�éY� [ ^±l2ÝGU�ÌqL�JML¯N aIÇ r [ d$c9^½L�}-OQhIU*ÌDO�RQh [ R�RPhILD�_cIc°L�Nlt y ~IH½dËL�FIOQH½UWFHjO'FIU6R'cINPL�OQL�N¥JML�~��IFI~IL�NÑcINPU � L�i¯RPHjUWFIODU6N-^½HjFIL [ NDd [ c9O�lÏ)Ä�½u¾ 7�È;:9:97p3 É°Ç43)3)7,¶M.YA)@¨ÇC8<=4@À¸Ñ798;,�@¨ÇC8o¬S Ê hIL�U6NPL�d�sÒÔ Ê hIL�U6NPL�d tWØ-RPhIL"�_cIc°L�N�i�U6NQNQL�^ [ RPH½UWF�~IH½dËL�FIOQH½UWF z � UWVßcINPU � L�i�RPH½UWFIOÔ�H½d [ ëWL�O��IFI~IL�N�^½HjFIL [ N�d [ c9O�ØËUWV [ oqU6NPL�^ÑcINPU6Í [ Í9H½^jHKRxSâdËL [ OQ�_NPLÒHjOi�UWFIO¥R [ FMR²­ [ ^½dËUWO¥ROQ�_NPL�^½S ®Il v FæRPhIH½OÎOQL�i�RPHjUWFæ̬L [ ~I~_NPL�OQO�RQhIL�cINPU6Í9^½L�d UWVÏ�9FI~IH½FIë0UW�_R�hIU�Ì OQd [ ^j^ÑRPhIH½Oi�UWFIO¥R [ FeR-i [ F�ͺL6l u U6NDRQhIHjO¤c9�_NQc°UWOQL�ÌqL��IO¥L�RPhIL�dËL�RPhIUê~IODVBNPUWd � u fU�±l��L�R@Z [ FI~a^�ͺL�H½FeRPL�ëWL�NPODÌDH½RQh |"þ Z þ ^2lL��L�R à Í°L [ oqU6NPL�^°cINPU6Í [ Í9Hj^½H½RTS�dËL [ OQ�_NQLUWF�á�ã¢Ì-HKRPh�i�UWd$c [ i�R�OQ�_cIc°U6N¥R»l���L�R iêa ~ aC[ FI~ÐdÑÍ°L"NPL [ ^ßFe�Id"ͺL�NPO�Ì-HKRPh |�þ­i�þ ~ þdËH½FYX�d�ù[Z]\ [ FI~�d þ ^2lyp¤UWFIOQH½~IL�N�NPL [ ^4Fê�Id#Í°L¯NPO gÎ[ FI~�$OQ�Ii�håRPh [ RÐÔ¡^aiTd¥Ø j Ô�^Ci ~ Ø þgËþ } ap|Ðþ þli j ^ ap[ FI~

Ô±s_í½}*Ø ^�Ôx}¥k{�Ø�Ôx}�i g Ø þli íu U6N [ ^½^)ú"ÌDH½RQh |"þ ú þ }�ÌqL�O¥L�RÔ±s_í��WØ Ñ ô ì ï ú � f � � Q � ô � � ã � f � � � ã � ô �`� � Q n ú n ú w n ú � ã � f � � � ã � ô � ï ì:ø ô í�ÛL�cINPUYi�L�L�~ [ ODHjF�ÖYL�i�RPHjUWF z l u U6N [ ^j^ |Æþ ú þ } [ FI~Îø�ûÎá ã ~IL��9FILT ô ÔÕø°Øqì ï ||Ò µ4ÓÔÓ ³ � µ4ÓÖÕ&× tYغdËHjFïXM}WùOÑ Qô ó ø�i�� ó � Q \ ~ à Ô¡�IØ�í��L�R Ä ï z d [ rËX z ã � � � � � ù[Z z ã � � � � � j Ô�^�Ô]}yk£�Ø�i¨ZÎØ�\Ml © c°UWHjFMR$ø ûäO¥cIR à HjOÐi [ ^j^½L�~Ô�ú*ùÄù[ZÎØ y NPL�ëW�I^ [ NqÌDH½RQh�NPL�O]c°L�i�RDRPU à HjV

Ô±s_í¶sMØ T ô ÔÕø)Ø nÄúJ�Pm à Ô±÷ÔÕø4ùPúWØQØ�í

v V'Ô±s_l«sMؤHjOÑFIU6RÑO [ RPHjO]�9L�~ a ÌqL#O [ SRPh [ RÑø�û�O]cIR à H½O"Ô�ú*ùÄù[ZÎØ y H½NQNPL�ëW�I^ [ N'Ì-HKRPhÎNPL�O]c°L�i�RGRQUà l Ê hIL#Ô�ú�ù

Ä j �Yù[ZÎØ y NPL�ëW�I^ [ NPHKRxS [ FI~ËRPhIL#Ô�ú*ù Ä j �Yù[Z�Ø y H½NQNQL�ëW�I^ [ NPHKRTSÏÌ-HKRPh$NPL�O¥c°L�i¯R¬RPU à [ NQL~IL��9FIL�~�OQHjd$Hj^ [ NQ^½SMl w U6RPL�RPh [ RGRQhIL�OQL#~IL��9FIHKRPHjUWFIO�~IL�cºL�FI~ [ ^½OQUÚUWF iêa ~ a d aÙgpa)[ FI~� a Í9�_RÍ°L�^jU�Ì RPhIL�OQL�Fê�Id#Í°L¯NPO [ NPL��_rIL�~)l�lÎIÊ�ÊwÉÂÚYÑCÒ5ÓàÔ i õ ~ õ�d�õ g õÂÕ%ÖY×�±Ø�ÓwÕ�Ü»ÕfØ�å�ò%Óîß áSÞâÓ�Ö±Ô¡ÞâÓ�Û-Ó�Óã:ÙÀÜ�Ô¡Ü»Õ�Ö,ëIê�Ø�Ó�Ûaó ö×fÓ(çïÓ�ÖY×�ÙÏÖ Ú�å�Ö,æ�è�å�Ö g õ&¹õ�Õ%ÖY×e^qÜ�ë,ì�Þ�Ô�Þ Õ%Ô,íîå�ÛuÕ%æÀæ0äyå�Û[Óàæ¹ç0Û[å:Ø-ÕfØàÙÏæ�ÙÏÔéè_ê�Ó[Õ�Ü�ëJÛ-Ó�Ü à å�Ö$á�ãÕ�ÖY×@íîå�Û_Õ�æÀæIú ö n } j �hÝ¥ÓeÞ Õ�ò:Ó

à Ô!X»ø�ûÎO¥cIR à ó ø²ÙÀÜ�Ô�ú*ù Ä j �Yù[Z�Øàþ.ÙÏÛ�Û-Ó(Ú�ëIæ�Õ�ÛyÝ#ÙÏÔ¡ÞaÛ-Ó�ÜðçËÓ�ì�Ô�Ô!å à \�Ø n �%ó ö ú ã � � � � w �öíîå�Û�Õ�æÀæ9úhÝ#ÙÏÔ¡Þ |Ðþ ú þ ú ö ß OÛ[å9å!íîß���L�R@ó ö Í°L [ ODH½FC��L�dËd [ }<�Yl v V�øåû�O¥cIR à H½OÏÔÕú�ùÄ j �Yù[Z�Ø y H½N¥NPL�ëW�I^ [ N�Ì-HKRPh�NPL�O¥c°L�i¯RRPU à a RQhIL�FÎRPhIL�NQL�H½O ª ÌDH½RPh�ú w n ª n }�O¥�Ii�h�RQh [ R

Ô±s_í z Ø à Ô±÷ÔÕø4ù ª Ø¥ØLv }� } z ªú � ã � � � � � à Ô±÷ÔÕø4ùPúWØQØ�í

Page 13: LINEAR MAPPINGS AND GENERALIZED UPPER SPECTRUM FOR

� ©¶­��Q���gªq������©¶­�²¯®4��­�§G�±�*��£����6�]���P¦B§�©¶ª¬�Q­*®±©��¯­ ¡�v FåV [ i�R a HjV2RPhIH½OGHjOGFIU6RGRPhIL"i [ O¥L a RPhIL�FåHjFMRPL�ë6N [ RPHjFIëÚÍeS�c [ NQRPO [ FI~Ò�IOQHjFIë�Ô±s_l½}*Ø [ FI~ÙÔ±s_l��WØÌqL�h [ JMLT ô ÔÕø)Ø ï Z�Ñ Qô | × tÛ t ª � Q � � à Ô±÷ÔÕø4ù ª Ø¥Ø ~«ª k·Ñ Qô ø � Qô à Ô±÷ÔÕøCù[ø ô ØQØn Ñ Qô à Ô�÷�Ô¸øCùPú6ØQØJ}�Z | ôÛ t ª � Q � � ~«ª k }� z ã � � � � �ðZ�ú ã � � � � � �xw � � �I| ô ¨ô ª � Q � � ~«ªk }� z ã � � � � � Z�úf� ã � � � � � | × tô ¨ ª ã � � � � � � Q � � ~+ª k }� z ã � � � � � úf� ã � � � � � ø ã � � � � � � Qô �n Ñ Qô à Ô�÷�Ô¸øCùPú6ØQØ } Ñ � Qô k }� z ã � � � � �±ú ã � � � � � �xw � � � � Qk }� z ã � � � � �ðZ Ô¡^�Ôx}¥k{�Ø¢i�ZÎØ � � ú � ã � � � � �(ø ã � � � � � � Qôk }� z ã � � � � �`ú � ã � � � � �ðø ã � � � � � � Qô �nçà Ô±÷ÔÕø4ùPúWØQØMvx}lk

Ä  Ñ Qô ú � Q �&m k

Äz Ñ Qô ú � ã � � � � �ðø ã � � � � � � Qô y

nÄ� úJ�Pm à Ô±÷ÔÕø4ùPúWإدùÔ±s_í«tWØ

ÌDhIHji�h�HjO [ i�UWFeRQN [ ~IHji�RPH½UWF)l�EGO¥HjFIë�Ô�s_l z Ø�RPhIL�i�^ [ Hjd V¨UW^½^jU�Ì-O¤ÍêSÂ��L�dËd [ }<�Yl�U�lÎIÊ�ÊwÉTÜWÑCÒ5Ó�Ô i õ ~ õ0d�õ g õ�Õ�ÖY×,]Ø�Ó_Õ�ܤչØ�å�ò:ÓîßOÒ5ÓàÔ à Ø�Ó�Õ�äyå�Û[Ó�æJçYÛ-å:Ø-Õ¹ØÙÏæ�ÙÏÔéè�ê�Ó-Õ�Ü�ëIÛ[Óå�ÖÎá�ã±Ý#Ù�Ô�Þ»ì-å�ê�ç,Õfì�Ô�Ü�ëîçfçËå�Û�Ô.ßUáSÞâÓàÖ�Ô�ÞâÓàÛ[Ó�Õ�Û[Ó k ��v | Õ�Ö0×Ýn���v | Ü�ëËì�ÞUÔ¡Þ Õ�Ô�í9å�Û�Õ�æÀæ|"þ ú þlk ��Ý¥ÓgÞ Õ�ò%Ó |�Þß t�T ô Ô¸ø)Ø ~ à ÔÕø°Ø n n�� |NÞs&t à Ô±÷ÔÕø4ùPúWØQØ ~ à ÔÕø°Ø¯ùÝ*ÞâÓàÛ[Ó�à� ô ï X»ø ûÛO¥cIR à ó ø²ÙÀÜ�Ô�ú*ù Ä ù[ZÎØàþ.Û[Ó.Ú�ëJæ�Õ%Û�Ý#ÙÏÔ¡ÞÂÛ[Ó�ÜðçËÓ-ì�ÔlÔ!å à \ÅÕ�ÖY×eàá ô ï O¥cIR à�âà� ô ß OÛ[å9å!íîß Ê hIHjO�i [ FÒÍ°L#cINPU�JML�~ [ Oe�4L�dËd [ } z �IO¥HjFIëÂ�4L�d$d [ �Yl v FIO¥RPL [ ~�UWVGÔ z l¶sMØDÌqLÐ�IO¥LRPhILÚV¨UW^½^jU�Ì-H½FIëÒH½FIL�éY� [ ^½H½RTS V¨U6N#RQhILV¨�IFIi¯RPHjUWF�T ô l u U6N"ú»v |_a ^½L�R_��× t ͺLËRPhIL�OQd [ ^j^½L�O¥RHjFMRPL�ëWL¯NÑOQ�Ii�hÎRPh [ R���× t v²ø ô j úYl v FÎRPhIL�O [ dËL�Ì [ S [ O'HjF Ô z l«sMØ�ÌqL�U6ÍIR [ HjF��IOQHjFIë�Ô±s_l��WØT ô ÔÕø)Ø n óY¬2ãSä t� I 8 � }|å úú � ã � � } Ñ ôå ú � Q ñ ô ï ó�¬¨ñ ô ú f � Q � � Q � ô � � ã � f � � � ã � ô � ãSä t� I 8 � å ã � � � Qn óY¬¨ñ ô ú f � ã � � Q � ô � � ã � f � � � ã � ô � ø ã � Qô ï óY¬¸ñ ô íÔ±s_í��MØÝGL�NPLË̬LÚ~IU�FIU6R"h [ JMLËRPhILÚ^½UWë [ NPH½RQhIdËHjiÆi�U6NQNPL�i�RPHjUWF [ OÏH½F Ô z l«sMØ�OQHjFIi�LÚÌqLÚi�UWFIOQHj~IL�N#RQhILRQNP�IFIi [ RPL�~Î\ML�NQFIL�^±lu U6N [ ^j^ å ï }Wù��Yù»í»í»í [ FI~ � ï }Wù��Yù»í»í»í�ÌqLÚ~IL��9FILÚRPhILÚOQL¯RPO�à�ÝIô [ FI~�àá3Iô � ÍêS NQL�c9^ [ i yHjFIëÎÔ�ú*ùqnÏØ y NPL�ëW�I^ [ NPH½RTS [ FI~ÛÔÕú�ùqn�Ø y HKNQNPL�ëW�I^ [ NPH½RTS$ÌDH½RQhÛÔ�ú*ù

Äù[ZÎØ y NPL�ëW�I^ [ NQH½RTS [ FI~ Ô�ú*ù Ä ù[ZÎØ yH½N¥NPL�ëW�I^ [ NQH½RTS [ FI~#RPhILqV¨�IFIi¯RPHjUWF u ô Ì-HKRPh_T ô H½F#RPhIL¬~IL��9FIHKRPHjUWFIOßUWV��ÝIô [ FI~Ýá�Iô � Ô�O¥L�L'ÖYL�i�RPH½UWFz دl�ÖYHjdËH½^ [ NP^KS [ ODHjFÛÔ z l��WØ�ÌqL�OQL�L�RQh [ R-V¨U6N [ ^j^ å ï }Wù��qí»í»í| Þß?æt T ô ÔÕø°Ø ~ à ÔÕø)Ø n z óY¬ } I�÷�8 � � ÷ � I | Þs 9t à Ô�÷�Ô¸øCùPú6ØQØ ~ à ÔÕø)Ø � ù

Page 14: LINEAR MAPPINGS AND GENERALIZED UPPER SPECTRUM FOR

¡ ¯ F������±©��4�¯­*§ D ®���G�H�������¥­*�IH��H�ÌDhIL�NPLTàá�Iô ïhç 6� 8 � àá�Iô � l Ê hIL�i�^ [ H½d V¨UW^j^½U*ÌDO [ ODHjFa�4L�d$d [ } z lÄUu U6N [ F [ ^jUWëW�IL�OßUWV9RPhILqRx̬U�V¨UW^½^jU�Ì-H½FIë�NQL�OQ�I^½RQO�H½FÆRQhIL¤c [ iP\YHjFIë�~IHjdËL�FIOQHjUWFÆi [ OQL¤OQL�Lg� u f ak�NQU6c°UWOQHKRPHjUWFC�Yl�� [ FI~ Ê hIL�U6NPL�d s_l«s��±l

[)\ËÌN\S]�^`_)^`\Saéè�ÑCÒ;Ó�ÔyZ Õ%ÖY×U^ Ø�ÓUÙÏÖËÔ.Ó(ÚJÓ�Û-ÜhÝ#ÙÏÔ¡Þ |æþ Z þ ^lß�Ò;Ó�Ô à Ø�Ó»Õwäyå�Û[Ó�æçYÛ-å:Ø-Õ¹ØàÙ�æ�ÙÏÔ¡è»ê�Ó[Õ�Ü�ëJÛ-Ó]å�Ö á�ã§Ý#Ù�Ô�Þrì-å�ê@ç,ÕJì�ÔgÜ�ëîçfçËå�Û�Ô.ß·ê�Ü-Ü�ëJê�ÓhÔ�ÞâÕ�Ô ~ Õ�Ö0× ~ Õ�Û[ÓhÛ[Ó[Õ�æÖ,ëIê�Ø�Ó�Û�ÜLÜ�ë,ì�Þ¤Ô¡Þ Õ�Ô;Ó�ÙÏÔ¡ÞâÓ�Û |"þ ~ þ z �*Ô à Ø�å�Û ~ ï z �*Ô à Ø ï |Æþ ~ õ¢Õ�ÖY× ~ n ~ þ z � Ô à Ø9ßëÀí ~ þ ZUõ�Ô�ÞâÓàÖbÔ�Þ,Ó�Û[Ó�ÙÀÜ�Õ�çËå<Ü�ÙÏÔéÙÏò%Ó_Û-Ó[Õ�æ*Ö,ëIê�Ø�Ó�Û�n�.]Õ�ÖY×]ÕaÜîÓ[öàëËÓ�Ö0ì�ÓÐÔ�Ñâ÷WØ�å�í@çËå<Ü�ÙÏÔéÙÏò%ÓÛ[Ó-Õ�æWÖ,ëIê�Ø�Ó�Û�ÜuÔ!Ó�ÖY×�Ù�ÖIÚ�Ô.å�ù9Ó�Û-å�Ü�ëËì�ÞaÔ¡Þ Õ�Ôïíîå�Û_Õ�æÀæP>Ñ Q ÷ | | 6Û 9 ª � Q � � à Ô�÷�Ô¸øCù ª ØQØ ~+ªY~ à ÔÕø°Ø n n�.CÑ Q ô � � ô � � Q � ô � � ã � ô�� � � ã � ô �Q�÷ í

OÛ[å9å!íîß���L�R |åþ°i�þ ~ [ FI~ ~ þ d þ z � Ô à دlÚÖYHjFIi�L ~ þ z �»Ô à Ø�U6N ~ ï |_a RPhIL�NQLËHjO [i�UWFIO¥R [ FeR�ó�O¥�Ii�h�RQh [ R-V¨U6N [ ^j^;ú�v |Ô±s_í��WØ Ä � Ô à ùPú6Ø n ó¯ú%ô íu �_NQRPhIL�N a OQH½FIi�L d þ z � Ô à Ø a RPhIL�NQL�H½O [ OQL�éY�IL�FIi�LÚÔ�ú9÷6ØqUWV�c°UWOQHKRPH½JML�NPL [ ^)Fe�Id#Í°L�NQODRPL�FI~IH½FIëRPU�s�L�NPUÐOQ�Ii�h�RPh [ R-V¨U6N [ ^j^�>Ô±s_í� MØ Ä � Ô à ùPú9÷6Ø n ú f÷ í��L�R gÎ[ FI~�"Í°LÏNQL [ ^4Fe�Id#Í°L�NQOGOQ�Ii�hÒRPh [ R"Ô¡^Ci�d¥Ø j Ô¡^ai ~ Ø þÇgÎþ } aº|Úþ þ£i j ^ aº[ FI~Ô±s_l½}*Ø'hIUW^j~IO�l��4L¯R3>�Í°L#^ [ NQëWLÏL�FIUW�IëWhåOQ�Ii�hÒRQh [ RGú9÷ þhk � a ÌDhIL�NPL k ��HjO [ OGH½FK�4L�dËd [  _l��L�RìÑ ÷�ì ï Ñ ô 9 [ FI~�øe÷Òì ï ø ô 9 ͺL [ OÏH½FâÔ±s_l��WØPl�EGOQHjFIë u �_Í9HjFIH¡ª¶O�RPhIL�U6NPL�d a �4L�d$d [   aÔ±s_l¶sMØ a Ô±s_l��WØ ap[ FI~ Ô±s_l� MØ a ÌqL�h [ JMLZ�Ñ Q ÷ | | 6Û 9 ª � Q � � à Ô±÷ÔÕø4ù ª Ø¥Ø ~«ªï~ à ÔÕø)Øn |ýT ô 9 Ô¸ø)Ø ~ à ÔÕø°Ø�k�Z�Ñ Q ÷ | 6× 9 ª � Q � �

Ä � Ô à ù ª Ø ~+ªn � Ä ú �Pm÷ Ä � Ô à ùPú9÷6Ø�kqó�Z�Ñ Q ÷ | 6× 9 ª ô � Q � � ~+ªn � Ä ú f �Pm÷ kqó�Z�Ô¡Zýi ~ Ø � � Ñ Q ÷ ø ô � Q÷n n�.�ú f �&m÷ n n�.)Ñ Q � f �Pm � � � f � � Q � ô � � ã � f � � � ã � ô �Q�÷ ùÔ±s_í�£MØÌDhIHji�h�HjdÐc9^jHjL�O¤RQhIL�i�^ [ Hjd ÌDhIL�F�ÌqL�i�hIUêUWOQL i OQd [ ^½^°L�FIUW�IëWh)l U!*�;Î<\S[<Î Ê Ð�íWÑaÒ5ÓàÔ à Ø�Ó�Õ�äyå�Û[Ó�æJçYÛ-å:Ø-Õ¹ØÙÏæ�ÙÏÔéè�ê�Ó-Õ�Ü�ëIÛ[Ó�å�ÖÚá�ãlß�ëÀí�z �*Ô à جò²Z]õlÔ¡ÞâÓ�ÖÔ±s_í½} | Ø z � Ô àWV Ø ï Zíîå�Û�N ã P Q þÕ�æ�ê�å�Ü�Ô�Õ�æÀæïRÓûKT ã P Q ß�ëÀí�z �*Ô à Ø þ Z]õlÔ¡ÞâÓ�ÖÔ±s_í½}W}*Ø z � Ô àWV جò z � Ô à Ø�Ô]}Oi�z �*Ô à Ø j ^�Ø}¥k Ô]} j Z¦i0} j ^�Ø z � Ô à Ø;i�z �*Ô à Ø j Z

Page 15: LINEAR MAPPINGS AND GENERALIZED UPPER SPECTRUM FOR

� ©¶­��Q���gªq������©¶­�²¯®4��­�§G�±�*��£����6�]���P¦B§�©¶ª¬�Q­*®±©��¯­ ¡�=íîå�ÛeN ãIP Q þÕ%æ�ê�å<Ü�ÔgÕ%æÀæ¢R û¨T ãIP Q ß áSÞâÓyíîå�Û�ê_ëIæ�ÕfÓÔ�s_lj} | ØaÕ�Ö0×åÔ±s_lj}W}*ØaÕ�Û[ÓaÕ�æ�ÜîåUÔéÛ�ëËÓyíîå�Û� ã%Q þÕ�æ�ê�å�Ü�Ô�Õ�æÀæWæ�ÙÏÖ0Ó-Õ�Ûeê�Õ�çïÜ��çì6á ã d á Q õLÝ#ÞâÓ�Ö]Ý�ÓuÛ[Ó(ç0æ�Õfì-Ó àWV Øàèg��c à ß OÛ[å9å!íîß v VWz ��Ô à جòxZ U6N z � Ô à Ø ï z ��Ô à Ø n Z a RPhILGi�^ [ Hjd V¨UW^j^jU�ÌDO¬VBNPUWd RPhILÑNQL�OQ�I^½RQOqV¨U6NRPhIL�^jU�ÌqL�NDi�U6NQNPL�^ [ RPHjUWF�~IHjdËL�FIOQHjUWF Ô�OQL�L��ÂÝy� a Ê hIL�U6NQL�d }Wl½}�¸Ø¯l© OQOQ�Id$LÑRPh [ ROz �*Ô à Ø þ dËHjF0X9ZÒù z � Ô à Ø-\ [ FI~ z � Ô à Ø¥v | l#�4L¯R ~ [ FI~ ~ Í°L�OQ�Ii�h�RPh [ R|<þ ~ þ z �*Ô à Ø�Ô�H½V�z �*Ô à Ø ï | ̬LR [ \ML ~ ï | Ø [ FI~ ~ þ ~ þ z � Ô à دí2o¬S u [ RPUW��ª¶O^jL�dËd [_a_u �_Í9HjFIH¡ª¶O¤RPhIL�U6NPL�d a � u f a �4L�d$d [ s_l���� a9[ FI~�k�NPU6cºUWOQH½RPH½UWFh£"ÌqL�h [ JML| ^jHjdçH½FIVÛ õ�ö Ñ � Q ô � � ô � � Q � ô � � ã � ô�� � � ã � ô �`� | àWV Ô�÷�Ô¸øCùOÑIØQØ ~ àWV Ô¸ø)Ø ~ N ã P Q ÔðR#Øn ^½HjdçHjFIVÛ õ�ö ÑY� Q ô � � ô � � Q � ô � � ã � ôO� � � ã � ô �`� ó2Ñ Q | | 6Û ª � Q � � à Ô�÷�Ô¸øCù ª ØQØ ~+ªY~ à ÔÕø°Ø þ ÿ íÊ hê�IODV¨U6N�N ã P Q yT[ ^½dËUWO¥R [ ^j^SR ûKT ãIP Q ÌqL�h [ JML

^½HjdçHjFIVÛ õ�ö Ñï� Q ô � � ô � � Q � ô � � ã � ô�� � � ã � ô �`� Ä � Ô àWV ùOÑIØ þ ÿ ùëWH½JYHjFIë z � Ô àWV جò ~ Ô]}�i ~ j ^�Ø}lk Ô]} j Zmi?} j ^�Ø ~ i ~ j Z íÊ hILËi�^ [ H½d V¨UW^j^jU�ÌDO�ÍeS�R [ \YHjFIë�OQL�éY�IL�FIi�L�OÔ ~ $ ØGRPL�FI~IHjFIë�RPUhz �*Ô à Ø [ FI~âÔ ~ $ ØÑRPL�FI~IHjFIë�RQUz � Ô à دlo¬S Ê hIL�U6NPL�dËO�s [ FI~ t RPhIL�i�^ [ Hjd H½O [ ^jOQU RQNP�ILÒV¨U6NC��ã%Q yT[ ^½dËUWO¥R [ ^j^G^½HjFIL [ Nd [ c9O�0ìWá�ãKd áLQl�U *\|[)\YÍ�Í�É [2î±ÐËÐïÑaÒ5Ó�Ô à Ø�Ó_Õ�äyå�Û-Ó�æIçYÛ-å:Ø-Õ¹ØàÙ�æ�ÙÏÔ¡è�ê�Ó[Õ�Ü�ëJÛ-Ó_å�Ö�á ã ß�ëÀí�z � Ô à جò²ZUõ¥Ô�Þ,Ó�ÖÔ±s_í½}<�WØ z � ÔÍȹc à Ø ï Zíîå�Û�ÕeçYÛ-Ó�ò�Õ�æ1ÓàÖ,Ô�Ü9Ó�ÔOå�í�í�ëIÖ0ì�ÔéÙðå�ÖâܾÈÒû'Å � ÔÕá ã ùQá Q Øîß�ë í�z � Ô à Ø þ ZUõlÔ¡ÞâÓ�ÖÔ±s_í½}�sMØ z � ÔFȹc à ؤò z � Ô à Ø�Ô]}Oi�z �*Ô à Ø j ^�Ø}lk Ô]} j Zmi?} j ^�Ø z � Ô à Ø¢i�z �*Ô à Ø j Zíîå�Û�ÕeçYÛ-Ó�ò�Õ�æ1ÓàÖ,Ô�Ü9Ó�ÔOå�í�í�ëIÖ0ì�ÔéÙðå�ÖâܾÈÒû'Å � ÔÕáßãßùQá#Q$ØîßW�Ó�ê�Õ�ÛYX9Üß�}*Ø Ê hIL^jU�ÌqL�N"ͺUW�IFI~ÙëWH½JML�F<H½F Ê hIL�U6NPL�d$O} |å[ FI~ }W}H½O#RPhILͺL�O¥R"c°UWO¥OQH½Í9^½LUWFIL6l Ê hIH½O-V¨UW^½^jU�Ì-O'VBNPUWd Ç r [ d$c9^jL"}Wl�WØ © VBRPL�N�RPhIHjO�ÌqU6NQ\GÌ [ O4i�UWd$c9^½L�RPL�~ a ÌqL�^½L [ NPFMR [ ͺUW�_R [ c [ c°L�N�ÍeS u [ ^ji�UWFIL�N [ FI~#m_ª w L�Hj^� u m�� a ÌDhIL�NPL"RPhIL�SåO¥RP�I~_S a�[ dËUWFIë�U6RPhIL¯N�RQhIHjFIëWO a RPhIL"cINPU � L�i�RPH½UWF [ ^CcINPU6cºL�NQRPH½L�O�UWV�ëWL�FIL�N y[ ^jH�s�L�~Åt y ~IH½dËL�FIOQH½UWF��IOQH½FIë [ cIcINPU6cINQH [ RPL�^KSå~IL��9FIL�~Ûi�UWFMJMUW^j�_RQHjUWF�\ML¯NPFIL�^jO�l v FÒc [ NQRPHji��I^ [ N aRPhIL�S�cINPU�JML�~ÎRQhIL�HjFIL�éY� [ ^jHKRTS

Á | 6ô ª � Q � � È2Ô ª Ø ~+ª � { n ó�úJ� � | 6ô ª � Q { � � È2Ô ª Ø { ~+ª ùÌDhIL�NPLÝÈåHjO'FIUWF y ~IL�i�NQL [ OQHjFIë [ FI~�i�UWFIO¥R [ FMRÑV¨U6N ª v²ø V¨U6NDOQUWdËLgøÆl�EGOQHjFIë$RQhIHjO'HjF Ô±s_l�£MØ aÌqLËi [ F OQhIU�Ì RPh [ R�RPhIL [ F [ ^½UWëW�IL�O�UWV Ê hIL�U6NQL�dËO$} |Î[ FI~0}W}$hIUW^j~ [ ^½OQUÎV¨U6N |Òþ t þ }Wl© ë [ HjF Ç r [ d$c9^½LÎ}�OQhIU�Ì-O#RPh [ R�RPhILÚi�U6NQNPL�O¥c°UWFI~IH½FIë�^½U*ÌqL¯N#Í°UW�IFI~¢HjOÏRPhILËͺL�O¥R#c°UWO¥OQH½Í9^½LUWFIL6lsMØ Ê hIL�L¯r [ dÐc9^jL�O'UWVu�ÂÝy� a ÖYL�i�RPHjUWF�tYl����;OQhIU�ÌèRPh [ RÏÔ±s_lj}W}*جHjO'FIU6R'RQNP�IL�V¨U6N�t_v }Wl

Page 16: LINEAR MAPPINGS AND GENERALIZED UPPER SPECTRUM FOR

¡ � F������±©��4�¯­*§ D ®���G�H�������¥­*�IH��H�ï ij:9:9798;=4@ñðNòÆ>?7I.4,�È;3)79,#ÇZó�ô .S¶¹¶M,Æ.�8;=?É°Ç48°5;Ç;¶MÈ)A)@¨ÇC8;,

v F<RPhIH½OÐO¥L�i�RPH½UWF<ÌqLcINPU*JML [ RQL�i�hIFIHji [ ^'^jL�d$d [ i�UWFIi�L�NPFIH½FIë i�UWFMJMUW^j�_RPH½UWFIOÆÌDhIHji�hÙ̬L�IOQL�RQU$i�h [ N [ i�RPL¯NPH1s�L�RPhIL��_cIcºL�Nyt y ~IHjdËL�FIOQHjUWF)l u U6N'RPhIHjO'c9�_NQc°UWOQL�ÌqL�FIL�L�~�RPhIL�V¨UW^½^jU�Ì-H½FIëNPL�O¥�I^½RÑUWFRPhIL�ͺL�h [ J_HjUW�_NDUWVCd$L [ OQ�_NPL�ODUWV4Í [ ^½^jO'VBNPUWd«� u fK�`l�lÎIÊ�ÊwɨÐ�b0ÑCÒ5Ó�Ô |�þ£g�þ }UÕ�ÖY×*�v | ßháSÞâÓ�Û-Ó�Ó�ã:ÙÀÜ�Ô¡Ü_ÕÂÖ,ëIê�Ø�ÓàÛ�ó ö õ@×fÓ(çïÓ�ÖY×�ÙÏÖ ÚCå�ÖËæ�èå�Ö g õ(¹õ�Õ�ÖY×_^¢õOÜ�ëËì�ÞKÔ�ÞâÕ�ÔWíîå�Û�Ó�ò%Ó�Û�è�äyå�Û[Óàæ,ç0Û[å:Ø-ÕfØàÙÏæ�ÙÏÔéè�ê�Ó[Õ�Ü�ëJÛ-Ó à å�ÖÎá�ã Õ�Ö0×eíîå�Û�Õ�æÀæú ö n } j �UÝ¥ÓeÞ Õ�ò:Ó

à } ð øåûÎá ã ó Ô¡ÞâÓ�Û-Ó�Õ�Û-ÓDú]Õ�ÖY× ª Ý#ÙÏÔ¡Þ |Æþ ú þ ú ö Õ�ÖY×Ïú w n ª n }�Ü�ë,ì�ÞaÔ�Þ Õ%Ôà Ô±÷ÔÕø4ù ª ØQØlv }� } z ªú � ã � � � � � à Ô±÷ÔÕøCùPú6ØQØ ü � n �%ó ö ú ã � � � � w �ö í

OÛ[å9å!íîß-ÖYL�L�� u f a ��L�dËd [ �Yl����±l�U��L�R à Í°L [ oqU6NPL�^�cINPU6Í [ Í9H½^jHKRxS�dËL [ OQ�_NQLÚUWF¢áßãßlK��L�R | þõgÙþ } [ FI~ | þ þ }WlpqUWFIOQHj~IL�NDøåûÎá ã [ FI~�ú�v | OQ�Ii�h�RPh [ RÔ z í½}*Ø à Ô±÷ÔÕø4ù ª Ø¥Ø n } z ªú � ã � � � � � à Ô±÷ÔÕøCùPú6ØQØV¨U6N [ ^j^ ª Ì-HKRPhÎú w n ª n }Wl v FMRPL�ë6N [ RQHjFIëÐÍeSc [ NQRPO¤ÌqL�h [ JML

u ô ÔÕø°Øqì ï | dËHjFïXM}WùPú ã ó ø�iw� ó � ã \ ~ à Ô��IØ ï ^)ú ã | 6ô ª � ã � � à Ô±÷ÔÕø4ù ª Ø¥Ø ~«ªï ^)ú ã }5| ô ¨ô ª � ã � � à Ô�÷�Ô¸øCù ª ØQØ ~+ª k²| �

ô ¨ ª � ã � � à Ô±÷ÔÕø4ù ª ØQØ ~+ªk | 6� ª � ã � � à Ô±÷ÔÕø4ù ª ØQØ ~+ª �nçà Ô±÷ÔÕø4ùPú w ØQØWk z ã � � � � � ¹� � à Ô±÷ÔÕø4ùPúWØ¥Ø]úJ� ã � k à ÔÕá ã Ø]ú ãn n-v à Ô±÷ÔÕø4ùPúWØQØxú �xw � � � ã � � � � �,k à Ô±÷ÔÕø4ùPúWØQØ]ú � ã � k<ú ã yÔ z í��WØV¨U6N�n ï z ã � � � � � j elw U6RPL¬RPh [ RßÍeSK� u f a p¤U6NPUW^½^ [ NQS��Yl«s��9V¨U6N à yT[ ^½dËUWO¥R [ ^½^YøåûÎá�ãRQhIL�NPL'HjOßú�³�OQ�Ii�hÆRPh [ R�V¨U6N[ ^j^ |Úþ ú n ú�³ËRQhIL#HjFIL�éê� [ ^jH½RTS Ô z lj}*ؤHjOÑO [ RPHjO¥�9L�~ÒV¨U6N [ ^½^ ª Ì-HKRPh�ú w n ª n }WlÑÝGU*̬L�JML�N aÌqL�i [ F0FIU6RË�IOQLÎRQhIHjOÐNQL�OQ�I^½R$Í°L�i [ �IO¥L�UWVÑRPhILÎ~IHxöËi��I^KRPHjL�O$i [ �IOQL�~0ÍeS<RPhILÎV [ i�R$RPh [ R$ú�³~IL�cºL�FI~IO�UWFÒøCl v FÒU6NP~IL�N�RPU [ JMUWH½~ÒRPhIL�OQLÐ~IHxöËi��I^KRPHjL�O�̬LÆcINPU�JML"Rx̬U�^½L�dËd [ OÑV¨U6N�ÌDhIHji�hÌqL�FIL�L�~ÎRPhIL�V¨UW^j^½U*ÌDHjFIëÆFIU6R [ RPH½UWF)l��L�R | þ�g¢þ } [ FI~ |�þ þ }WlC��L�R"ú]v |�[ FI~{n ï z ã � � � � � j elC�ÛLË~IL�FIU6RPLËÍêSO¥cIR à RQhIL�OQ�_cIcºU6NQRÐUWV [ o¤U6NQL�^¬cINPU6Í [ Í9Hj^½H½RTS dËL [ O¥�_NPL à l © c°UWH½FeR"øèû O¥cIR à H½O"i [ ^j^½L�~Ô�ú*ùqnÏØ y NPL�ëW�I^ [ NDÌDH½RPhNPL�O¥cºL�i�RDRPU à HjVu ô ÔÕø°Ø ï | dËH½FYXM}WùPú ã ó ø�i�� ó � ã \ ~ à Ô¡�IØn nKv à Ô±÷ÔÕøCùPú6ØQØ]ú �zw � � � ã � � � � � k à Ô±÷ÔÕø4ùPúWØ¥Ø]úJ� ã � kÙú ã y í

v VCøÒû�O¥cIR à H½O-FIU6R"ÔÕú�ùqn�Ø y NPL�ëW�I^ [ N'Ì-HKRPh�NPL�O¥c°L�i¯RÑRPU à ÌqL�O [ SRPh [ RÑHKR-H½OÏÔ�ú*ùqnÏØ y H½N¥NPL�ëW�I^ [ NÌDH½RPh#NPL�O]c°L�i�R�RPU à l Ê hILÏÔ�ú*ùqn j �WØ y NPL�ëW�I^ [ NQH½RTS [ FI~"RPhIL�Ô�ú*ùqn j �WØ y HKNQNPL�ëW�I^ [ NPH½RTS�ÌDH½RQhÆNPL�O¥c°L�i¯RRPU à [ NPLG~IL��9FIL�~�HjFÚRPhIL�O [ dËLDÌ [ SMl w U6RPLÑRQh [ R¬RPhIL�OQL�~IL��9FIHKRPHjUWFIO¤~IL�cºL�FI~ [ ^½OQU"UWF gË[ FI~el�ÝGU�ÌqL�JML¯N a ̬L�ÌDHj^½^)�IOQL�RPhIL�d UWFI^KS�V¨U6N'�_rIL�~ g�[ FI~Kel

Page 17: LINEAR MAPPINGS AND GENERALIZED UPPER SPECTRUM FOR

� ©¶­��Q���gªq������©¶­�²¯®4��­�§G�±�*��£����6�]���P¦B§�©¶ª¬�Q­*®±©��¯­ ¡÷R�lÎIÊ�ÊwÉ Ð��WÑCÒ5Ó�Ô |þøgåþ }]Õ�ÖY× |�þ þ }Iß]á5ÞâÓ�Û[Ó�Ó�ã:ÙÀÜ�Ô¡Ü�Õ�ÖËëJê�Ø�Ó�Ûeó ö õ�×fÓ(çËÓàÖY×�ÙÏÖIÚå�Ö,æ�èÂå�Ö g õÙ¹õ¥Õ�ÖY×e^¢õ#Ü�ë,ì�Þ�Ô¡Þ Õ�ÔËíîå�Û¤Óàò%Ó�Û�è_äyå�Û[ÓàæJçYÛ[å:Ø�Õ¹ØàÙÏæ�ÙÏÔéè_ê�Ó-Õ�Ü�ëIÛ[Ó à å�Ö$á ã Õ�Ö0×�íîå�ÛÕ�æÀæpú ö n } j �hÝ¥ÓgÞ Õ�ò%Ó

à Ô!X»ø�û�O]cIR à ó ø²ÙÀÜ�Ô�ú*ùqn j �WØàþ.Ù�Û�Û[Ó.Ú�ëJæ�Õ%Û�Ý#ÙÏÔ¡ÞÂÛ[Ó�ÜðçËÓ-ì�ÔlÔ!å à \�Ø n �%ó ö ú ã � � � � w �öíîå�Û�Õ�æÀæ9úhÝ#ÙÏÔ¡Þ |Ðþ ú þ ú ö ß OÛ[å9å!íîß���L�R�ó ö Í°L [ O-H½FK�4L�d$d [ }<�Yl v VgøÒûåO¥cIR à H½O#Ô�ú*ùqn j �WØ y HKNQNPL�ëW�I^ [ NDÌDH½RQhÎNPL�O¥cºL�i�RÑRQUà a RPhIL�F�H½FeRQL�ë6N [ RPH½FIëÐÍeSc [ N¥RPO [ ODHjF Ô z l��WØ�ÌqL�OQL�L�RQh [ Rà Ô�÷�Ô¸øCù ª ØQØlv }� } z ªú � ã � � � � � à Ô±÷ÔÕø4ùPúWØ¥Ø

V¨U6NDOQUWdËL�ú w n ª n }Wl Ê hIH½O-ëWH½JML�O'RPhIL�i�^ [ H½d ÍêSa��L�dËd [ }<�Yl�U�lÎIÊ�ÊwɱÐ<�SÑCÒ;Ó�Ô |�þùg þ }%õ |�þ þ }%õeÕ�ÖY×�thv | ß�Ò;Ó�Ô à Ø�Ó�ÕÂäyå�Û[Ó�æYç0Û[å:Ø-ÕfØàÙÏæ�ÙÏÔéèê�Ó[Õ�Ü�ëJÛ-Ó_å�Ö$á�ãxÝ�ÙÏÔ�ÞUì-å�ê@ç,ÕJì�Ô#Ü�ë9ç¹çËå�Û�Ô.ß�áSÞâÓ�Û-Ó�Óã:ÙÀÜ�Ô k � v | Õ�ÖY×�n � v | Ü�ëËì�Þ�Ô¡Þ Õ�Ôïíîå�ÛÕ�æÀæpúCÝ#Ù�Ô�Þ |Ðþ ú þ·k � Ý�ÓuÞ Õ�ò:Ó| ß t u ô Ô¸ø)Ø { ~ à Ô¸ø)Ø n n � Ô�^jUWë �ô Ø { |�sPt u ô ÔÕø°Ø { ~ à ÔÕø°Ø¯ùÝ*ÞâÓàÛ[Ó¾� ô ï X»ø�û�O¥cIR à ó øqÙÀÜ�ÔÕú�ùqn�Øàþ.Û-Ó(Ú�ëIæ�Õ�ÛeÝ�ÙÏÔ�ÞaÛ[Ó�ÜðçïÓ-ì�ÔlÔ.å à \UÕ�ÖY× á ô ï O¥cIR à�â � ô ß OÛ[å9å!íîß Ê hILåÍ [ OQHjiåHj~IL [ ͺL�hIHjFI~âRPhIHjOÚcINPUêUWV�HjOÚRPh [ RÚRQhILÒcºUWHjFMRPOÚø ûÓO¥cIR à V¨U6NÌDhIHji�hà Ô±÷ÔÕø4ùPúWØQØqHjO­]Í9H½ë:®ÒÔ�V¨U6NG�_rIL�~Òú6Ø [ NPLÏNPL�ëW�I^ [ N [ FI~ [ c°UWH½FeR���û�O¥cIR à i [ F�Í°L#H½N¥NPL�ëW�I^ [ NUWFI^½SËHjV)RPhIL�NPL [ NPLGcºUWHjFMRPOqø�û�O¥cIR à OQ�Ii�h�RPh [ R à Ô±÷ÔÕø4ùPúWØ¥ØLv à Ô±÷Ô¡�ºùPú6ØQدlS�ÛLGÌ-H½^j^°~IHKJ_Hj~ILO¥cIR à HjFMRPU�c [ NQRPO#OQ�Ii�h¢RPh [ R#ÌqLÚi [ F¢i�UWFMRQNPUW^�RPhIUWO¥LËc°UWHjFMRPOÏÌDhIHji�hÆ­¥d [ HjFI^KS ®�i [ �IO¥L�RQhILH½N¥NPL�ëW�I^ [ NQH½RTS�UWV [ ëWHKJML�F�cºUWHjFMR@�ÚûÎO¥cIR à l��L�R |þ ú ö n } j �$Í°L#OQ�Ii�h�RQh [ Ru�%ó ö ú ã � � � � w �ö n } j � a ÌDhIL�NPL�ó ö H½O [ OGHjF]�4L�dËd [ }�s_l��L�R |Æþ ú þ ú ö l b L��9FIL ñ ô ï OQ�_c³r¸+¹Qº�»�¼ à Ô±÷ÔÕø4ùPúWØQدíu U6NqL [ i�høåûÎO¥cIR à ̬LGi [ F�L�O¥RPH½d [ RPL u ô ÔÕø°Ø�HjFËRPhIL�V¨UW^½^jU�Ì-H½FIëÏÌ [ Sºì u H½NPO]RqÌqLGi�U�JML�N'O¥cIR àÍeS�Í [ ^½^jOCÌDH½RPh�N [ ~IHj�IO4ú [ FI~ÏÌDH½RPh�i�L�FMRQNPL�O [ R4RPhIL¬~IHjO]R [ FIi�L å ú¤VBNPUWd ø a ÌDhIL�NPL å ï }Wù»í»í»í�ù[� ô[ FI~�� ô HjO4RQhIL¬OQd [ ^½^jL�O¥RCHjFMRPL�ëWL�NCÌ-HKRPh¤� ô v0~IH [ d�Ô�O¥cIR à Ø j úêl Ê hIL�F#ÌqL¬i�U�JML�N2L [ i�h"UWV_RPhIL�O¥LÍ [ ^j^½Oq÷ Ì-HKRPhËÍ [ ^j^½O¬UWV)N [ ~IHj�IOqú [ FI~�i�L�FeRQNQL�O¤HjF÷ G O¥cIR à l Ç O¥RPH½d [ RPHjFIë�RPhILGHjFMRPL�ë6N [ FI~�H½FL [ i�h�UWVCRQhIL�OQL�Í [ ^½^jO a ̬L�U6ÍIR [ H½FÔ z í¶sMØ u ô Ô¸ø)Ø n ó ¬ ã

t� I 8 � } å úú � ã � � } úå ú � ã ñ ô n ó ¬ú¬ ñ ô ^½UWë �ô íÝGL�NPL¤ó ¬ H½O [ i�UWFIO]R [ FeRÑ~IL�c°L�FI~IH½FIë�UWFI^KSÎUWFK^ [ FI~Åó ¬ú¬ HjO [ i�UWFIO¥R [ FMRG~IL�c°L�FI~IHjFIë�UWFI^½S�UWF^ [ FI~åRPhIL#~IH [ dËL¯RPL�NÑUWVßO¥cIR à ~IL�FIU6RPL�~ÒÍeS�~IH [ d�Ô�O¥cIR à دl w U6RPL�RPh [ RÑRQhIL#OQL�i�UWFI~ÒO¥RPL�c�HjOFIU6RDFIL�i�L�OQO [ NQS�hIL�NQL�Í9�_R'Ì-H½^j^°Í°L�FIL�L�~IL�~�H½F�RPhIL�^ [ RPL¯N [ cIc9^½Hji [ RPH½UWFIODUWV¬Ô z l«sMØ�lu U6N [ ^½^�o¤U6NQL�^)OQL�RPO�û��?á�ã ~IL��9FIL��Ô�û�Ø ï � |�ü dËHjFïXM}WùPú ã ó ø�iw� ó � ã \ ~ à Ô��Iدí

Page 18: LINEAR MAPPINGS AND GENERALIZED UPPER SPECTRUM FOR

¡�6 F������±©��4�¯­*§ D ®���G�H�������¥­*�IH��H�ÖYL�R � �ô ï ð9øåû�O]cIR à ó øåHjO�Ô�ú*ùqnÏØ y NPL�ëW�I^ [ N'ÌDH½RPhNPL�O¥cºL�i�R-RQU à [ FI~ à Ô±÷ÔÕøCùPú6ØQØ�ò ñ ô� ü[ FI~ á �ô � ï X»ø�û�O¥cIR à ó øåH½O#Ô�ú*ùqnÏØ y H½NQNQL�ëW�I^ [ N¤ÌDH½RPhNPL�O¥cºL�i�R-RQU à [ FI~ u ô Ô¸ø)Ø n ��Ô�� �ô Ø-\Míu �_NQRPhIL�N a V¨U6N [ ^j^ � ï �Yù�s_ù»í»í»í�ÌqL�~IL��9FIL�H½FI~I�Ii�RPHKJML�^½SOQL�RQO�á �ô � ÍeSá �ô � ï ð øåû�O]cIR à ó øåHjO�Ô�ú*ùqnÏØ y H½NQNPL�ëW�I^ [ N¤Ì-HKRPh�NQL�O¥c°L�i�RÑRPU à

[ FI~ u ô ÔÕø°Ø n � } � �ô ç } � � �=$ 8 � á �ô $ �0� ü íu HjF [ ^½^½S a ̬L�~IL��9FIL á �ô ï 6=� 8 � á �ô �[ FI~ � �ô ï � �ô ç á �ô íÖYHjFIi�L"ø��d à Ô±÷ÔÕø4ùPúWØ¥Ø'HjOG�_cIcºL�N�OQL�dËH½i�UWFMRPHjFe�IUW�IO [ FI~åø½�d u ô ÔÕø°Ø-HjOGi�UWFMRPHjFe�IUW�IO a RPhIL�O¥LOQL�RQO [ NPL�oqU6NPL�^;OQL�RPO�l�ÛL��INPO¥R-OQhIU*ÌèRPh [ RÔ z í z Ø | ß ;t u ô ÔÕø°Ø { ~ à ÔÕø°Ø n Ôð�%ó�¬ý¬»^jUWë �ô Ø { | s ;t u ô ÔÕø°Ø { ~ à ÔÕø°Ø¯í�ÛL"d [ S [ O¥OQ�IdËL#RPh [ R à Ô�� �ô Ø�v | l���L�RGø û�á �ô l Ê hIL�F u ô Ô¸ø)Ø n ��Ô�� �ô ØDÌDhIHji�hÒHjd$c9^½HjL�ORPh [ RgøÐHjOqÔ�ú�ùqn j �WØ y H½N¥NPL�ëW�I^ [ N4Ì-HKRPhÏNPL�O]c°L�i�R2RQU à Ô�� �ô Ø � � à óJþ ;t lCÝGL�NPL à ó þ ;t HjOCRPhIL¬NQL�O¥RQNPH½i�RPH½UWFUWV à RPUÐRPhIL�OQL¯RH� �ô l¤EGOQHjFIë��4L�d$d [ }�sÆÌqLÏU6ÍIR [ H½FÎRPh [ R à ÔQá �ô Ø n à Ô�� �ô ؤÌDhIHji�håHjd$c9^½HjL�ORPUWëWL�RQhIL�NqÌ-HKRPhÒÔ z l¶sMØ [ FI~ÚÌ-HKRPhËRPhILGV [ i�RqRQh [ R u ô Ô¸ø)جò à Ô±÷ÔÕø4ùPúWØQØ�òâñ ô j ��V¨U6N [ ^½^ ø�û�� �ôRPh [ R | ß ;t u ô ÔÕø)Ø { ~ à ÔÕø)Ø n ÔéóY¬ú¬Bñ ô ^½UWë �ô Ø { à Ô�á �ô Ø n Ôð�%óY¬ú¬�^½UWë �ô Ø { | s ;t u ô ÔÕø°Ø { ~ à ÔÕø)Ø�íÊ hê�IOÏÔ z l z جhIUW^j~IO�lu U6N [ ^½^ å ï �Yù�s_ù»í»í»í�ÌqL�~IL��9FIL� Iô ï X»øåû�O¥cIR à ó øåHjO�Ô�ú�ùqn�Ø y NQL�ëW�I^ [ N'Ì-HKRPh�NQL�O¥c°L�i�R-RPU à

[ FI~U� � I ñ ô nçà Ô±÷ÔÕø4ùPúWØQØ þ � � I � � ñ ô \[ FI~á Iô � ï ð ø�û�O¥cIR à�â I � �=÷�8 � � ÷ô ó øåH½OÏÔÕú�ùqn�Ø y HKNQNPL�ëW�I^ [ NDÌDH½RPh�NPL�O]c°L�i�R-RPU à

[ FI~ u ô Ô¸ø)Ø n � }0} I � �=÷�8 � � ÷ô � ç � Iô � ü í

Page 19: LINEAR MAPPINGS AND GENERALIZED UPPER SPECTRUM FOR

� ©¶­��Q���gªq������©¶­�²¯®4��­�§G�±�*��£����6�]���P¦B§�©¶ª¬�Q­*®±©��¯­ ¡�u �_NQRPhIL�N a V¨U6N [ ^j^ � ï �Yù�s_ù»í»í»í�ÌqL�~IL��9FILá Iô � ï ð øåû�O]cIR à/â I � �=÷�8 � � ÷ô ó øåHjO�Ô�ú*ùqnÏØ y H½NQNQL�ëW�I^ [ N'Ì-HKRPh�NQL�O¥c°L�i�R-RPU à

[ FI~ u ô ÔÕø°Ø n � }0} I � �=÷�8 � � ÷ô � ç � Iô ç } � � �=$ 8 � á Iô $ �0� ü í��L�R á Iô ï 6=� 8 � á Iô �[ FI~ � Iô ï � Iô ç á Iô í© O¤Í°L�V¨U6NPL a RPhIL�OQL�OQL¯RPO [ NPL�oqU6NPL�^;OQL�RQO�l�ÛL�FIU*ÌècINPU�JML�RPh [ R-V¨U6N [ ^j^ å ï }Wù��Yù»í»í»í�̬L�h [ JMLÔ z í«tWØ à Ô�÷�Ô¸øCùPú6ØQØ þ �J� I ñ ôV¨U6N [ ^j^9øåû�O¥cIR à â#ç I÷�8 � � ÷ô l v F�U6NP~IL�NqRQUÏcINPU*JMLÑRQhIHjO¬ÌqLÑcINPUêi�L�L�~ÚÍeSÚHjFI~I�Ii¯RPHjUWF�UWF å l#� Ld [ S [ O¥OQ�IdËL$RPh [ R z ó ¬ÿ¬ ^jUWëpÔ]} j ú6Ø n n�ú � ã � ÍeS d [ \YHjFIë�úOQd [ ^½^jL�N�HjVqFIL�i�L�OQO [ NQSMl v F RQhILi [ OQL å ï } a ÌqL#U6ÍIR [ HjF�VBNPUWd Ô z l«sMجRPh [ RGH½Vgø�ûåO¥cIR à [ FI~ à Ô±÷ÔÕø4ùPúWإؤòýñ ô j � a RPhIL�F�ø�HjOÔ�ú*ùqnÏØ y NPL�ëW�I^ [ NDÌDH½RPhNPL�O¥cºL�i�RDRPU à l Ê he�IODøåûK� �ô � � �ô ap[ FI~�OQUåÔ z l«tWØqhIUW^½~IODV¨U6N å ï }Wl�ÛL�FIU*Ì [ OQOQ�Id$LRPh [ R à Ô±÷ÔÕø4ùPúWØQØ þ � � � I � � � ñ ô V¨U6N [ ^j^qø(ûäO¥cIR à�â3ç I � �÷�8 � � ÷ô l��4L¯RøÛûÒá�ãÛÌDH½RPh à Ô�÷�Ô¸øCùPú6ØQØ'ò � � IÕñ ô l@�ÛLÏÌDHj^j^;cINPU�JMLÏRPh [ RGø û?ÔÕá�ã â O¥cIR à Ø ç Ô ç I÷�8 � � ÷ô دlu U6N#RPhIHjO#c9�_NQcºUWOQL̬Ld [ S [ O¥OQ�IdËLÚRPh [ R#ø û O¥cIR à-â�ç I � �÷�8 � � ÷ô l v VDøÙHjO�Ô�ú*ùqnÏØ y NPL�ëW�I^ [ NÌDH½RPhÛNPL�O]c°L�i�RÆRPU à a RPhIL�F øæû � Iô �õ� Iô l v V'øÙHjO�Ô�ú*ùqnÏØ y H½NQNPL�ëW�I^ [ NÏÌDH½RPhÛNPL�O¥c°L�i¯RÐRPU à aRPhIL�F u ô ÔÕø°Ø n ��ÔÕO¥cIR à,â�ç I � �÷�8 � � ÷ô Ø a OQHjFIi�LÑÌqLGd [ S [ OQO¥�IdËL-RPh [ R¬ø jû-á3Iô l u �_N¥RPhIL�N a ÍeS$RQhILHjFI~I�Ii¯RPHjUWFÚhMSYcºU6RPhIL�OQH½OqO¥cIR à"â(ç I � �÷�8 � � ÷ô �¬X9�Úû�O]cIR à ó à Ô�÷�Ô��ºùPúWØ¥Ø þ � � � I � � � ñ ô \ [ FI~ËRQhILO [ dËL�i [ ^½i��I^ [ RPH½UWF [ ODHjF Ô z l«sMجëWH½JML�O

u ô Ô¸ø)Ø n � ||Ò µ)¸+¹`ºO»�¼�Ó ¼ � û � µ P ô �`� � � :�� æ :<;�� � tYغdËHjFïXM}WùPú ã ó ø�iw� ó � ã \ ~ à Ô��IØn z ó�¬ÿ¬À�J� I ñ ô ^½UWë �ô n z ó�¬ÿ¬�^jUWë �ô à Ô±÷ÔÕø4ùPúWØQØ n n�úJ� ã � à Ô�÷�Ô¸øCùPú6ØQدùÌDhIHji�h<H½O [ i�UWFeR¥N [ ~IHji�RQHjUWF OQHjFIi�L�øÙHjOÔ�ú*ùqnÏØ y H½N¥NPL�ëW�I^ [ N�Ì-HKRPhÛNPL�O¥cºL�i�R"RPU à l Ê he�IO�Ô z lntWØhIUW^j~IO�lo¬S�Ô z lntWØ�ÌqL�h [ JMLÔ z í��MØ á ô ï 6=I 8 � á Iô ùOQH½FIi�LÏHjV4ø�û�á ô â�ç 6I 8 � á Iô a RPhIL�F¢Ô z l«tWؤH½d$c9^jH½L�O'RQh [ R à Ô±÷ÔÕø4ùPúWØQØ ï |_a ÌDhIHji�h�H½O [ i�UWFMRQN [�y~IHji¯RPHjUWF�OQHjFIi�L�øåû�O¥cIR à lv R'OQ�5öËi�L�O-RPUÐcINQU*JML�RPh [ R-V¨U6N [ ^j^ å ï }Wù��Yù»í»í»íÔ z í��WØ | ß1æt u ô ÔÕø°Ø { ~ à ÔÕø°Ø n Ô z ó�¬ÿ¬»^jUWë �ô Ø { Á I�÷�8 � � { � ÷ � I �J|5s 9t u ô ÔÕø°Ø { ~ à ÔÕø)Ø©��í

Page 20: LINEAR MAPPINGS AND GENERALIZED UPPER SPECTRUM FOR

4�� F������±©��4�¯­*§ D ®���G�H�������¥­*�IH��H�v FÐV [ i¯R a OQHjFIi�L¾� ô ïhç 6I 8 � �ÝIô ÌDhIL�NPL¤RPhILDoqU6NPL�^_OQL¯RPO#�ÝIô [ NPL¤~IHjO � UWH½FeR a ÌqL¤U6ÍIR [ H½FÆRPhIL¤i�^ [ Hjd�IOQH½FIëåÔ z l��MØ [ FI~ Ô z l��WجOQH½FIi�L| ß t u ô ÔÕø°Ø { ~ à ÔÕø°Ø n Ô z ó�¬ý¬»^jUWë �ô Ø { Á 6�÷�8 ö �J� { ÷ � Á 6� I 8 � |5s æt u ô Ô¸ø)Ø { ~ à Ô¸ø)Ø��n n � ÔÕ^jUWë �ô Ø { |�s&t u ô Ô¸ø)Ø { ~ à Ô¸ø)دív V å ï } a RPhIL�F�Ô z l��WØ�V¨UW^½^jU�Ì-OqVBNPUWd�Ô z l z Ø�l���L�R å òr�Yl v FU6NP~IL�N'RPU#cINPU�JMLÐÔ z l��WØ�ÌqL�d [ S[ OQOQ�Id$L�RPh [ R à Ô ç I÷�8 � � ÷ô Ø¥v | l#�4L�RDø�û-á�Iô l Ê hIL�F�V¨U6N [ ^½^ � ï }Wù��Yù»í»í»í¬ÌqL�h [ JML

u ô ÔÕø)جò � Á¥} I � �=÷�8 � � ÷ô � ç � I � �ô ç } � � �=$ 8 � á I � �ô $ � �ëWH½JYHjFIë u ô ÔÕø°Øqò¬��Ô ç I � �÷�8 � � ÷ô دí Ê hê�IO u ô Ô¸ø)Ø n ��Ô�O¥cIR à*â�ç I � �÷�8 � � ÷ô دlßEGOQHjFIëÎÔ z lntWØ a RPhIL�O [ dËLi [ ^½i��I^ [ RPH½UWF [ O'HjF Ô z l«sMجëWH½JML�O'RPh [ RDV¨U6N [ ^½^°øåû-á3IôÔ z í� MØ u ô ÔÕø°Ø n z ó�¬ú¬ �J� I ñ ô ^jUWë �ô íu �_NQRPhIL�N a HjVeø�û ç I÷�8 � á ÷ô a RPhIL�F u ô ÔÕø°Ø n ��Ô ç I÷�8 � � ÷ô Ø°Ì-hIH½i�h�Hjd$c9^½HjL�O)RPh [ R�ø#H½O¬Ô�ú*ùqn j �WØ yH½N¥NPL�ëW�I^ [ N�Ì-HKRPh NPL�O¥cºL�i�R#RPU�� ï à Ô ç I÷�8 � � ÷ô Ø � � à ó� æ 9��; þ 9t l Ê he�IO���L�dËd [ }�s�ëWHKJML�O�RPh [ R

�)Ô ç I÷�8 � á ÷ô Ø n } j � a ÌDhIHji�h�H½d$c9^jH½L�O�)Ô I=÷�8 � � ÷ô Ø ï �)Ô I=÷�8 � � ÷ô Ø¢i��)Ô I=÷�8 � á ÷ô Øqò } j �Yí

Ê hê�IO à Ô�á Iô Ø n à Ô ç I÷�8 � � ÷ô دl�ÖYHjFIi�LÐñ ô n � ÷ u ô Ô¸ø)ØDV¨U6N [ ^½^CøÛû�� ÷ô [ FI~T> ï }Wù»í»í»í¯ù å a ̬Lh [ JML�ÍeS Ô z l� MØ| ß1æt u ô ÔÕø°Ø { ~ à ÔÕø°Ø n Ô z ó ¬ÿ¬ � � I ^½UWë �ô Ø { | � æ 9��; s 9t ñ {ô ~ à ÔÕø°Øn Ô z ó�¬ÿ¬»^jUWë �ô Ø { I�÷�8 � � { � ÷ � I �I| s 9t u ô ÔÕø°Ø { ~ à ÔÕø°Ø¯ùÌDhIHji�h�i�UWd$c9^½L�RPL�O¤RPhIL�cINPUYUWV4UWVqÔ z l��WØ�lVU

#7 óê7p3)798;É°79,� ³�� ³ �Q¹��%8��CÀ�8������<���L��3���+�2��Y���l&(2�$93��� L2�3u� ���! S��3#"��$��� ����3� .��2�3< l��3��������l$93�� %!2��.��� �&"y2'%���$9+��1�)(

%*�ð�à���À��+� .8,+º��·�®xÀ�-/.�0�¼�¡�������¿(8 � RQ¦`R-6�À� >°¡ � >���´¶¹x��­��¥��8,1¤À9G*À�8�2��é����À��+435�(2����!�&��"65;�¤��� ���������1���(��+7%!2�$93��à���1��2�3< ���3������6�%+����(���1��2�3< .89G¯�¯��­Ñ��©Â¦´��]·98�:��¯­�®!8��°�*©¶¹T�*�¥®����¥�8e¡������*À� >,4 � >���´¶¹x��­��¥��8#1qÀ�G�À�8$;%�(�ð�<3��=<�$��! l� 3>2��é����À��+�35�(2������=�?"�8�G¯�¯��­G��©¶´��]·@8�:���­*®�8��°��©�¹T�*�¥®��±�]�8e¡���OR»À� >BA�¡ � >���´¶¹x��­��¥��8,1¤À9G*À��¯­*§9A��DC����P·�§f8<G*ÀE-¬À�8�FG�é2¡'��(���1��2�3e� ���(2��ð�!�¥ H%!2���&(2JI���3��!�<�à�*K�� 3ML9��� ���!3� .��2�3� .8F����`�MÀ,+ ���»¹¯À��°��ª�µ*�`©�§*²��N+º�*©¶´Â��®xÀO:»��¹¯ÀO0�0EP#¼�¡���� � ¿.8:4-6OR�Qî4���=»À� >BA;4 � >���´¶¹x��­��¥��8�1qÀâG*À9�¯­*§RA��DC;�±�P·�§¹8YG*ÀH-¬À�8$F5��SK�� 3MLR��� ���!3� .��2�3� y2T%U���é2¡'��(�����12�3� ���3��V��� ����3� .��2�3

���é2¡/Y+��! .8�F����`�MÀ,+ ���»¹¯À��°��ª�µ*�`©�§*²��W+º�*©¶´��¯®]À�:»��¹¯ÀO0,XO0�¼�¡����OR¯¿(8:4 � �JQî4�6 � À� >âG � >���´¶¹x��­��¥��8O1¤À¹G*Àe�¯­*§¤G%H���±�¯�Q­�� H��H�98 F"À�8YF5��SK�� 3MLZ��� ����3< .��2�3� �2T%l ��(�!�1��2�3� O2T%l ����� .8IF����`�MÀ[+ ���»¹¯À

�°��ª�µ��±©¶§�²¯�N+º�*©¶´Â��®xÀ[:��»¹¯Àê¼½���'���*�6�Q���T¿{À

Page 21: LINEAR MAPPINGS AND GENERALIZED UPPER SPECTRUM FOR

� ©¶­��Q���gªq������©¶­�²¯®4��­�§G�±�*��£����6�]���P¦B§�©¶ª¬�Q­*®±©��¯­ 4�¡� >,F � >���´¶¹x��­��¥��8�1qÀËG�À �¯­*§�F����±�±©¶´¶�98G+IÀ�8\;f���]�<��*K�� 36L^��� ����3� .��2�3h2T%\���é2¡'��(���1��2�3< e��3��� ��(���1��2�3< g2T%���(�� .$E�é�! .8�F����`�MÀ#+p�±�»¹�À��°�¯ª�µ*�`©¶§�²��U+ ��©�´��¯®]À[:��»¹�ÀY0�0MPƼա��� � ¿.8 � ��=�Q)R�¡��À� >`_ � >���´¶¹x��­��¥��8[1qÀ:G�ÀM��­�§Z_Uab4�¥©¶´�8MÁ2À�8Y)Y2�3#c-2�+�$9�1��2�3< O��3��������WLà�(2������&��"e2T%L��$<+��1� %*�é����À��+â���(�� .$E�ð�! .8F����`�MÀ,bC��¹T�*�]ÀY¼K�±�¤�����6�¥���{¿TÀ�ed � d �`��®±®Õµ6�¥��²¯�]�8`+IÀ�8f35��3:�?�ð��+��hg��*�Z��� ����3< .�12�3� �2T%l .�=�ð��36L�y�����&�é����À27�. .8Y+º��·�®xÀ � �¥���¥À¹7iP�j¼Õ¡��6���¿(84�4OR�Q94-���*À� Ak+ � A4�¥­��`®�¹x���Q´ 8[A�À d À D ÀM��­�§Z+p�±�»¹]�¯¹]¹¥©��<8Yl`À�8k;f����� 3/ï3�� �À�@3�$9��&(����2T%!L��3%���é��+ �hg��*�m��� ����3� .��2�3< �2T%

%*�ð�à���À��+� L��3��� .�&�é��3MLà�����1�&�é���� 2��( .8#+ ��·�®]À,-on"¼�¡���6���¿.8 ¯ ��=JQ ¯�¯�¯ À� A4Á � AC£¹8âE�À_�¯­*§ÐÁY�Q·�´����8�:WÀIG�À�8p2��ð��!�À��+`�#�ð2��<�?�!�����! @2T%W���é2D��$���� ���3��q���é2¡'��(���1��2�3< y2'%����(�� .$E�é�! �� 3

r�s 8�F����`�MÀ#+p�±�»¹�À��°�¯ª�µ��±©¶§�²¯�N+ ��©�´��¯®]À[:��»¹�ÀY0�0Et"¼Õ¡��� ¯ ¿.8%=�4�R�Q9= ¯�¯ À� Ak1 � AC£*­��8 ³ ÀJu�ÀP��­*§W1ß��´��¯®±�*©¶­¹8�v�À�wº£MÀ�8�x�2��@���é2¡'��(���1��2�3< S�Ty5�(���J�����z��� ���!3� .��2�3� {�<�(���&�.$9��2'%�%��é���� ��+���(�� .$E�é�! .8,b4�¯­�´�©¶­��Q���`©��¸·|0E}"¼�¡�����R�¿.8ê¡¥���*¡*Q�¡¥� ¯ � À� A~:6w � AC£*­��8 ³ À�u�À�8�:»��£*�]�4Á2À�8���­�§qwp���S�¯��G*À�7�À�8#FG�é��c-��+���3:�(��5L���=�ð��3� .+����1��2�3�(�� 3�c-���.����3��W��+ ��2� .�0��c-����"

%!2��#� 3/Y3�� �À�]��� ����3� .��2�3%��+J {�<�à�(�! .8 ³ £�´�´½À%7CªÑÀ%F����±�eÀ#:��»¹¯À�X�j#¼�¡�����4�¿(8:4»¡÷R�Qî4���6*À� G*¡ � G%H���±�¯�Q­�� H��H�98fFÆÀ�8$�S3_�����O$��6�<�?�L�u� 3MKà2��, TK��f��� ����3< .�12�3��W� ���!�<��SK�� 3ML>��� ����3� .��2�3,�5��3���2��.� ��2*L�(2�3%��+��#�ð2¡'��(������2�3� .8<7C­*­MÀ%7C¹]�¯§MÀ�:�¹]©½À�>��Q­*­MÀ%F����±�eÀ,-2©¶®�®]À�P�PƼա��� ¯ ¿(8Y¡*Qî� ¯ À� Gà4 � G%H���±�¯�Q­�� H��H�98 F"À�8Bx���$î ��2���yM��3����<��*K�� 3ML���� ����3� .��2�3< *�;� 3�� ���( ��(���1��2�3����(�- .$,�é�! *�;��3��e .� ��� +����.�)(�1���! .8�7C­*­MÀ%7C¹]�¯§MÀ�:�¹]©½À�>��Q­*­MÀ�F����±�eÀê¼½���¤���*�6�¥���{¿TÀ� GàF � G%H���±�¯�Q­�� H��H�98-FÆÀ]�¯­*§LF����±�`©�´¶�f+IÀ�8�x���$î ��2���y���3����<�à�*K�� 3ML!��� ����3< .�12�3��J��3��¢ ��(�!�1��2�3� �2T%,���(�� .$E�ð�! .8F����`���Pªq���`©��P�G¼½�±�¤���*�6�Q���T¿{À� 1ß��� � 1ß�¯�*��­*��G*À ¦�+IÀ�8�"%$E�¤+������ ����3� .��2�3/��! g� 3��À�?�. ��(�!�1��2�3� .8�©�­h7C®Õ�6�Q¹x�`®Ñ���¥F����±�*�Qª¤���`©�¹¥®G��­�§Î©��`®������´¶©�¹¥���`©��¯­*®�¼�¡���6 � ¿.8 D ´¶®Õ�¥��©Â�¥�~:�¹]©��Q­*¹]�!+ £*µ�´¶©�®±���¥�±® ³ ÀEv�À�8 ¯ ¡�JQ ¯ ���*À� 1ß��£ � 1ß�¯£��½ªq�¯­*­f86u�À�8Y�W39x���$î ��2���y���� ����3< .��2�3u2T%$���é2¡'��(�����12�3� .8<F����`���Pªq���`©��P��0,t�¼�¡�� � 6�¿(8M¡�=-�DQ�¡=�=»À� F���� � F����`®Õ�±�±�¯­*§f8âFÆÀ�8Ë"¹2�����%($93�������!3��À��+BL�(2����!�&�.���(��+����é2��<���.���1�. �2'%N�%+���3%�� ����� �2T%f%��é�������2�3:��+p���)(����3< .�12�3�8�+ �±��¹¯À � �¯­�§��¯­uF����±�eÀ�:��»¹¯Àê¼1��¿G.Ƽ�¡���= ¯ ¿.8%4�=�R�Q���à4»À� F����]¡ � F����±�±©¶´��<84+IÀ�84x���$î ��2���y���� ����3< .��2�3��L2��.����2*L2�3%��+����é2¡'��(�����12�3� ���3��g� 3�� ���( ��(���1��2�3< 9�Ë� � ���%+���3:�! .87C­*­MÀ�74¹¥��§eÀ�:�¹]©½À�>��¥­�­MÀ%F����±�eÀ[0�¼�¡���R-=¯¿.8:4�4�R�Qî4 ¯�¯ À� F����ð4 � F����±�±©¶´��<8Y+IÀ�8�xL��$î ��2���yi��� ����3� .��2�3���3��u�(���<���� ���1�. �2T%�� 3��À�?�. ��(�!�1��2�3�2T%l ��!�� O� 3Z�B(À {�<��(��8f7C¹x�`�F����`�MÀ[0,t�X#¼�¡���6 ¯ ¿.8+RYR�Q6¡]�à=»À� F����é� � F����±�±©¶´��<8E+IÀ�8`35�(2����!�&��"�2T%5"¹�!�� ���3����¤�(�� .$E�é�! ;� 39�W$���+��=��(��3y"6�<��(�. �5f2��é����À��+� *��3��\�ð�(�!�1� /��&!� +�(� �&"�8#�)�¯ª�µ*�`©�§*²��U�4­�©Â���]�`®±©Â�¸·9+ ���Q®�®�8��)�¯ª�µ*�`©�§*²��à8e¡����=�À� _ � _g´¶®Õ�Q­¹8 � À�8#�M��$9+ ��� %*�é���� ��+6%!2��.����+��� .��8�74§*�WÀ�F����`�MÀO0�0E�"¼�¡�����=�¿(8%6�4JQ�¡��à=»À� + � +I�Q®�©¶­¹87w ��À�8Y�W39�.��L2��é2�$î W�����������������1�(��+����/ï3�� �1��2�3< ;2T%;�(27�?�é��+����1��2�39��� ����3� .��2�3y��3��~Là��3:���é��+��hg��*� {�<�(���=�!$9��%!2��U��� ����3� .��2�3< .8�G*À�:��`���`©�®��]À�+º��·�®xÀ#jO0�¼Õ¡������¿(8:=�4��JQ9= ¯ R»À� :�w � :»�¯£��¥��8�Á2À-¬À���­�§Nw ���S���à8�G�À 7�À�86�~�é�5��������� ����3� .��2�3< 52T%S�� ����,��3��L� �� W� ���JLà�*�*<�$���+�$93�����W�&"?�%���(��+ .��2[2����N%($93:�����12�3� ��!8 D �±²��»§*©¶¹�Á;�*�]�¯��·9-g·�­*��ªÑÀ#:»·�®Õ�±�Qª¤®�0,j#¼�¡�����R�¿.8�� ¯ ¡�Q��= � À� :»© � :»©«ª¬��­¹8 ³ À�84;f���*"%�À���1�� .�1���(��+:�¤�(�(����3��1�. �2T%z�J�����1���(��35�� ��! *�!��2�+���8E+p�`©�­�¹x�¥�±�¯­q�4­�©Â���]�`®±©Â�¸·�+p�±�¥®±®�8

+ �±©¶­*¹]�]�±�¯­f8,b4��CUG¯�]�`®��]·:8M¡������À� :�� � :��±��C°�����8 d À ��À�8��(3��=�ð2D��$�������2�3¤�À2O�����=�.�hI¤)ï2����%$9� ���1��2�3< .8�74¹¥��§*�Qªq©¶¹N+ �±�¥®±®!8�b���C�wp���S�%8e¡�OR-��À